Causal Geometry of Einstein-Vacuum Spacetimes

Qian Wang

A Dissertation Presented to the Faculty of Princeton University

in Candidacy for the Degree of Doctor of Philosophy

Recommended for Acceptance

by the Department of Mathematics

September 2006 \begin{matrix}c \\ ◯ \end{matrix} Copyright by Qian Wang, 2007.

All Rights Reserved

Abstract

In this dissertation, we study the geometry of null cones in smooth Einstein vacuum spacetimes. We provide the L^{∞} estimate for the trace of the null second fundamental form, as well as other geometric quantities such as the shear and torsion tensors, only in term of the L^{2} curvature flux.

In [12], [14] and [15], the authors have established similar estimates on truncated null hypersurfaces. The method in this dissertation is based on the above three papers, however, with particular attention to the portion near the vertices of null cones, which requires us to prove weighted version of the main estimates in [12], [14] and [15]. In addition, we provide the proofs of the boundedness of 0-order Hodge operators in Besov spaces. Such estimates were stated and used in [12] without a proof. We present a modified version of these estimates. Modification adds complexity to error estimates in Chapter 7.

Acknowledgements

I would like to thank, first and foremost, my advisor, Professor Sergiu Klainerman, for his guidance and support throughout my graduate career and during the completion of this thesis. I also would like to thank Professor Igor Rodnianski, for his time and patience in reading this thesis and for his many helpful suggestions.

Finally, I thank my family for the support and in particular, I acknowledge my husband, Qinian Jin, without whose love and encouragement, I would not have finished this thesis.

Contents

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv

1 Introduction 1 1.1 Discussion of the proof of main results . . . . . . . . . . . . . . . . . . . 2 1.2 Organization of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 General results in 2-D surfaces 7 2.1 Equivalent Besov norms on 2-D surfaces . . . . . . . . . . . . . . . . . . 7 2.2 Hodge systems on 2-D surfaces . . . . . . . . . . . . . . . . . . . . . . . 17

3 Null Hypersurfaces 19 3.1 Null Hypersurfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.2 Initial Data and Structure Equations . . . . . . . . . . . . . . . . . . . . 21 3.3 Commutation Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 3.4 Preliminary Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.4.1 Norms on H . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

4 Bootstrap Assumptions 29 4.1 Main Theorem and Bootstrap Assumptions . . . . . . . . . . . . . . . . . 29 4.2 Weakly spherical surfaces and Equivalent Besov norms on H . . . . . . . 31 4.3 Elliptic estimates on 2-D surfaces . . . . . . . . . . . . . . . . . . . . . . 35 4.4 Commutation relation between ∇_{L} and Hodge operators D^{−1} . . . . . . . 43 4.5 Estimates of second derivatives . . . . . . . . . . . . . . . . . . . . . . . 44

5 Transport-Product Estimates in Besov Norms 47

5.1 Product Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 5.1.1 Trace Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 5.1.2 Proof of Sharp Trace Estimate . . . . . . . . . . . . . . . . . . . . 52 5.1.3 Preliminary Estimates . . . . . . . . . . . . . . . . . . . . . . . . 54 5.1.4 Estimates of B_{k}^{3} . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 5.1.5 Estimates of B_{k}^{1} and B_{k}^{2} . . . . . . . . . . . . . . . . . . . . . . . 61 5.1.6 Estiamtes of A_{k} . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

5.2 Non-Sharp Product Estimates . . . . . . . . . . . . . . . . . . . . . . . . 66 6 Elliptic P^{σ} Estimates of Hodge operators on H 73

6.1 Preliminary Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 6.2 Hodge-Elliptic P^{σ} Estimates . . . . . . . . . . . . . . . . . . . . . . . . . 76

6.2.1 Sketch of the proof . . . . . . . . . . . . . . . . . . . . . . . . . . 77 6.2.2 Step 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 6.2.3 Step 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

7 Error Estimates 90 7.1 Structure of error terms . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 7.2 Proof of Proposition 7.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 7.3 L_{t}^{a}L_{x}^{2} Estimates of D^{−1}C_{0}(Ř) . . . . . . . . . . . . . . . . . . . . . . . . 99

7.3.1 Error Type Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 7.3.2 Proof of Lemma 7.3 and Proposition 7.5 . . . . . . . . . . . . . . 103

7.4 L_{t}^{a}L_{x}^{2} Estimates for D^{−1}(Err · F) . . . . . . . . . . . . . . . . . . . . . . 106 7.5 L_{t}^{a}L_{x}^{2} Estimates for ∇_{L}D^{−2}Ř . . . . . . . . . . . . . . . . . . . . . . . . . 109 7.6 Decomposition and correction estimates for C_{2}(Ř) and C_{3}(Ř). . . . . . . 114

8 Main Estimate 123 8.1 Estimates for trχ, χ̂ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 8.2 Estimates for μ, and ∇ζ . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 8.3 Estimates for (tr\underline{χ}, \underline{χ̂}) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

9 Appendix 134

9.1 Initial Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 9.1.1 Local expansions of χ, \underline{χ}and ζ . . . . . . . . . . . . . . . . . . . . 134 9.1.2 Transformation of angular coordinates . . . . . . . . . . . . . . . 141

9.2 Commutator Estimates: Proof of Lemma 5.1 . . . . . . . . . . . . . . . . 141

Chapter 1

Introduction

This thesis is concerned with the geometry of null cones in 3+1 smooth Einstein vacuum spacetimes, i.e. 3 + 1 Lorentzian manifolds (M, g) with the Ricci flat metric,

R_{αβ}(g) = 0.  (1.1)

Let H be a null hypersurface on (M, g) generated by null geodesics starting from an arbitrary point p. More precisely, let T be a fixed, future directed, timelike vector at p satisfying ⟨T, T ⟩ = −1. Consider all future directed null vectors L_{ω} ∈ T_{p}M with ω ∈ 𝕊^{2} such that ⟨L_{ω}, T ⟩ = −1 and ⟨L_{ω}, L_{ω}⟩ = 0. Let Γ_{ω} be the future null geodesic initiating at p verifying

\frac{d}{ds}Γ_{ω}(0) = L_{ω} and Γ_{ω}(0) = p.

We define a null vector field L to be L|_{Γ_{ω}(s)} = \frac{d}{ds}Γ_{ω}. Clearly L(s) = 1 and s(p) = 0. The affine parameter s can be regarded as a function along H, whose level surfaces, denoted by S_{s}, form a geodesic foliation. Let H_{t} be the portion of H when s varies in (0, t] and denote still by H the portion H_{1}. We parameterize every point q in H_{t} by the coordinates (s, ω) corresponding to the values of s for which q = Γ_{ω}(s). Let D denote the Levi-Civita connection of Einstein vacuum metric g. Let γ be the induced metric on S_{s} and ∇ its induced covariant derivative. At any point q ∈ S_{s} ⊂ H we denote by \underline{L} the null vector conjugate to L relative to the S_{s} foliation, i.e. ⟨L, \underline{L}⟩ = −2, ⟨\underline{L}, X⟩ = 0 for all X ∈ T_{q}(S_{s}).

The total curvature flux R_{0} is then defined by

R_{0} = (∥α∥_{L^{2}(H)}^{2} + ∥β∥_{L^{2}(H)}^{2} + ∥ρ∥_{L^{2}(H)}^{2} + ∥σ∥_{L^{2}(H)}^{2} + ∥\underline{β}∥_{L^{2}(H)}^{2})^{\frac{1}{2}} ,

where

α(X, Y) = R(L, X, L, Y), β(X) = \frac{1}{2}R(X, L, \underline{L}, L), ρ = \frac{1}{4}R(\underline{L}, L, \underline{L}, L), · · · .

We refer the reader to Chapter 3 for the complete definition.

The geometry of null cones depends in particular on the null second fundamental form χ(X, Y) = ⟨D_{X}L, Y ⟩ with X, Y arbitrary vector fields tangent to S_{s}. We will denote trχ and χ̂ the trace and traceless part of χ respectively. Other important geometric quantities are the dual null second fundament form \underline{χ}(X, Y) = ⟨D_{X} \underline{L}, Y ⟩ and the torsion ζ(X) = \frac{1}{2}⟨D_{X}L, \underline{L}⟩.

In [12, 14, 15] Klainerman and Rodnianski developed systematic methods to prove that, on truncated null hypersurfaces initiating from a 2-D surface which is diffeomorphic to 𝕊^{2}, within the radius of injectivity, trχ can be controlled by appropriate norms of the small initial data and small total curvature flux R_{0}. In this work, we extend their result to null cones in smooth Einstein vacuum spacetimes. We will prove that trχ − \frac{2}{s} can be bounded only by curvature flux before the formation of caustics or geodesic loops. Our result is used in [16] to provide the uniform lower bound on the radius of injectivity of null boundaries in Einstein vacuum spacetimes. Such lower bound is essential in understanding the causal structure and propagation properties of solutions to the Einstein equations, and is important in construction of a Kirchoff-Sobolev type parametrix for solutions of wave equations on M, (see [17],) which is used in [18] to prove a large data break-down criterion for solutions of the Einstein vacuum equations.

In the following, we state a simplified version of the main theorem,

Theorem 1.1. Consider an outgoing null hypersurface H initiating from a given point p with a geodesic foliation which is associated to the affine parameter s with s|_{p} = 0. When R_{0} is small enough, the following estimate holds true on H:

∥trχ − \frac{2}{s}∥_{L^{∞}(H)} ≲ R_{0}

Additional estimates hold true for χ̂, ζ, \underline{χ}.

The notation A ≲ B in the above results means A ≤ C · B for some constant C depending only on appropriate universal constants. We will use this notation throughout the thesis.

1.1 Discussion of the proof of main results

It can be shown, see [12], that trχ verifies the Raychadhouri equation

\frac{d}{ds}trχ = −\frac{1}{2}(trχ)^{2} − |χ̂|^{2}.  (1.2)

By integrating (1.2) along Γ_{ω}(s), it is clear that to bound the L^{∞} norm of trχ − \frac{2}{s}, it is necessary to have a uniform bound for ∫_{0}^{1} |χ̂|^{2}. To obtain such an estimate, we are allowed to rely only on the curvature flux R_{0}. In view of the ellipticity of Codazzi equation

div χ̂ = \frac{1}{2}∇trχ − \frac{1}{2}trχ · ζ − χ̂ · ζ − β  (1.3)

∇χ̂ can be viewed of having the same differentiability as β. If the following trace type inequality holds

(∫_{0}^{1} |χ̂|^{2})^{\frac{1}{2}} ≤ C∥∇χ̂∥_{L^{2}(H)},  (1.4)

we may control ∫_{0}^{1} |χ̂|^{2} by the curvature flux.

The main difficulty encountered by such method is that the inequality (1.4), similar to the sharp Sobolev embedding into L^{∞}, is not true in general. Note, by (1.3), we may write

χ̂ = D_{2}^{−1}(−β − χ · ζ + \frac{1}{2}∇trχ),

where the operator D_{2} takes any S-tangent symmetric traceless 2-tensor F into the S-tangent one form div F. If the second and the third terms are under control, we may expect that χ̂ behaves like D_{2}^{−1}β. The curvature component β can be written as ∂^{2}g + (∂g)^{2}, where g is the spacetime metric. In wave coordinates the Einstein metric g satisfies a nonlinear wave equation of the following form

g^{μν}∂_{μν}^{2}g = (∂g)^{2}.

Since D_{2}^{−1} is a pseudodifferential operator of order −1, we can view D_{2}^{−1}β roughly as ∂f, where f, in the first approximation, is a H^{2} solution to the wave equation in Minkowski spacetime □f = 0. Thus we need to control ∫_{Γ} |∂f|^{2}. However this quantity can be arbitrarily large, independent of the size of the initial data and the length of the geodesic segment (see [8]).

To overcome this difficulty, we note that in ℝ^{3}, the following trace inequality holds for ϕ = ∂_{x_{1}}f(x) ∈ C_{c}^{∞}(ℝ^{3}),

∫_{ℝ^{1}} |ϕ(x_{1}, 0, 0)|^{2}dx_{1} ≲ ∫_{ℝ^{3}} |∇^{2}f(x)|^{2}dx.

This suggests us to find a special structure so that D_{2}^{−1}β can be written as ∇_{L}Π, where Π is a tensor verifying \overline{∇}^{2}Π ∈ L^{2}, where \overline{∇} represents ∇_{L} and ∇. Then we will establish the trace estimate

∥∇_{L}Π∥_{L_{ω}^{∞}L_{t}^{2}} ≲ ∥\overline{∇}^{2}Π∥_{L^{2}(H)} + l.o.t.  (1.5)

where “l.o.t” represents lower order terms. In view of the null Bianchi identities

L(ρ) = div β + ⋯ L(σ) = − curl β + ⋯

β can be expressed as β = D_{1}^{−1}∇_{L}(ρ, −σ)+⋯, where the operator D_{1} takes any 1-form F into the pairs of functions ( div F, curl F). Thus we may write

χ̂ = −D_{2}^{−1}D_{1}^{−1}∇_{L}(ρ, −σ) + D_{2}^{−1}∇trχ + ⋯
= −∇_{L}D_{2}^{−1}D_{1}^{−1}(ρ, −σ) + [∇_{L}, D_{2}^{−1}D_{1}^{−1}](ρ, −σ) + D_{2}^{−1}∇trχ + ⋯,  (1.6)

where the “⋯ ” in the above inequalities represents error terms. ^{1}

As to the first term ∇_{L}D_{2}^{−1}D_{1}^{−1}(ρ, −σ) in (1.6), let Q = D_{2}^{−1}D_{1}^{−1}(ρ, −σ), we may use the trace estimate (1.5) to get

∥∇_{L}Q∥_{L_{ω}^{∞}L_{t}^{2}} ≲ ∥\overline{∇}^{2}Q∥_{L^{2}(H)} + l.o.t,

the right hand side of which may be bounded by curvature flux R_{0} since ∇^{2}D_{2}^{−1}D_{1}^{−1} can be viewed as a 0-order operator. However, the fact that 0-order operator D_{2}^{−1}∇ is not bounded in L_{ω}^{∞} brings us difficulty when we try to bound the third term of (1.6), D_{2}^{−1}∇trχ, in L_{ω}^{∞}. Hence, instead of estimating D_{2}^{−1}∇trχ in L_{ω}^{∞}, we try to provide the estimate in the Besov space B_{2,1}^{1} due to the fact that B_{2,1}^{1} ↪ L^{∞} and the 0-order operator is stable in B_{2,1}^{1}. Thus we are forced to provide a series of estimates in Besov spaces which usually are defined by Littlewood Paley theory. The classical Littlewood Paley theory, based on Fourier transform, requires us to use local coordinates on H. By using the coordinate frame transported by the Hamiltonian flow generated by L, we may lose 1/2 derivative relative to what we expect from the boundedness of curvature flux. Thus we need to use an invariant, geometric version of Littlewood Paley theory, which was introduced in [15] based on the heat flow approach of [22]. Unlike [12] where the trace type estimates are proved by using geometric Littlewood Paley (GLP) theorey only, we combine both types of Littlewood Paley decompositions to avoid the non-commutativity between the GLP projector and ∫_{Γ}.

To prove the aforementioned estimate for trχ and χ̂, we need the estimates of ζ, μ and \underline{χ}. Actually, both ζ and χ̂ verify a Hodge equation or a Hodge system. By providing appropriate estimates of μ and ∇trχ, we will provide the L_{ω}^{∞}L_{s}^{2} estimates of ζ and χ̂ by a trace theorem. Heuristically, ζ and χ̂, μ and ∇trχ play similar roles in the proof of main theorem and will be treated in the same way. None of the estimates can be established independent of others. The whole proof relies on an elaborate bootstrap principle. Note that we will integrate along null geodesics Γ_{ω}, we have to determine the limit of the geometric quantities χ, ζ, μ etc. at the vertex. This will be studied in Section 9.1.

1In fact, some of these error terms have the same order as the leading term and have to be treated in a different way.

To control terms such as the commutator and error terms in (1.6), we need to establish the Besov estimate of the 0-order Hodge operator ∇ · D_{2}^{−1}. The estimate ∥∇·D_{2}^{−1}F ∥_{P^{0}} ≲ ∥F ∥_{P^{0}} with P^{0} being a certain Besov norm (see [12, Proposition 5.11]), holds true only when an additional term involved with the L_{s}^{∞}L_{ω}^{2} norm of D_{2}^{−1}F is added, due to the limited regularity of Gauss curvature K of each slice S_{s}. In this thesis we give the correct P^{0} estimates of the 0-order Hodge operators. The corrections which we make on the P^{0} estimates of 0-order Hodge operators and some other product estimates, add much complexity to the commutator estimates. Moreover, since various quantities behave like s^{−a} for some number a as s → 0, we have to keep track of the weight s^{a} in each step.

1.2 Organization of the thesis

The main body of the thesis begins in Chapter 2, we revisit the Geometric Littlewood Paley decomposition (GLP) and define Besov norms on 2-D surfaces by using the GLP and the classical LP decompositions respectively. Then we establish an equivalence result between them.

In Chapter 3, we introduce structural equations and provide the results about initial data whose proofs are given in Appendix. Then we make a preliminary bootstrap assumption and draw some simple consequences from it. We also introduce important norms and some preliminary inequalities in this chapter.

In Chapter 4, we provide the precise version of the main theorem. We make a complete set of bootstrap assumptions and prove the equivalence result of two type of Besov norms defined on the null cone. We also provide the L^{2} type estimates of the operators of the type ∇ · D^{−1} by establishing Propositions 4.5 and Proposition 4.6 on Gaussian curvature K of each leave S_{s}, where the Hodge operator D is one of the operators D_{1}, D_{2} or their L^{2} adjoint operators. At last we provide the commutation formula for [∇_{L}, D^{−1}] and estimates of the terms ∇^{2}D^{−1}F for appropriate S-tangent tensor fields F, and some other related estimates.

In Chapter 5, we prove the trace theorem and some important product estimates. In Section 5.1.2, we provide the proof of the sharp trace estimate. We basically follow the treatment in [14], but also take advantage of technical simplifications, which are based on the equivalence results established in Chapter 4.

In Chapter 6, Theorem 6.1, which gives the P^{0} estimates of 0-order Hodge operators, is proved in Section 6.2 by using the result in Section 4.3 and geometric Littlewood Paley decomposition (GLP). This theorem, which has special significance to Chapter 7, is a correction of [12, Proposition 5.11]. The proof is based on the unpublished notes of Klainerman and Rodnianski [9].

In Chapter 7, we make full use of the elliptic estimates in Chapter 4 and the P^{0} estimates in Chapter 6 to fulfill the decomposition of the commutators such as the one in (1.6).

In Chapter 8, we use the results in Chapters 2–7 to complete the proof of the main theorem.

In Appendix, Section 9.1 is devoted to the study of initial data, where we provide a few local estimates of important geometric quantities around the vertex. This section depends heavily on [20, Chapters 2 and 3]. In the rest of the appendix, we prove Lemma 5.1 which are important for the proofs of the sharp trace estimate.

General results in 2-D surfaces

2.1 Equivalent Besov norms on 2-D surfaces

Let S be a 2-dimensional closed manifold with a metric g whose Gaussian curvature is denoted by K. We say S is a weakly regular surface (WR) if it can be covered by a finite number of coordinate charts U with coordinates ω^{1}, ω^{2} relative to which

C_{1}^{−1}|ζ|^{2} ≤ g_{ab}(p)ζ^{a}ζ^{b} ≤ C_{1}|ζ|^{2}, uniformly for all p ∈ U  (2.1)

and

∑_{a,b,c} ∫_{U} |∂_{c}g_{ab}|^{2}dx^{1}dx^{2} ≤ C_{1}^{−1},  (2.2)

where C_{1} is a positive constant.

As discussed in [15], a natural way to extend classical Littlewood Paley (LP) theory to S is to exploit heat flow. Below is a geometric Littlewood Paley (GLP) theory on S ( see [15]) which is based on the weakly regular assumption on S.

Definition 2.1. Consider the class M of smooth functions m on [0, ∞), vanishing sufficiently fast at infinity, and verifying the vanishing moments property

∫_{0}^{∞} τ^{k_{1}}∂^{k_{2}}m(τ)dτ = 0, |k_{1}| + |k_{2}| ≤ N.  (2.3)

Set m_{k}(τ) = 2^{2k}m(2^{2k}τ), define the geometric Littlewood Paley (GLP) projections P_{k} associated to the LP-representative function m ∈ M, for arbitrary tensor field F on S, to be

P_{k}F = ∫_{0}^{∞} m_{k}(τ)U(τ)F dτ,  (2.4)

where U(τ)F is the heat flow defined on (S, g) by

∂_{τ}U(τ)F − Δ_{g}U(τ)F = 0, U(0)F = F,  (2.5)

and Δ_{g} is the Laplace-Beltrami operator on (S, g).

Then we record the properties of the heat equation (2.5) for arbitrary tensorfields F on S. Observe that the operators U(τ) are selfadjoint and form a semigroup for τ > 0 i.e. for all real valued, smooth tensorfelds F, G

∫_{S} U(τ)F · G = ∫_{S} F · U(τ)G, U(τ_{1})U(τ_{2}) = U(τ_{1} + τ_{2}).

We have the following estimates for the operator U(τ),

Proposition 2.1.

∥U(τ)F ∥_{L^{2}(S)} ≤ ∥F ∥_{L^{2}(S)} ∥∇U(τ)F ∥_{L^{2}(S)} ≤ ∥∇F ∥_{L^{2}(S)}

∥∇U(τ)F ∥_{L^{2}(S)} ≤ \frac{\sqrt{2}}{2}τ^{− \frac{1}{2}} ∥F ∥_{L^{2}(S)}

∥U(τ)∇F ∥_{L^{2}(S)} ≤ \frac{\sqrt{2}}{2}τ^{− \frac{1}{2}} ∥F ∥_{L^{2}(S)}

∥ΔU(τ)F ∥_{L^{2}(S)} ≤ \frac{\sqrt{2}}{2}τ^{−1}∥F ∥_{L^{2}(S)}

∥U(τ)F ∥_{L^{p}(S)} ≤ ∥F ∥_{L^{p}}, with 2 ≤ p ≤ ∞

We refer the reader to [15] for the proof.

From Definition 2.1, it is obvious that P_{k} is a self-adjoint operator and commutes with Δ_{g}. Given an interval I ⊂ ℤ, we define P_{I} = ∑_{k∈I} P_{k}. We shall also use the notation P_{<k}, P_{≤k}, P_{>k}, P_{≥k} frequently.

Definition 2.2. Consider GLP projection associated to a fixed m ∈ M so that ∑_{k} P_{k}^{2} = Id. For 0 ≤ α ≤ 1, we define the Besov space B_{2,1}^{α} consisting of all tensor fields on S verifying

∥F ∥_{B_{2,1}^{α}} = ∑_{k>0} 2^{αk}∥P_{k}F ∥_{L^{2}} + ∥P_{≤0}F ∥_{L^{2}}.  (2.6)

Now we recall the basic properties of geometric Littlewood Paley decomposition on weakly regular 2-D closed surfaces, for more properties please refer to [15].

Proposition 2.2. There exists m ∈ M such that the GLP projections P_{k} associated to m verify ∑_{k∈ℤ} P_{k}^{2} = Id. By f we denote a scalar function on S and F a tensor field, the GLP projections P_{k} associated to arbitrary m ∈ M verify the following properties:

(i) (L^{p}-boundedness) For any 1 ≤ p ≤ ∞, and any interval I ⊂ ℤ,

∥P_{I}F ∥_{L^{p}(S)} ≲ ∥F ∥_{L^{p}(S)}

(ii) (L^{p}-almost orthogonality) Consider two families of LP-projections P_{k} and P̃_{k} associated to m and respectively m̃, both in M. For any 1 ≤ p ≤ ∞:

∥P_{k}P̃_{k′}F ∥_{L^{p}(S)} ≲ 2^{−2|k−k′|}∥F ∥_{L^{p}(S)}

(iii) (Bessel inequality) For any tensorfield F on S,

∑_{k} ∥P_{k}F ∥_{L^{2}(S)}^{2} ≲ ∥F ∥_{L^{2}(S)}^{2}, ∑_{k} 2^{2k}∥P_{k}F ∥_{L^{2}(S)}^{2} ≲ ∥∇_{g}F ∥_{L^{2}(S)}^{2}

(iv) (Finite band property) For any 1 ≤ p ≤ ∞, k ≥ 0,

∥Δ_{g}P_{k}F ∥_{L^{p}(S)} ≲ 2^{2k}∥F ∥_{L^{p}(S)}
∥P_{k}F ∥_{L^{p}(S)} ≲ 2^{−2k}∥Δ_{g}F ∥_{L^{p}(S)}.

Moreover given m ∈ M we can find m̄ such that Δ_{g}P_{k} = 2^{2k}P̄_{k} with P̄_{k} the geometric Littlewood Paley projections associated to m̄. In addition, the L^{2} estimates

∥∇_{g}P_{k}F ∥_{L^{2}(S)} ≲ 2^{k}∥F ∥_{L^{2}(S)}
∥∇_{g}P_{≤0}F ∥_{L^{2}(S)} ≲ ∥F ∥_{L^{2}(S)}
∥P_{k}F ∥_{L^{2}(S)} ≲ 2^{−k}∥∇_{g}F ∥_{L^{2}(S)}

hold together with the dual estimate

∥P_{k}∇_{g}F ∥_{L^{2}(S)} ≲ 2^{k}∥F ∥_{L^{2}(S)}.

For any 1-form F, we have

∥P_{k} div F ∥_{L^{2}(S)} ≲ C_{1}(2^{\frac{2}{p} k} + 1)∥F ∥_{L^{p}(S)} with 2 ≥ p > 1.  (2.7)

(v) (Bernstein Inequality) For appropriate tensor fields F on S, we have weak Bernstein inequalities 2 ≤ p < ∞

∥P_{k}F ∥_{L^{p}(S)} ≲ C_{1} (2^{(1− \frac{2}{p})k} + 1) ∥F ∥_{L^{2}(S)},
∥P_{≤0}F ∥_{L^{p}(S)} ≲ C_{1}∥F ∥_{L^{2}(S)}

together with the dual estimates

∥P_{k}F ∥_{L^{2}(S)} ≲ C_{1} (2^{(1− \frac{2}{p})k} + 1) ∥F ∥_{L^{p′}(S)}
∥P_{≤0}F ∥_{L^{2}(S)} ≲ C_{1}∥F ∥_{L^{p′}(S)}.

where 1 < p′ ≤ 2 and \frac{1}{p} + \frac{1}{p′} = 1. In addition we have the following curvature dependent estimates.

(vi) (Dyadic Böchner inequality) For any tensor field F and 2 ≤ p < ∞, k ≥ 0,

∥∇_{g}^{2}P_{k}F ∥_{L^{2}(S)} ≲ (2^{2k} + 2^{k}∥K∥_{L^{2}(S)} + 2^{k \frac{p−2}{p−1}} ∥K∥L^{2}(S)^{\frac{p}{p−1}}C_{1}) · ∥F ∥_{L^{2}(S)},

∥P_{k}F ∥_{L_{x}^{∞}} ≲ 2^{k}C_{1}(1 + 2^{−θk}∥K∥_{L_{x}^{2}}^{θ})∥F ∥_{L_{x}^{2}},  (2.8)

and

∥P_{≤0}F ∥_{L_{x}^{∞}} ≲ C_{1}(1 + ∥K∥_{L_{x}^{2}}^{θ})∥F ∥_{L_{x}^{2}}

where K is the Gaussian curvature of (S, g) and 0 < θ < 1.

If in addition

K_{α} = ∥Λ^{−α}K∥_{L^{2}(S)} ≤ M with α > 1/2,  (2.9)

where M is a positive constant and the operator Λ^{−α} is defined by

Λ^{−α}F = (Id − Δ_{g})^{− \frac{α}{2}} F = \frac{1}{Γ(α/2)} ∫_{0}^{∞} τ^{\frac{α}{2} −1}e^{−τ}U(τ)F dτ,  (2.10)

then we have the sharp Bernstein inequalities

∥P_{k}f∥_{L^{∞}(S)} ≲ 2^{k}C̃∥f∥_{L^{2}(S)}, ∥P_{k}f∥_{L^{2}(S)} ≲ 2^{k}C̃∥f∥_{L^{1}(S)},

and

∥∇^{2}P_{k}f∥_{L^{2}(S)} ≲ 2^{2k}C̃∥f∥_{L^{2}(S)},

where C̃ depends on M in (2.9) and C_{1}.

We now briefly review the definition and basic properties of classical Littlewood Paley theory. If f is a function on ℝ^{2}, we have a Littlewood Paley partition of unity by decomposing it in infinite dyadic sums f = ∑_{k∈ℤ} f_{k} with frequency localized components f_{k}, i.e. suppf̂_{k} = {ξ ∈ ℝ^{2} : 2^{k−1} ≤ |ξ| ≤ 2^{k+1}} . Such a decomposition can be achieved by choosing a test function η, belonging to the Schwartz class, whose support is a dyadic shell \{\frac{1}{2} ≤ |ξ| ≤ 2\} and ∑_{k∈ℤ} η(2^{−k}ξ) = 1, if ξ ≠ 0. Indeed choose β(ξ) to be a real radial bump function

β(ξ) = \{ \begin{matrix}1 on B_{1}(0) \\ 0 on ℝ^{2}\B_{2}(0). \end{matrix}

Then η(ξ) = β(ξ) − β(2ξ) verifies the desired properties. We define

E_{k}f = F^{−1}(η_{k} · f̂) = \frac{1}{(2π)^{2}} ∫_{ℝ^{2}} η(ξ/2^{k})f̂(ξ)e^{ixξ}dx = ψ_{k} ∗ f

where

η_{k}(ξ) = η(2^{−k}ξ) and ψ_{k}(x) = 2^{2nk}ψ(2^{k}x)

with ψ being the Fourier inversion of η. Clearly f = ∑_{k∈ℤ} E_{k}f for all f ∈ L^{2}, where the summation converges in L^{2} sense.

The Besov spaces B̃_{p,q}^{γ}(ℝ^{2}) consist of all functions for which

∥f∥B̃_{p,q}^{γ} = (∑_{k>0} 2^{kγq}∥E_{k}f∥_{p}^{q})^{1/q} + ∥E_{≤0}f∥_{p}  (2.11)

is finite. They have the localization property, i.e. if f ∈ B̃_{p,q}^{γ}, then ϕf ∈ B̃_{p,q}^{γ} for every ϕ ∈ C_{0}^{∞}. They also have the diffeomorphism property, i.e. if χ is a diffeomorphism in ℝ^{2} which equals the identity outside a compact set, then ∥f ∘ χ∥B̃_{p,q}^{γ} ≤ c∥f∥B̃_{p,q}^{γ}. Consequently we can define Besov spaces on compact manifolds by using classical Littlewood Paley decomposition: let (U_{α}, χ_{α}) be a finite atlas with charts χ_{α} mapping U_{α} into the unit ball in ℝ^{2}, and let {ϕ_{α}} be a subordinated partition of unity on S. Then we are ready to define the Besov spaces on S.

Definition 2.3. We define B̃_{p,q}^{γ}(S) as the space of functions on S for which

∥f∥B̃_{p,q}^{γ} := ∑_{α} ∥(ϕ_{α}f) ∘ χ_{α}^{−1}∥_{B̃_{p,q}^{γ}(ℝ^{2})}  (2.12)

is finite.

The definition does not depend on the choices of the coordinate system and the partition of unity (see [25, Chapter 2.10]). Now we revisit the important facts about classical LP theory (see [22, 24]).

Proposition 2.3. (i) (Almost orthogonality) The operators E_{k} verify the orthogonality E_{k_{1}}E_{k_{2}} = 0 for all pairs of integers k_{1}, k_{2} with |k_{1} − k_{2}| ≥ 2, and E_{k} = ∑_{|k−k′|≤1} E_{k}E_{k′}. Since E_{k−1}, E_{k}, E_{k+1} do not differ too much among themselves, we may write

E_{k} = ∑_{|k−k′|<1} E_{k′}E_{k} ≈ E_{k}^{2}.

(ii) The operators E_{k} are self-adjoint and for any function f on ℝ^{2} there holds

∥f∥_{L^{2}}^{2} ≈ ∑_{k} ∥E_{k}f∥_{L^{2}}^{2}.

(iii) (L^{p} boundedness) For any 1 ≤ p ≤ ∞ and any interval J ⊂ ℤ there holds

∥E_{J}f∥_{L^{p}} ≲ ∥f∥_{L^{p}},

where E_{J} = ∑_{j∈J} E_{j}.

(iv) (Finite band property) We can write ∂E_{k}f in the form ∂E_{k}f = 2^{k}Ẽ_{k}f, where Ẽ_{k} is a cut-off operator induced by a slightly different cut-off function η̃. ^{1} For 1 ≤ p ≤ ∞, there hold

∥∇E_{k}f∥_{L^{p}} ≲ 2^{k}∥f∥_{L^{p}}
2^{k}∥E_{k}f∥_{L^{p}} ≲ ∥∇f∥_{L^{p}}

(v) (Bernstein inequalities) For any 1 ≤ p ≤ q ≤ ∞ we have the following inequalities

∥E_{k}f∥_{L^{q}} ≲ 2^{2k(1/p−1/q)}∥f∥_{L^{p}},  ∀k ∈ ℤ,
∥E_{≤0}f∥_{L^{q}} ≲ ∥f∥_{L^{p}}.

In particular,

∥E_{k}f∥_{L^{∞}} ≲ 2^{2k/p}∥f∥_{L^{p}}.

Now we are ready to state the main result of this chapter.

Theorem 2.1. Let S be a 2-dimensional closed manifold verifying (WR) conditions. Then, for each 0 ≤ α < 1, there exists a positive constant C_{0} such that

C_{0}^{−1}∥f∥_{B_{2,1}^{α}} ≤ ∥f∥B̃_{2,1}^{α} ≤ C_{0}∥f∥_{B_{2,1}^{α}}

for all scalar functions f.

Remark 2.1. C_{0} depends on the L^{∞}(S) norms of partition of unity ϕ and ∇_{g}ϕ, as well as C_{1}.

Proof of Theorem 2.1. It suffices to consider the case when α = 0. According to Definition 2.3, in order to calculate B̃_{2,1}^{0} norm of functions defined on S, with the coordinate charts (U, χ) we have chosen and their associated partition of unity {ϕ}, we need to compute B̃_{2,1}^{0}(ℝ^{2}) norms for (ϕf) ∘ χ^{−1}. With the help of the identity ∑_{j} P_{j}^{2} = Id, we may write

(ϕf) ∘ χ^{−1} = ∑_{j}(ϕP_{j}^{2}f) ∘ χ^{−1},

according to which, we have

E_{k}((ϕf) ∘ χ^{−1}) = E_{k} ((ϕ (∑_{j>0} + ∑_{j≤0}) P_{j}^{2}f) ∘ χ^{−1}) .

1It verifies Bernstein inequality, finite band property and L^{p} boundedness with 1 ≤ p ≤ ∞.

Consequently, when k ≤ 0, it is easy to see that

∥E_{≤0}((ϕf) ∘ χ^{−1})∥_{L^{2}(ℝ^{2})} ≲ ∥ϕ∥_{L^{∞}(S)}∥f∥_{L^{2}},

then we only need to consider terms of higher frequency, i.e. when k > 0,

∥E_{k}((ϕf) ∘ χ^{−1})∥_{L^{2}(ℝ^{2})}

≤ ∑_{j>0} ∥E_{k}((ϕP_{j}^{2}f) ∘ χ^{−1})∥_{L^{2}(ℝ^{2})} + ∥E_{k}((ϕ ∑_{j≤0} P_{j}^{2}f) ∘ χ^{−1})∥_{L^{2}(ℝ^{2})}.

We note that the C_{0}^{∞} functions ϕ and diffeomorphisms χ are fixed as long as the coordinate chart is fixed. For simplicity, we will identify a function on S with its local expression by suppressing χ^{−1} and identify E_{k}(ϕf) with E_{k}((ϕf) ∘ χ^{−1}), whenever there is no confusion. Thus, in order to prove ∥f∥B̃_{2,1}^{0} ≲ ∥f∥_{B_{2,1}^{0}}, it suffices to establish the following two inequalities for any scalar function f on S:

∑_{k>0} ∑_{j>0} ∥E_{k}(ϕP_{j}^{2}f)∥_{L^{2}(ℝ^{2})} ≤ C_{0} ∑_{j>0} ∥P_{j}f∥_{L^{2}(S)}  (2.13)

and

∑_{k>0} ∥E_{k}(ϕP_{≤0}^{2}f)∥_{L^{2}(ℝ^{2})} ≤ C_{0}∥f∥_{L^{2}(S)}.  (2.14)

It is easy to derive (2.14). Indeed by Proposition 2.3(iv) and Proposition 2.2(iv) we have

∑_{k>0} ∥E_{k}(ϕP_{≤0}^{2}f)∥_{L^{2}(ℝ^{2})} ≲ ∑_{k>0} 2^{−k}∥∇(ϕP_{≤0}^{2}f)∥_{L^{2}(ℝ^{2})}
≲ ∥∇_{g}(ϕP_{≤0}^{2}f)∥_{L^{2}}
≲ (∥∇_{g}ϕ∥_{L^{∞}} + ∥ϕ∥_{L^{∞}})∥f∥_{L^{2}}

We will prove (2.13) shortly.

On the other hand, to prove

∥f∥_{B_{2,1}^{0}} ≤ C_{0}∥f∥B̃_{2,1}^{0},

in view of the definition (2.6), since ∥P_{≤0}f∥_{L^{2}} ≲ ∥f∥_{L^{2}}, it suffices to consider the terms of higher frequency in (2.6). According to the finitely many coordinate charts (U_{α}, χ_{α}) we have chosen, there associates a partition of unity {ϕ_{α}} verifying ∑_{α} ϕ_{α} = 1 on S. Note that for scalar function f defined on S,

f = ∑_{α,β} ϕ_{β} ∑_{j} E_{j}(ϕ_{α}f ∘ χ_{α}^{−1}) ∘ χ_{α},

thus

P_{k}f = ∑_{j} P_{k} (∑_{α,β} ϕ_{β}(E_{j}(ϕ_{α}f ∘ χ_{α}^{−1}) ∘ χ_{α})) , when k > 0.

Since ϕ_{β}(E_{j}(ϕ_{α}f ∘ χ_{α}^{−1}) ∘ χ_{α}) is defined on S globally, P_{k}(ϕ_{β}(E_{j}(ϕ_{α}f ∘ χ_{α}^{−1}) ∘ χ_{α})) is well-defined.

As before it suffices to consider those terms with k > 0. Consequently, it remains only to establish the following two inequalities

∑_{k>0} ∑_{j>0} ∥P_{k} (∑_{α,β} ϕ_{β}(E_{j}(ϕ_{α}f ∘ χ_{α}^{−1}) ∘ χ_{α}))∥_{L^{2}} ≤ C_{0} ∑_{j>0} ∥E_{j}f∥_{L^{2}}  (2.15)

and

∑_{k>0} ∥P_{k} (∑_{α,β} ϕ_{β} (E_{≤0} (ϕ_{α}f ∘ χ_{α}^{−1}) ∘ χ_{α}))∥_{L^{2}} ≤ C_{0}∥f∥_{L^{2}}  (2.16)

As to E_{j}f on the right hand sides of (2.15) and (2.16), we have actually identified f with its local representation ϕ_{α}f ∘ χ_{α}^{−1}, where we also suppress the summation over α and β. Whenever there is no danger of confusion, we will not distinguish ϕ_{α} and ϕ_{β}, and denote them simply by ϕ. We will also suppress the push-forward, pull-back and extension procedures again in a fixed coordinate chart and denote E_{j}(ϕf ∘ χ^{−1}) ∘ χ by E_{j}f, and ϕE_{j}(ϕf ∘χ^{−1})∘χ by ϕE_{j}f. (2.16) can be derived in the same way as for (2.14) by using Proposition 2.3(iv) and Proposition 2.2(iv). As to (2.15), we enlarge the left hand side and consider

∑_{α,β} ∑_{k>0} ∑_{j>0} ∥P_{k}(ϕ_{β}(E_{j}(ϕ_{α}f ∘ χ_{α}^{−1}) ∘ χ_{α}))∥_{L^{2}}.  (2.17)

Since there are finitely many choices of (α, β), it suffices to show (2.17) for fixed pairs (α, β). Noting that we can write E_{k} as E_{k}E_{k} by Proposition 2.3 (i), then it is equal to establishing

∑_{k>0} ∑_{j>0} ∥P_{k}(ϕE_{j}^{2}f)∥_{L^{2}} ≲ C_{0} ∑_{j>0} ∥E_{j}f∥_{L^{2}}.  (2.18)

Thus to complete the proof of Theorem 2.1, it suffice to prove (2.13) and (2.18), which can be done by establishing the following two lemmas. For simplicity, we set M_{1} = ∥∇_{g}ϕ∥_{L^{2}(S)} + ∥ϕ∥_{L^{∞}(S)}.

Lemma 2.1. For any given function f on S, when k ≥ j > 0

∥E_{k}(ϕP_{j}^{2}f)∥_{L^{2}(ℝ^{2})} ≲ C_{1}M_{1}2^{j−k}∥P_{j}f∥_{L^{2}(S)}  (2.19)

and

∥P_{k}(ϕE_{j}^{2}f)∥_{L^{2}} ≲ C_{1}M_{1}2^{j−k}∥E_{j}f∥_{L^{2}}.  (2.20)

Proof. By using Proposition 2.3(iv) and Proposition 2.2(iv) we have

∥E_{k}(ϕP_{j}^{2}f)∥_{L^{2}(ℝ^{2})} ≤2^{−k}∥∇(ϕP_{j}^{2}f)∥_{L^{2}(ℝ^{2})}

≤ 2^{−k}C_{1}∥∇_{g}P_{j}^{2}f∥_{L^{2}(U)}∥ϕ∥_{L^{∞}} + 2^{−k}C_{1}∥P_{j}^{2}f∥_{L^{2}(U)}∥∇_{g}ϕ∥_{L^{∞}} ≲ 2^{j−k}C_{1}(∥P_{j}f∥_{L^{2}(S)}∥ϕ∥_{L^{∞}} + 2^{−k}C_{1}∥P_{j}^{2}f∥_{L^{2}(S)}∥∇_{g}ϕ∥_{L^{∞}})

≲ 2^{j−k}C_{1}M_{1}∥P_{j}f∥_{L^{2}(S)},

which gives (2.19). It follows from Proposition 2.2(iv) and Proposition 2.3(iv) that

∥P_{k}(ϕE_{j}^{2}f)∥_{L^{2}(S)} ≤2^{−k}∥∇_{g}(ϕE_{j}^{2}f)∥_{L^{2}(S)}
≲ 2^{−k}∥∇_{g}ϕ · E_{j}^{2}f∥_{L^{2}(S)} + 2^{−k}∥ϕ · ∇_{g}(E_{j}^{2}f)∥_{L^{2}(S)}
≲ 2^{−k}∥∇_{g}ϕ∥_{L^{∞}(S)}∥E_{j}^{2}f∥_{L^{2}(S)} + 2^{j−k}C_{1}∥ϕ∥_{L^{∞}}∥E_{j}f∥_{L^{2}(S)}
≲ 2^{j−k}C_{1}M_{1}∥E_{j}f∥_{L^{2}(ℝ^{2})}.

This gives (2.20).

By Lemma 2.1, we have the following result by a standard dual argument. Lemma 2.2. When k ≤ j, for appropriate scalar functions f,

∥E_{k}(ϕP_{j}^{2}f)∥_{L^{2}} ≲ 2^{k−j}C_{1}M_{1}∥P_{j}f∥_{L^{2}}  (2.21)
∥P_{k}(ϕE_{j}^{2}f)∥_{L^{2}} ≲ 2^{k−j}C_{1}M_{1}∥E_{j}f∥_{L^{2}}  (2.22)

Proof. From Lemma 2.1 it follows for g ∈ C_{c}^{∞}(ℝ^{2}) that

⟨E_{k}(ϕP_{j}^{2}f), g⟩ = ⟨ϕP_{j}^{2}f, E_{k}g⟩ = ⟨P_{j}f, P_{j}(ϕ(E_{k}g))⟩
≲ C_{1}M_{1}∥P_{j}f∥_{L^{2}}∥g∥_{L^{2}}2^{−j+k},

which gives (2.21). Similarly for any scalar function g ∈ C_{c}^{∞}(S) we have

⟨P_{k}(ϕE_{j}^{2}f), g⟩ = ⟨P_{k}(ϕE_{j}^{2}(ϕf ∘ χ^{−1}) ∘ χ), g⟩ = ⟨ϕE_{j}^{2}(ϕf ∘ χ^{−1}) ∘ χ, P_{k}g⟩

= ⟨E_{j}^{2}(ϕf ∘ χ^{−1}), (ϕP_{k}g) ∘ χ^{−1}⟩ = ⟨E_{j}(ϕf ∘ χ^{−1}), E_{j}((ϕP_{k}g) ∘ χ^{−1})⟩

≲ 2^{−j+k}C_{1}M_{1}∥E_{j}(ϕf ∘ χ^{−1})∥_{L^{2}}∥g∥_{L^{2}}

which implies (2.22).

Hence we complete the proof of Theorem 2.1.

We are going to extend Theorem 2.1 to tensor fields on a weakly spherical 2-D closed surface which is defined as follows, Definition 2.4. A 2-dimensional closed surface S is said to be weakly spherical if it is a weakly regular manifold verifying the following weakly spherical conditions:

∑_{a,b,c} ∥∂_{c} (g_{ab} − g_{ab}^{(0)})∥_{L^{2}(U)} ≤ I_{0} and ∑_{a,b} ∥g_{ab} − g_{ab}^{(0)}∥_{L^{∞}(U)} ≤ I_{0},

where g^{(0)} is a fixed smooth background metric on S and I_{0} is a sufficiently small number.

For each 0 ≤ α < 1 we have defined the B_{2,1}^{α}-norms for tensor fields in Definition 2.2. In the following we will use the classical LP decomposition to define the B̃_{2,1}^{α}-norms for tensor fields on weakly spherical surfaces S. We recall [14, Proposition 3.28] as follows,

Proposition 2.4. Let S be a weakly spherical surface, then there exist a finite number of vector fields X_{1}, ⋯, X_{l} verifying the following properties,

∥X∥_{L^{∞}} ≤ C_{2}, ∥∇_{0}X∥_{L^{∞}} ≤ C_{2}, ∥(∇_{g}−∇_{0})X∥_{L^{2}} ≤ I_{0}C_{2} and ∥∇_{g}(∇_{0}X)∥_{L^{2}} ≤ C_{2},

where ∇_{0} represents ∇_{g^{(0)}}, and C_{2} is a positive constant, I_{0} is a small positive number. The X_{i}’s can be chosen to be coordinate frames.

For each 0 ≤ α < 1, an S-tangent tensor F ∈ L^{2} is in B_{2,1}^{α} if and only if F ·X_{i} ∈ B_{2,1}^{α} for all 1 ≤ i ≤ l, and

C_{3}^{−1}∥F · X_{i}∥_{B_{2,1}^{α}} ≤ ∥F ∥_{B_{2,1}^{α}} ≤ C_{3}∥F · X_{i}∥_{B_{2,1}^{α}},

where C_{3} depends on C_{1}, C_{2} and M in (2.9).

With the help of these vector fields we have the the following characterization of the L^{2} norm for tensor fields F: for simplicity of notations, we restrict F to be 1-form. Then

∥F ∥_{L^{2}} ≈ ∑_{1≤i≤l} ∥F · X_{i}∥_{L^{2}}.  (2.23)

Now we are able to introduce the B̃_{2,1}^{α}-norms for tensor fields on weakly spherical 2-D closed surface S as follows.

Definition 2.5. Let S be a weakly spherical surface. For each 0 ≤ α < 1, by the classical LP projection E_{k} with ∑_{k∈ℤ} E_{k} = Id, we define, for each tensor field F on S, the norm

∥F ∥B̃_{2,1}^{α} = ∑_{1≤i≤l} ∥X_{i} · F ∥B̃_{2,1}^{α}.

Note that by Proposition 2.4 and Definition 2.5, we can obtain the equivalence result between B_{2,1}^{α}-norm and B̃_{2,1}^{α}-norm for tensor fields with 0 ≤ α < 1 by considering scalar functions X_{i} · F. Thus by using Theorem 2.1, we conclude Theorem 2.2. Let (S, g) be a 2-dimensional closed manifold verifying (WS) conditions and the Gauss curvature K verifying the condition (2.9) with some α > 1/2, then there exists a positive constant C_{4}, such that

C_{4}^{−1}∥F ∥_{B_{2,1}^{a}} ≤ ∥F ∥B̃_{2,1}^{a} ≤ C_{4}∥F ∥_{B_{2,1}^{a}}

for any tensor field F on S, where 0 ≤ a < 1.

Remark 2.2. C_{4} depends on C_{1}, C_{2}, M in (2.9) and the L^{∞}(S) norms of partition of unity ϕ and ∇_{g}ϕ.

2.2 Hodge systems on 2-D surfaces

We consider the following Hodge operators acting on 2-D surface S diffeomorphic to the standard 𝕊^{2} sphere.

(1) The operator D_{1} takes any 1-form F into the pairs of functions ( div F, curl F).

(2) The operator D_{2} takes any symmetric traceless 2-tensor F on S into the one form div F.

(3) The operator ^{⋆}D_{1} takes the pair of scalar functions (ρ, σ) into the 1-forms −∇ρ + (∇σ)^{⋆} on S.

(4) The operator ^{⋆}D_{2} takes 1-forms F on S into the 2-covariant, symmetric, traceless tensors −\frac{1}{2} L̂_{F} γ with L̂_{F} γ the traceless part of the Lie derivative of the metric γ relative to F, i.e.

(L̂_{F} γ)_{ab} = ∇_{b}F_{a} + ∇_{a}F_{b} − ( div F)γ_{ab}.  (2.24)

Then the kernels of D_{1} and D_{2} in L^{2}(S) are trivial. ^{⋆}D_{1} and ^{⋆}D_{2} are the L^{2} adjoints of D_{1}, respectively D_{2}. The kernel of the ^{⋆}D_{1} consists of pairs of constant functions (ρ, σ), and the kernel of ^{⋆}D_{2} consists of the set of all conformal killing vectorfields on S. In particular the L^{2} range of D_{1} consists of all pairs of functions ρ, σ on S with vanishing mean. The L^{2} range of D_{2} consists of all L^{2} integrable 1-forms on S which are orthogonal to the Lie algebra of all conformal Killing vectorfields on S. Denote by D one of the operators of D_{1}, D_{2} or their adjoint operators. Let us denote by R(D) the L^{2}(S) range of D and denote by ker(D) its L^{2}(S) kernel. We then define the operator P to be the projection operator from L^{2}(S) to R(D). Considering the inverse of the Hodge operators are only well-defined on R(D), we always denote by D^{−1} the operator D^{−1}P. With its aid, the following relation holds true for appropriate tensor fields on S, F = F_{1} + F_{2},

D^{−1}F = D^{−1}F_{1} + D^{−1}F_{2}.

Moreover we have the following identities,

⋆D_{1} · D_{1} = −Δ + K, D_{1} ·^{⋆}D_{1} = −Δ  (2.25)
⋆D_{2} · D_{2} = −\frac{1}{2}Δ + K, D_{2} ·^{⋆}D_{2} = −\frac{1}{2}(Δ + K).  (2.26)

and we record the following identities for Hodge operators Proposition 2.5. Let (S, g) be a compact 2-D closed surface with Gauss curvature K.

(i) The following identity holds for vector fields F on S:

∫_{S} (|∇F |^{2} + K|F |^{2}) = ∫_{S} (| div F |^{2} + |curlF |^{2}) = ∫_{S} |D_{1}F |^{2}  (2.27)

(ii) The following identity holds for symmetric, traceless, 2-tensor fields F on S:

∫_{S} (|∇F |^{2} + 2K|F |^{2}) = 2 ∫_{S} | div F |^{2} = 2 ∫_{S} |D_{2}F |^{2}  (2.28)

(iii) The following identity holds for any pair of scalar functions (ρ, σ) on S:

∫_{S} (|∇ρ|^{2} + |∇σ|^{2}) = ∫_{S} | − ∇ρ + (∇σ)^{⋆}|^{2} = ∫_{S} |^{⋆}D_{1}(ρ, σ)|^{2}  (2.29)

where (∇σ)^{⋆} denotes the Hodge dual of ∇σ. (iv) The following identity holds for vector fields F on S:

∫_{S} (|∇F |^{2} − K|F |^{2}) = 2 ∫_{S} |^{⋆}D_{2}F |^{2}.  (2.30)

For the proof, refer to [2, Chap.2 Sec.2.2].

Null Hypersurfaces

In this chapter, we recall the basic geometric notions about null hypersurfaces. We will make a preliminary bootstrap assumption, based on which we draw some useful results.

3.1 Null Hypersurfaces

Let p be a fixed point in a smooth 3 + 1 Einstein vacuum spacetime (M, g). Consider a null hypersurface H on (M, g) generated by null geodesics starting from p. To be more precise, let T be a fixed timelike vector at p satisfying ⟨T, T ⟩ = −1 and choose all future null vectors L_{ω}, ω ∈ 𝕊^{2} at p such that ⟨L_{ω}, T ⟩ = −1 and ⟨L_{ω}, L_{ω}⟩ = 0. For each ω ∈ 𝕊^{2} let Γ(s, ω) denote the outgoing null geodesic parametrized by the affine parameter s with the initial data

Γ(0, ω) = p and \frac{d}{ds}Γ(0, ω) = L_{ω}.

The union of all these outgoing null geodesics initiating from the point p forms a 3-D null hypersurface which is denoted by H.

We define the vector field L by

L := \frac{d}{ds}Γ.

Obviously L(0, ω) = L_{ω} and L satisfies

g(L, L) = 0 and D_{L}L = 0.

The parameter s can be regarded as a function on H verifying L(s) = 1 and s(p) = 0. We introduce the one parameter flow Γ_{s}(ω) := Γ(s, ω). It generates a family of 2-D closed surfaces {S_{s}} by S_{s} := Γ_{s}(𝕊^{2}), which form the geodesic foliation of H. It is clear that each S_{s} is diffeomorphic to 𝕊^{2} for s > 0 sufficiently small. By rescaling the metric g we may assume without loss of generality that for 0 < s ≤ 1 each slice S_{s} is diffeomorphic to 𝕊^{2}. Let H_{t} be the portion of H when s varies in (0, t]. For simplicity, we still denote by H the portion H_{1}. Every point q in H can be parametrized by the coordinates (s, ω) corresponding to the values of s for which q = Γ_{s}(ω). We then call (s, ω) the transport local coordinates.

Let D denote the Levi-Civita connection of Einstein vacuum metric g. Let γ be the induced metric on S_{s}, and ∇ its induced covariant derivative. At any point q ∈ S_{s} ⊂ H we denote by \underline{L} the null vector conjugate to L relative to the S_{s} foliation, i.e. ⟨L, \underline{L}⟩ = −2, ⟨\underline{L}, X⟩ = 0 for all X ∈ T_{q}(S_{s}). A smooth choice of an orthonormal frame \{e_{a}\}_{a=1,2} in T_{q}(S_{s}) combined with L, \underline{L} forms the null frame associated to the foliation.

We introduce the null components of the curvature tensor R of the spacetime metric g relative to L and \underline{L} as follows:

α_{ab} = R(L, e_{a}, L, e_{b}), β_{a} = \frac{1}{2}R(e_{a}, L, \underline{L}, L),
ρ = \frac{1}{4}R(\underline{L}, L, \underline{L}, L), σ = \frac{1}{4}^{⋆}R(\underline{L}, L, \underline{L}, L),  (3.1)
\underline{β}_{a} = \frac{1}{2}R(e_{a}, \underline{L}, \underline{L}, L), \underline{α}_{ab} = R(\underline{L}, e_{a}, \underline{L}, e_{b}),

where ^{⋆}R_{μνγδ} = \frac{1}{2}ϵ_{μνλτ}R^{λτ}_{γδ} and ϵ^{μνλτ} are the components of the volume element in (M, g). We refer the reader to [2, Sec.7.3] and [7, Sec.3.12] for details.

The total curvature flux R_{0} is then defined by

R_{0} = (∥α∥_{L^{2}(H)}^{2} + ∥β∥_{L^{2}(H)}^{2} + ∥ρ∥_{L^{2}(H)}^{2} + ∥σ∥_{L^{2}(H)}^{2} + ∥\underline{β}∥_{L^{2}(H)}^{2})^{\frac{1}{2}}

The geometry of H in particular depends on the null second fundamental form

χ(X, Y) = ⟨D_{X}L, Y ⟩

with X, Y arbitrary vector fields tangent to S_{s}. We will denote trχ and χ̂ the trace and traceless part of χ respectively. Other important geometric quantities are the dual null second fundament form

\underline{χ}(X, Y) = ⟨D_{X} \underline{L}, Y ⟩

and the torsion

ζ(X) = \frac{1}{2}⟨D_{X}L, \underline{L}⟩.

Given an orthonormal frame (e_{1}, e_{2}) on S_{s}, we will use the notations

χ_{ab} = ⟨D_{a}L, e_{b}⟩, \underline{χ}_{ab} = ⟨D_{a}\underline{L}, e_{b}⟩ and ζ_{a} = \frac{1}{2}⟨D_{a}L, \underline{L}⟩,

where a, b = 1, 2. Then

trχ = δ^{ab}χ_{ab}, χ̂_{ab} = χ_{ab} − \frac{1}{2}δ_{ab}trχ.

The mass aspect function μ is defined by

μ = − div ζ + \frac{1}{2}χ̂ · \underline{χ̂} − ρ + |ζ|^{2}.  (3.2)

Let r be the function on H defined by

r = r(s) = \sqrt{(4π)^{−1}|S_{s}|},

with |S_{s}| being the area of S_{s}.

Observe that

L_{L}γ(X, Y) = L(g(X, Y)) − g(L_{L}X, Y) − g(X, L_{L}Y)
= g(D_{X}L, Y) + g(X, D_{Y} L)
= 2χ(X, Y),

we have L_{L}γ = 2χ. We can further conclude

Lemma 3.1. Relative to the null geodesic foliation on H, for any scalar function f the following equation holds,

\frac{d}{ds} ∫_{S_{s}} fdA_{s} = ∫_{S_{s}} (\frac{d}{ds}f + trχf) dA_{s}  (3.3)

where dA_{s} = dA_{γ_{s}} denotes the volume element on S_{s}. Thus \frac{dr}{ds} = \frac{r}{2} \overline{trχ}.

We refer the reader to [12, Lemma 2.26] or [7, Lemma 3.1.3] for the detail of the proof.

3.2 Initial Data and Structure Equations

Now we record the initial data of χ, \underline{χ} and ζ etc, and refer the reader to Appendix 9.1 for the proofs.

Proposition 3.1. Let \{e_{1}, e_{2}\} is an orthonormal frame on S_{s} which, together with L, \underline{L} forms the null frames on H. We have the following local results of components of tensor fields χ, \underline{χ} and ζ relative to the null frames, near the vertex of H.

strχ = 2 + O(s^{3}) and str\underline{χ} → −2 as s → 0,  (3.4)

χ̂_{ab} = −\frac{1}{3}sS_{ab} + O(s^{2}) and \underline{χ̂}_{ab} → 0 as s → 0,  (3.5)
ζ_{a} = −\frac{1}{6}sS_{a} + O(s^{2})  (3.6)

where S_{ab} with a, b = 1, 2 is a symmetric traceless 2-tensor and S_{a} with a = 1, 2 is a 1-form, both of which are finite at the vertex, depending on curvature tensors.

Proposition 3.2. Relative to null frames \{e_{1}, e_{2}, L, \underline{L}\},

s∇_{c}trχ → 0, s∇_{c}χ̂_{ab} → 0  (3.7)
s div ζ → 0, s∇_{a}ζ_{b} → 0
sμ → 0 as s → 0.

where a, b, c = 1, 2.

Proposition 3.3. Let γ^{(0)} be the canonical metric on the standard 2-sphere 𝕊^{2}, and let

γ^{∘}= \frac{1}{s^{2}} γ and a_{s} = ^{\begin{matrix}√ & \overline{|γ_{s}|} \\ √ & \overline{|γ^{(0)}|} \end{matrix}}. Then

γ^{∘}_{ij}= γ_{ij}^{(0)} + O(s^{2})  (3.8)
s^{−2}a_{s} → 1, \frac{r}{s} → 1 as s → 0.  (3.9)

We will rely on the structure equations to prove our main theorem. The structure equations of the geodesic foliation take the following form:

\frac{d}{ds}trχ = −\frac{1}{2}(trχ)^{2} − |χ̂|^{2}  (3.10)
∇_{L}χ̂ = −trχ · χ̂ − α  (3.11)
∇_{L}(∇trχ) = −\frac{3}{2}trχ · ∇trχ − χ̂ · ∇trχ − 2χ̂ · ∇χ̂  (3.12)
∇_{L}ζ = −trχζ − 2χ̂ · ζ − β  (3.13)
\frac{d}{ds}tr\underline{χ} = −\frac{1}{2}trχtr\underline{χ} − 2 div ζ − χ̂ · \underline{χ̂} + 2|ζ|^{2} + 2ρ  (3.14)
∇_{L}\underline{χ̂} = −∇ˆ⊗ζ − \frac{1}{2}(trχ · \underline{χ̂} + tr\underline{χ} · χ̂) + ζ ˆ⊗ζ  (3.15)
div χ̂ = \frac{1}{2}∇trχ − \frac{1}{2}trχ · ζ − χ̂ · ζ − β  (3.16)
curl ζ = −\frac{1}{2}χ̂ ∧ \underline{χ̂} + σ  (3.17)
div \underline{χ̂} = \frac{1}{2}∇tr\underline{χ} − \frac{1}{2}tr\underline{χ}ζ + ζ · \underline{χ̂} + \underline{β}  (3.18)
K = −\frac{1}{4}trχtr\underline{χ} + \frac{1}{2}χ̂ · \underline{χ̂} − ρ  (3.19)

We also have the renormalized Bianchi identities,

∇_{L}β = div α + ζ · α  (3.20)
L(ρ̌) + \frac{3}{2}trχ · ρ̌ = div β − ζ · β + \frac{1}{2}χ̂ · (∇ˆ⊗ζ + \frac{1}{2}tr\underline{χ} · χ̂ − ζ ˆ⊗ζ)  (3.21)
L(σ̌) + \frac{3}{2}trχ · σ̌ = − curl β + ζ ∧ β + \frac{1}{2}χ̂ ∧ (∇ˆ⊗ζ − ζ ˆ⊗ζ)  (3.22)
∇_{L} ˇ\underline{β} = −∇ρ + (∇σ)^{⋆} − 2(∇ˆ⊗ζ) · ζ + 3(ζ · ρ − ζ^{⋆}σ) − trχ\underline{β}
+ 2ζ · (−\frac{1}{2}trχ · \underline{χ̂} − \frac{1}{2}tr\underline{χ} · χ̂ + ζ ˆ⊗ζ) − 4χ · \underline{χ̂} · ζ  (3.23)

where

ρ̌ = ρ − \frac{1}{2}χ̂ · \underline{χ̂} , σ̌ = σ − \frac{1}{2}χ̂ ∧ \underline{χ̂} , ˇ \underline{β} = \underline{β} + 2\underline{χ̂} · ζ.

The transport equation for the mass aspect function μ defined by (3.2) takes the form

\frac{d}{ds}μ + \frac{3}{2}trχμ = χ̂ · (∇ˆ⊗ζ) + \frac{1}{2}trχρ̌ + 2ζ · ∇trχ − 4χ̂ · ζ · ζ + trχ|ζ|^{2} − \frac{1}{4}tr\underline{χ}|χ̂|^{2}. (3.24)

For the derivation of (3.10)-(3.24) refer to [2] or [12, Sec. 2.12].

3.3 Commutation Relations

We will use the following commutation formula frequently (refer to [7, P183-185] and [12, Section 2.15] for the derivation).

Proposition 3.4. Consider an arbitrary k-covariant, S-tangent tensorfield F\underline{a} = F_{a_{1}···a_{k}}, then

∇_{L}∇_{b}F\underline{a} − ∇_{b}∇_{L}F\underline{a} = −χ_{bc}∇_{c}F\underline{a} + ∑_{i}(−χ_{a_{i}b}ζ_{c} + χ_{bc}ζ_{a_{i}} − ϵ_{a_{i}c}^{⋆}β_{b})F_{a_{1}···c···a_{k}} (3.25)

In particular for scalar functions f we have

∇_{L}∇_{b}f − ∇_{b}∇_{L}f = −χ_{bc}∇_{c}f.  (3.26)

For a 1-form F,

L( div F) − div (∇_{L}F) = −χ · ∇F + \frac{1}{2}trχ · ζ · F + χ̂ · ζ · F − β · F  (3.27)

and for scalar functions f,

L(Δf) − Δ(Lf) = −2χ · ∇^{2}f + (2χ̂ · ζ + ∇trχ + trχζ) · ∇f  (3.28)

where ^{⋆}β is the Hodge dual of one form β with ^{⋆}β_{a} = ϵ_{ab}β_{b}, and ϵ_{ab} are the components of the area element relative to the frame {e_{a}}_{a=1,2}.

3.4 Preliminary Estimates

In this section, we will provide preliminary estimates under a preliminary bootstrap assumption of trχ, the full set of bootstrap assumptions will be presented in Chapter 4.

Definition 3.1. Given a function f on H_{t} we denote by ∫_{H_{t}} f the integral

∫_{H_{t}} f = ∫_{0}^{t} ∫_{S_{s}} fdA_{s} = ∫_{0}^{t} ∫_{|ω|=1} f\sqrt{|γ|}(s, ω^{1}, ω^{2})dsdω^{1}dω^{2}.

The proof of the main theorem is based on a bootstrap argument. For the first approximation, the bootstrap assumption requires that there exists a sufficiently small positive constant 0 < Δ_{0} < 1/2 such that

∥V ∥_{L_{ω}^{∞}L_{t}^{∞}} ≲ Δ_{0},  (3.29)

where V (s, ω) = trχ− \frac{2}{s}. We also set \underline{V} (s, ω) = tr\underline{χ}+ \frac{2}{s}, which will be used in the future.

Since L_{L}γ = 2χ, then \frac{d}{ds}a_{s} = trχa_{s}. In view of (3.9) which says lim_{t→0} \frac{a_{t}}{t^{2}} = 1, we get \frac{a_{t}}{t^{2}} = exp ∫_{0}^{t} V_{s}ds. Therefore when 0 < t ≤ 1, for small Δ_{0} there holds

|t^{−2}a(t, ω) − 1| ≲ ∫_{0}^{t} |V (s, ω)|ds ≲ tΔ_{0},

thus

\frac{1}{2}t^{2} ≤ a(t, ω) ≤ 2t^{2}.  (3.30)

In view of Lemma 3.1 which says \frac{dr}{ds} = \frac{r}{2} \overline{trχ}, it is easy to check

\frac{d}{ds} log \frac{r}{s} = \frac{1}{2}\overline{V_{s}} and \frac{d}{ds} (\frac{1}{r} − \frac{1}{s}) + \frac{1}{s} (\frac{1}{r} − \frac{1}{s}) = −\frac{1}{2r}\overline{V_{s}}

By (3.9), lim_{s→0} \frac{r}{s} = 1 and lim_{s→0} \frac{1}{r} − \frac{1}{s} = 0, integrating the above transport equations along any geodesic yields \frac{r}{s} = exp(s · O(Δ_{0})) and |\frac{1}{r} − \frac{1}{s}| = O(Δ_{0}). Therefore we get the following

Proposition 3.5. Under the bootstrap assumption (3.29), the radius r(s) of each leave S_{s} and the affine parameter s are always comparable in the sense that |\frac{1}{r} − \frac{1}{s}| = O(Δ_{0}) for 0 < s ≤ 1 on H.

3.4.1 Norms on H

In view of (3.30), we have

\frac{1}{2} ∫_{|ω|=1} |F (s, ω)|^{p}s^{2}dω ≤ ∥F ∥_{L^{p}(S_{s})}^{p} ≤ 2 ∫_{|ω|=1} |F (s, ω)|^{p}s^{2}dω.

Now for 1 ≤ p, q ≤ ∞ we define the L_{t}^{q}L_{x}^{p} norm

∥F ∥_{L_{t}^{q}L_{x}^{p}} := (∫_{0}^{1} (∫_{|ω|=1} |F (s, ω)|^{p}s^{2}dω)^{\frac{q}{p}} ds)^{\frac{1}{q}}  (3.31)

and the L_{x}^{p}L_{t}^{∞} norm

∥F ∥_{L_{x}^{p}L_{t}^{∞}} := (∫_{|ω|=1} (sup_{s} s^{\frac{2}{p}} |F (s, ω)|)^{p})^{\frac{1}{p}} .

It is easy to verify that

∥F · G∥_{L_{t}^{2}L_{x}^{2}} ≲ ∥F ∥_{L_{x}^{∞}L_{t}^{2}} · ∥G∥_{L_{x}^{2}L_{t}^{∞}}.

We also define the norms

N_{1}(F) := ∥t^{−1}F ∥_{L_{t}^{2}L_{x}^{2}} + ∥∇_{L}F ∥_{L_{t}^{2}L_{x}^{2}} + ∥∇F ∥_{L_{t}^{2}L_{x}^{2}}

and

N_{2}(F) := ∥t^{−2}F ∥_{L_{t}^{2}L_{x}^{2}} + ∥t^{−1}∇_{L}F ∥_{L_{t}^{2}L_{x}^{2}} + ∥∇∇_{L}F ∥_{L_{t}^{2}L_{x}^{2}} + ∥∇^{2}F ∥_{L_{t}^{2}L_{x}^{2}}.

We will modify the definition (2.10) to define the operator Λ^{a} with a ≤ 0. For any S_{s} tangent tensor fields F, we define

Λ^{a}F = s^{−a}(Id − Δ_{γ^{∘}})^{\frac{a}{2}} F = \frac{s^{−a}}{Γ(−a/2)} ∫_{0}^{∞} τ^{− \frac{a}{2} −1}e^{−τ}U(τ)F dτ,  (3.32)

where the heat flow U(τ)F is defined according to the heat equation on (S_{s}, γ^{∘})

\frac{d}{ds}U(τ)F − Δ_{γ^{∘}}U(τ)F = 0, U(0)F = F.

Clearly Δ_{γ^{∘}} = s^{2}Δ_{γ}, thus we can rewrite the above heat equation as follows

\frac{d}{ds}U(τ)F − s^{2}ΔU(τ)F = 0, U(0)F = F.  (3.33)

We then extend the definition (3.32) to the range of a with a > 0 by defining for 0 < a ≤ 2 according to the formula

Λ^{a}F = Λ^{a−2} · (s^{−2}Id − Δ_{γ})F,

and then for 0 < a ≤ 2m according to

Λ^{a}F = Λ^{a−2m} · (s^{−2}Id − Δ_{γ})^{m}F.

Remark 3.1. We still denote by P_{k} the geometric Littlewood Paley decomposition associated to U(τ) given in (3.33).

Definition 3.2.

∥F ∥_{H^{a}(S)} = ∥Λ^{a}F ∥_{L^{2}(S)} ≈ (∑_{k≥0} 2^{2ka}s^{−2a}∥P_{k}F ∥_{L^{2}(S)}^{2})^{\frac{1}{2}} + ∥s^{−a}P_{≤0}F ∥_{L^{2}(S)}

We then have the Sobolev embedding in the following, (for its proof, see [15, Proposition 8.4])

Proposition 3.6. For arbitrary tensorfield F on a surface S = S_{s}, 0 < s ≤ 1, with 2 < p < ∞ and a > 1 − \frac{2}{p}

∥F ∥_{L^{p}(S)} ≲ ∥Λ^{a}F ∥_{L^{2}(S)}.

Moreover we define the Besov norms B^{α} and P^{α} with 0 ≤ α < 1 as follows,

∥F ∥_{B^{α}} = ∑_{k>0} ∥(2^{k}t^{−1})^{α}P_{k}F ∥_{L_{t}^{∞}L_{x}^{2}} + ∥t^{−α}P_{≤0}F ∥_{L_{t}^{∞}L_{x}^{2}},
∥F ∥_{P^{α}} = ∑_{k>0} ∥(2^{k}t^{−1})^{α}P_{k}F ∥_{L_{t}^{2}L_{x}^{2}} + ∥t^{−α}P_{≤0}F ∥_{L_{t}^{2}L_{x}^{2}}.  (3.34)

We collect some notations which will be used repeatedly. We denote by γ^{∘}= s^{−2}γ the rescaled metric on S_{s} and use the following conventions

• We denote by Λ^{∘} the operator defined by (2.10) with g =γ^{∘}, by K^{∘} the Gauss curvature of (S_{s}, γ^{∘}). By Λ we denote the operator given by (3.32) and by K the Gauss curvature of (S_{s}, γ).

• We denote by B^{α} and P^{α} the Besov norms defined by the geometric Littlewood Paley decompositions P_{k} which is given by the heat flow (3.33). We denote by B̃^{α} and P̃^{α} the Besov norms defined by the classical Littlewood Paley decompositions E_{k}. ^{1}

Remark 3.2. Considering that two metrics, γ and γ^{∘}, are used frequently, we denote a norm relative to the rescaled metric γ^{∘} by adding the subscript γ^{∘}. For example, | · |_{γ^{∘}} is the norm of a tensor relative to the rescaled metric γ^{∘}. By | · | and ∥ · ∥_{L_{t}^{q}L_{x}^{p}} we denote the norms of a tensor with respect to the metric γ.

Remark 3.3. Note that Proposition 2.2 holds for P_{k} induced by the heat flow defined in (3.33) with each 2^{ak} replaced by 2^{ak}s^{−a} for appropriate numbers a and with (S, g) being each slice (S_{s}, γ).

1These norms will be defined soon in Chapter 4.

On each slice S = S_{s}, we have Sobolev inequalities for scalar functions f and tensor fields F

∥f∥_{L^{2}(S)} ≲ ∥∇f∥_{L^{1}(S)} + ∥s^{−1}f∥_{L^{1}(S)}  (3.35)
∥f∥_{L^{∞}(S)} ≲ ∥∇^{2}f∥_{L^{1}(S)} + ∥∇f∥_{L^{2}(S)} + ∥s^{−2}f∥_{L^{1}(S)},  (3.36)
∥F ∥_{L^{p}(S)} ≲ ∥∇F ∥_{L^{2}(S)}^{1− \frac{2}{p}} · ∥F ∥L^{2}(S)^{\frac{2}{p}} + ∥s^{\frac{2}{p} −1}F ∥_{L^{2}(S)},  (3.37)
∥F ∥_{L^{∞}(S)} ≲ ∥∇^{2}F ∥L^{2}(S)^{\frac{1}{p}}(∥∇F ∥L^{2}(S)^{\frac{p−2}{p}}∥F ∥L^{2}(S)^{\frac{1}{p}} + s^{\frac{2}{p} −1}∥F ∥L^{2}(S)^{\frac{p−1}{p}}) + ∥∇F ∥_{L^{2}(S)}. (3.38)

We refer the reader to [15] for the proof.

The following preliminary estimates will be used routinely.

Lemma 3.2. Let F be an arbitrary S-tangent tensorfield on H, then

\begin{matrix}∥t^{− \frac{1}{2}} F ∥_{L_{t}^{∞}L_{ω}^{∞}} + ∥t^{−1}F ∥_{L_{t}^{2}L_{ω}^{∞}} ≲ N_{2}(F), \\ ∥F ∥_{L_{x}^{4}L_{t}^{∞}} + ∥F ∥_{L_{t}^{6}L_{x}^{6}} + ∥t^{−1/2}F ∥_{L_{x}^{2}L_{t}^{∞}} ≲ N_{1}(F). \end{matrix}  (3.39)

Proof. The proof of the estimate in the first line can be found in [2].

Now we show estimates in the second line of (3.39). Observe that it suffices to prove them for scalars f. First,

t^{2}|f(t, ω)|^{4} ≲ ∫_{0}^{t} s|f|^{4}ds + ∫_{0}^{t} |f|^{3} · |∇_{L}f(s, ω)|s^{2}ds

≲ x ∫_{0}^{t} s^{−1}|f|^{4}s^{2}ds + (∫_{0}^{t} |f(s, ω)|^{6}s^{2}ds)^{\frac{1}{2}} · (∫_{0}^{t} |∇_{L}f(s, ω)|^{2}s^{2}ds)^{\frac{1}{2}}

≲ (∫_{0}^{t} |f(s, ω)|^{6}s^{2}ds)^{\frac{1}{2}} (∫_{0}^{t} s^{−2}|f|^{2}s^{2}ds + ∫_{0}^{t} |∇_{L}f(s, ω)|^{2}s^{2}ds)^{\frac{1}{2}} .

Hence, integrating on |ω| = 1, we obtain

∥f∥_{L_{x}^{4}L_{t}^{∞}} ≲ ∥f∥L^{6}(H)^{\frac{3}{4}} (∥s^{−1}f∥_{L^{2}(H)} + ∥∇_{L}f∥_{L^{2}(H)})^{\frac{1}{4}} .  (3.40)

On the other hand, by Sobolev embedding we have on each S_{s} that

∥f∥_{L_{x}^{2}} ≲ ∫_{S_{s}}(|∇f| + s^{−1}|f|)dA_{s}.

Combine this with ∫_{S_{s}} |f|^{6}dA_{s} ≲ ∥f^{3}∥_{L_{x}^{2}}^{2}, we deduce

∫_{S_{s}} |f|^{6}dA_{s} ≲ (∫_{S_{s}}(|∇f| + s^{−1}|f|)|f|^{2}dA_{s})^{2}
≲ ∫_{S_{s}} (|∇f|^{2} + s^{−2}|f|^{2}) dA_{s} · ∫_{S_{s}} |f|^{4}dA_{s}.

Then integrating in s yields

∥f∥_{L^{6}(H)} ≲ ∥f∥L_{t}^{∞}L_{x}^{4}^{\frac{2}{3}} (∥s^{−1}f∥_{L^{2}(H)} + ∥∇f∥_{L^{2}(H)})^{\frac{1}{3}} .  (3.41)

At last ∥t^{−1/2}F ∥_{L_{x}^{2}L_{t}^{∞}} can be estimated by noting it is bounded by ∥ ∫_{t}^{1} \frac{d}{ds}(s|F |^{2})ds∥_{L_{ω}^{2}}^{1/2} + ∥F (1)∥_{L_{ω}^{2}}. The first term is bounded by N_{1}(F) by using Hölder inequality. The second term can be estimated by taking a test function 0 ≤ θ ≤ 1 supported in [1/2, 1] with θ(1) = 1 and vanishing identically at 0 < t < 1/2, and sup_{t≤s≤1} |θ′(s)| < \frac{1}{1−t}. Since

∥F (1)∥_{L_{ω}^{2}}^{2} = ∫_{|ω|=1} |∫_{t}^{1} \frac{d}{ds}(sθ|F |^{2})ds| dω ≲ ∥∇_{L}F ∥_{L^{2}(H)}^{2} + ∥s^{−1}F ∥_{L^{2}(H)}^{2} ≲ N_{1}(F)^{2}

we conclude ∥t^{−1/2}F ∥_{L_{x}^{2}L_{t}^{∞}} ≲ N_{1}(F). Combining (3.40) with (3.41), we conclude that the estimates in the second line of (3.39) holds true.

Bootstrap Assumptions

4.1 Main Theorem and Bootstrap Assumptions

Theorem 4.1 (Main Theorem). Consider an outgoing null hypersurface H in a smooth 3+1 Einstein vacuum spacetime (M, g), initiating from a point p and foliated by the geodesic foliation associated to the affine parameter s with s|_{p} = 0. ^{1} With the total curvature flux R_{0} sufficiently small, we have

∥trχ − \frac{2}{t}∥_{L_{t}^{∞}L_{ω}^{∞}} ≲ R_{0}  (4.1)

and

∥∫_{0}^{1} |χ̂|^{2}dt∥_{L_{ω}^{∞}} + ∥∫_{0}^{1} |ζ|^{2}dt∥_{L_{ω}^{∞}} ≲ R_{0},

∥sup_{t≤1} |t∇trχ|∥_{L_{ω}^{2}} + ∥sup_{t≤1} t^{\frac{3}{2}} |μ|∥_{L_{ω}^{2}} + ∥μ∥_{L^{2}(H)} ≲ R_{0},  (4.2)

N_{1}(χ̂) + N_{1}(ζ) + N_{1} (trχ − \frac{2}{t}) ≲ R_{0},

∥∇trχ∥_{B^{0}} + ∥t^{1/2}μ∥_{B^{0}} + ∥μ∥_{P^{0}} ≲ R_{0},  (4.3)
∥sup_{t≤1} t^{\frac{1}{2}} |tr\underline{χ} + \frac{2}{t}|∥_{L_{ω}^{2}} + ∥sup_{t≤1} t^{\frac{1}{2}} |\underline{χ̂} |∥_{L_{ω}^{2}} ≲ R_{0},  (4.4)
∥tr\underline{χ} + \frac{2}{t}∥_{L_{t}^{2}L_{ω}^{2}} + ∥\underline{χ̂} ∥_{L_{t}^{2}L_{ω}^{2}} ≲ R_{0},  (4.5)

∥∇_{L}(tr\underline{χ} + \frac{2}{t})∥_{L^{2}(H)} ≲ R_{0}

∥t^{−1/2}(tr\underline{χ} + \frac{2}{t})∥_{B^{0}} + ∥t^{−1/2}\underline{χ̂} ∥_{B^{0}} ≲ R_{0},  (4.6)
∥t^{−1}(tr\underline{χ} + \frac{2}{t})∥_{P^{0}} + ∥t^{−1}\underline{χ̂} ∥_{P^{0}} ≲ R_{0}.  (4.7)

1The parameter t is identical to s.

Remark 4.1. In order to get the main estimate (4.1), we are forced to prove the stronger estimate ∥∇trχ∥_{B^{0}} ≲ R_{0} in (4.3), which will be obtained by using the transport equation (3.12). However, none of the above estimates can be derived independent of others, thus the whole process of proof necessitates the full set of estimates for μ, ζ, χ̂, \underline{χ}. We will prove them by bootstrap argument.

Recall that we have made a bootstrap assumption (3.29) and have provided some preliminary estimates based on it. We shall state a more complete set of bootstrap assumptions based on the following notations.

• We denote by R the full collection of null curvature components α, β, ρ, σ, \underline{β}.

• We denote by R_{0} the collection of the null curvature components β, ρ, σ, \underline{β}.

• We denote by Ř the collection of the renormalized null curvature components (ρ̌, −σ̌), \underline{β̌}.

• By \underline{A} we denote the collection V,χ̂, ζ, \underline{V}, \underline{χ̂}

• We denote by A the collection of V, χ̂, ζ.

• By M we denote the collection ∇trχ, μ.

• By ∇A we denote the collection ∇trχ, ∇χ̂, ∇ζ.

We shall make the following bootstrap assumptions

∥trχ − 2/t∥_{L_{t}^{∞}L_{ω}^{∞}} ≤ Δ_{0}, ∥A∥_{L_{ω}^{∞}L_{t}^{2}} ≤ Δ_{0} and N_{1}(A), ∥∇trχ∥_{L_{x}^{2}L_{t}^{∞}} ≤ Δ_{0}, (BA1)
∥t^{−1/2}\underline{A}∥_{L_{x}^{2}L_{t}^{∞}} ≤ Δ_{0}, ∥∇_{L}\underline{A}∥_{L^{2}(H)} ≤ Δ_{0}  (BA2)

and the auxiliary bootstrap assumption

K_{a} := ∥Λ^{−a}(K − s^{−2})∥_{L_{t}^{∞}L_{x}^{2}} ≤ Δ_{0} with \frac{1}{2} < a < 1,  (BA3)

where Λ = (s^{−2}Id − Δ)^{\frac{1}{2}} . Note that we have

∥R∥_{L^{2}(H)} ≤ R_{0}  (4.8)

where R_{0} is a small constant. With its aid, also using (BA1) and (BA2), we get

∥Ř∥_{L_{t}^{2}L_{x}^{2}} ≤ Δ_{0}^{2} + R_{0}.  (4.9)

Remark 4.2. (BA3) implies (2.9) holds true with g =γ^{∘} uniformly in s. Since by rescaling, from (BA3), ∥s^{−1+a}Λ^{∘}^{−a}(K^{∘} −1)∥_{L_{s}^{∞}L_{ω}^{2}} ≤ Δ_{0} with 1 > a > 1/2, we infer there exists a positive universal constant M_{0} such that

K^{∘}_{a}= ∥Λ^{∘}^{−a} K^{∘} ∥_{L_{s}^{∞}L_{ω}^{2}} < M_{0}, with 1 > a > 1/2.

In Section 4.3, we will rely on (BA1) and (BA2) to verify the auxiliary assumption (BA3). We are going to show by choosing the free constant R_{0}, the bootstrap assumptions (BA1) and (BA2) can be recovered with Δ_{0} replaced by Δ_{0}/2.

4.2 Weakly spherical surfaces and Equivalent Besov norms on H

Note Proposition 3.3 says that

lim_{s→0} γ^{∘}_{ij}= γ_{ij}^{(0)}, lim_{s→0} ∂_{k} γ^{∘}_{ij}= ∂_{k}γ_{ij}^{(0)}  (4.10)

where i, j, k = 1, 2. With its aid, using bootstrap assumptions (BA1), we will prove for each 0 < s ≤ 1 the leave S_{s} is a weakly spherical surface. Lemma 4.1. For the transport local coordinates (s, ω), the following properties hold true for all surfaces S_{s} of the geodesic foliation on the null hypersurface H: the metric γ^{∘}_{ij} (s) on each S_{s} verifies weakly spherical conditions i.e.

∥∂_{k} γ^{∘}_{ij} (s) − ∂_{k}γ_{ij}^{(0)}∥_{L_{ω}^{2}} ≲ Δ_{0}  (4.11)
∥ γ^{∘}_{ij} (s) − γ_{ij}^{(0)}∥_{L_{ω}^{∞}} ≲ Δ_{0}  (4.12)

where Δ_{0} is a small constants. Proof. Recall that we parametrize the null hypersurface H, by (s, ω) with ω = (ω_{1}, ω_{2}) on 𝕊^{2} and the null geodesic Γ(s, ω) satisfying

\frac{d}{ds}Γ = L on H and Γ(0, ω) = p.

Relative to the coordinates s, ω_{1}, ω_{2} on H the metric γ verifies \frac{d}{ds}γ_{ij} = 2χ_{ij} with i, j = 1, 2, which follows immediately from L_{L}γ = 2χ.

Consequently, \frac{d}{ds}(γ^{∘}_{ij}) =γ^{∘}_{ij} V + s^{−2}χ̂_{ij}. With the initial condition given by (4.10), we integrate the transport equation along any null geodesic on H,

∑_{i,j} sup_{t} ∥ γ^{∘}_{ij} −γ_{ij}^{(0)}∥_{L_{ω}^{∞}} ≲ ∑_{i,j} sup_{t} ∥∫_{0}^{t} γ^{∘}_{ij}V + s^{−2}χ̂_{ij}ds∥_{L_{ω}^{∞}}

≲ ∥V ∥_{L_{t}^{∞}L_{ω}^{∞}} (∑_{ij} ∥ γ^{∘}_{ij} −γ_{ij}^{(0)}∥_{L_{t}^{∞}L_{ω}^{∞}} + 1) + ∑_{ij} sup_{t} ∥∫_{0}^{t} s^{−2}χ̂_{ij}ds∥_{L_{ω}^{∞}} .

Moreover, we have

∑_{ij} ∥∫_{0}^{t} s^{−2}χ̂_{ij}ds∥_{L_{ω}^{∞}} ≲ ∥χ̂∥_{L_{ω}^{∞}L_{t}^{2}} (1 + ∑_{i,j} ∥ γ^{∘}_{ij} −γ_{ij}^{(0)}∥_{L_{t}^{∞}L_{ω}^{∞}}) .

According to (BA1) with Δ_{0} ≤ 1/2, (4.12) is proved. As to (4.11), it is easy to get the following transport equation:

\frac{d}{ds}∂_{k}γ^{∘}_{ij} = ∂_{k}γ^{∘}_{ij}V + γ^{∘}_{ij}∂_{k}V + s^{−2}∂_{k}χ̂_{ij},

where i, j, k = 1, 2. Integrating the transport equation in s with the initial condition given by (4.10) implies

∂_{k} γ^{∘}_{ij} (t) − ∂_{k}γ_{ij}^{(0)} = ∫_{0}^{t} (∂_{k}V · γ^{∘}_{ij} + ∂_{k}γ^{∘}_{ij} · V + s^{−2}(∇_{k}χ̂_{ij} − Γ · χ̂)) ds

where Γ represents Christoffel symbols, and Γ · χ̂ stands for the terms ∑_{l=1}^{2} Γ_{ki}^{l}χ̂_{lj} with l = 1, 2. Then

∥ sup_{0<t≤1} |∂_{k} γ^{∘}_{ij} (t) − ∂_{k}γ_{ij}^{(0)}|∥_{L_{ω}^{2}}

≲ ∥∇trχ∥_{L_{t}^{2}L_{x}^{2}} + ∥∇χ̂∥_{L_{t}^{2}L_{x}^{2}} + ∥Γ∥_{L_{ω}^{2}L_{t}^{∞}}∥χ̂∥_{L_{ω}^{∞}L_{t}^{2}} + Δ_{0}∥∂_{k} γ^{∘}_{ij} −∂_{k}γ_{ij}^{(0)}∥_{L_{ω}^{2}L_{t}^{∞}} + Δ_{0},

which, combined with (BA1), gives ∥∂_{k}(γ^{∘}_{ij} −γ_{ij}^{(0)})∥_{L_{ω}^{2}L_{t}^{∞}} ≲ Δ_{0}. ^{2} Therefore for each (S_{s}, γ^{∘}), the weakly spherical condition holds.

Next we will prove an equivalence argument between the Besov norms B^{α} and P^{α} defined by the geometric Littlewood Paley projection and their counterparts defined on H by the classical LP projection.

We first define the Besov norms by the classical LP decomposition for scalars.

Definition 4.1. With the classical Littlewood Paley decomposition which is denoted by E_{k}, We define the norms B̃^{α} and P̃^{α} for scalar functions f with 0 ≤ α < 1

∥f∥B̃^{α} = ∑_{k>0} ∥(2^{k}t^{−1})^{α}E_{k}f∥_{L_{t}^{∞}L_{x}^{2}} + ∥t^{−α}E_{≤0}f∥_{L_{t}^{∞}L_{x}^{2}},
∥f∥P̃^{α} = ∑_{k>0} ∥(2^{k}t^{−1})^{α}E_{k}f∥_{L_{t}^{2}L_{x}^{2}} + ∥t^{−α}E_{≤0}f∥_{L_{t}^{2}L_{x}^{2}}.  (4.13)

2This result is stronger than (4.11).

The classical Littlewood Paley projection E_{k}f is well-defined because of the following: the geodesic flow Γ_{s} : 𝕊^{2} → S_{s} for each s > 0 is a diffeomorphism. Let (U_{i}, η^{(i)}) be a finite atlas with charts η^{(i)} mapping U_{i} into the unit disc in ℝ^{2}, and let {ϕ} be a subordinated partition of unity on 𝕊^{2}. Then {ϕ ∘ Γ_{s}^{−1}} is the partition of unity in the slice S_{s} for 1 ≥ s > 0 which will be denoted as ϕ_{s}. Denote η^{(i)} ∘ Γ_{s}^{−1} by η_{s}^{(i)}, E_{k}f is defined by E_{k}((ϕ_{s}f) ∘ η_{s}^{(i)}^{−1}) on S_{s}. The L_{ω}^{2} norm in Definition 4.1 is understood to be the L^{2} norm on ℝ^{2}. For convenience, we will adhere to this notation whenever there is no confusion.

Using Lemma 4.1 and the bootstrap assumption (BA3) which implies (2.9) holds for every (S_{s}, γ^{∘}), we can adapt [14, Proposition 3.28] to obtain the following lemma.

Lemma 4.2. There exists {X_{i}}_{i=1}^{l} to be a frame on H verifying the conditions

∥X, t∇_{0}X∥_{L_{t}^{∞}L_{ω}^{∞}} ≲ 1, ∥t^{2}∇(∇_{0}X)∥_{L_{x}^{2}L_{t}^{∞}} ≲ 1, ∥t(∇ − ∇_{0})X∥_{L_{x}^{2}L_{t}^{∞}} ≲ Δ_{0}, ∇_{L}X = 0

(4.14)

where ∇_{0} represents the covariant derivative induced by the background metric s^{2}γ^{(0)}. For appropriate S_{s}-tangent tensor F ∈ L_{t}^{∞}L_{x}^{2}, F ∈ B^{α} if and only if ^{3} F · X_{i} ∈ B^{α}, and

c^{−1} ∑_{i} ∥F · X_{i}∥_{B^{α}} ≤ ∥F ∥_{B^{α}} ≤ c ∑_{i} ∥F · X_{i}∥_{B^{α}}, with 0 ≤ α < 1,  (4.15)

where c depends on the uniform bound M_{0} in Remark 4.2. The same results hold for the spaces P^{α}. Moreover

N_{1}[F ˆ⊗X] + ∥F ˆ⊗X∥_{L_{ω}^{∞}L_{t}^{2}} ≲ N_{1}(F) + ∥F ∥_{L_{ω}^{∞}L_{t}^{2}},

where ˆ⊗ stands for either a tensor product or a contraction.

To prove Lemma 4.2, we can adapt the proof of [14, Proposition 3.28] by keeping track of the weight of t in calculations. For details, refer to [14].

Lemma 4.2 allows us to define Besov norms for arbitrary S_{s}-tangent tensor fields F on H by classical Littlewood Paley decomposition.

Definition 4.2. Suppose F is an (m, n) S_{s}-tangent tensor on H and F has the local expression F_{i_{1}i_{2}···i_{n}}^{j_{1},j_{2},···j_{m}} under the frame {X_{i}}_{i=1}^{l}, we define B̃^{α} by

∥F ∥B̃^{α} = ∑ ∥F_{i_{1}i_{2}···i_{n}}^{j_{1}j_{2}···j_{m}}∥B̃^{α}
∥F ∥P̃^{α} = ∑ ∥F_{i_{1}i_{2}···i_{n}}^{j_{1}j_{2}···j_{m}}∥P̃^{α}.

where the summation is taken over all possible (i_{1} ⋯ i_{n}; j_{1} ⋯ j_{m}).

3Let us suppose F to be one forms without loss of generality.

At last we prove the following equivalence result.

Proposition 4.1. For arbitrary S_{s}-tangent tensor fields F,

∥F ∥B̃^{α} ≈ ∥F ∥_{B^{α}}
∥F ∥P̃^{α} ≈ ∥F ∥_{P^{α}},

with 0 ≤ α < 1.

Proof. We observe that it suffices to prove Proposition 4.1 for scalar functions and for α = 0. From Lemma 2.1 and Lemma 2.2, we know that since (S_{s}, γ^{∘}) is weakly regular, for each s, there exists a positive C_{0} for all scalar functions f on S_{s} such that

∥E_{k}(ϕP_{j}^{2}f)∥_{L_{ω}^{2}} ≤ C_{0}2^{−|k−j|}∥P_{j}f∥_{L_{ω}^{2}}, ∥P_{k}(ϕE_{j}^{2}f)∥_{L_{ω}^{2}} ≤ C_{0}2^{−|k−j|}∥E_{j}f∥_{L_{ω}^{2}}, (4.16)

where C_{0} depends on sup_{|ω|=1} |∇_{γ^{∘}}ϕ_{s}|_{γ^{∘}} and ∥ϕ_{s}∥_{L_{ω}^{∞}}. Let ϕ(s, ω) = ϕ_{s}(ω), then it suffices to show, under the bootstrap assumptions (BA1), that sup_{s,|ω|=1} |∇_{γ^{∘}}ϕ_{s}|_{γ^{∘}} and ∥ϕ∥_{L_{t}^{∞}L_{ω}^{∞}} are bounded by a universal constant M.

According to the definition of ϕ, ∥ϕ∥_{L_{t}^{∞}L_{ω}^{∞}} is obviously bounded by some constant M. By (3.26), we derive the following transport equation,

∇_{L}∇ϕ = −χ · ∇ϕ.

By a straightforward computation, we derive the following

\frac{d}{ds}|∇ϕ|^{2} + \frac{2}{s}|∇ϕ|^{2} = −2χ̂ · ∇ϕ · ∇ϕ − V |∇ϕ|^{2}.  (4.17)

Note the fact that lim_{s→} s^{2}|∇ϕ|^{2} ≈ 1, we infer from (4.17) that

∥t∇ϕ∥_{L_{t}^{∞}L_{ω}^{∞}}^{2} ≲ 1 + ∥A∥_{L_{ω}^{∞}L_{t}^{2}}∥t∇ϕ∥_{L_{ω}^{∞}L_{t}^{2}}^{2}.

By (BA1), ∥A∥_{L_{ω}^{∞}L_{t}^{2}} ≲ Δ_{0}, thus |∇_{γ^{∘}}ϕ|_{γ^{∘}} ≲ 1 for any s > 0 and ω ∈ 𝕊^{2}. Therefore by (4.16), we conclude that there exists a positive universal constant C for all appropriate scalar functions f on H such that

C^{−1}∥f∥P̃^{α} ≲ ∥f∥_{P^{α}} ≲ C∥f∥P̃^{α},
C^{−1}∥f∥B̃^{α} ≲ ∥f∥_{B^{α}} ≲ C∥f∥B̃^{α},

where 0 ≤ α < 1.

4.3 Elliptic estimates on 2-D surfaces

In this section, we will establish the following inequalities.

Proposition 4.2. (Elliptic Estimates.) The following estimates hold on each leaf, 2surface S = S_{s} ⊂ H:

(i) The operator D_{1} is invertible on its range and its inverse D_{1}^{−1} takes pair of function f = (ρ, σ) in the range of D_{1} into S-tangent 1-forms F with div F = ρ, curl F = σ. The following estimate holds

∥∇D_{1}^{−1}f∥_{L^{2}(S)} + ∥s^{−1}D_{1}^{−1}f∥_{L^{2}(S)} ≲ ∥f∥_{L^{2}(S)}

(ii) The operator D_{2} is invertible on its range and its inverse D_{2}^{−1} takes S-tangent 1-forms F (in the range of D_{2}) into S-tangent symmetric, traceless, 2-tenorfields Z with div Z = F. The following inequality holds,

∥∇D_{2}^{−1}F ∥_{L^{2}(S)} + ∥s^{−1}D_{2}^{−1}F ∥_{L^{2}(S)} ≲ ∥F ∥_{L^{2}(S)}

(iii) The operator (−Δ) is invertible on its range and its inverse (−Δ)^{−1} verifies the

estimate

∥∇^{2}(−Δ)^{−1}f∥_{L^{2}(S)} + ∥s^{−1}∇(−Δ)^{−1}f∥_{L^{2}(S)} ≲ ∥f∥_{L^{2}(S)},

(iv) The operator ^{⋆}D_{1} is invertible as an operator defined for pairs of H^{1} functions with mean zero (i.e. the quotient of H^{1} by the kernel of ^{⋆}D_{1}) and its inverse ^{⋆}D_{1}^{−1} takes S-tangent L^{2} 1-forms F (i.e. the full range of ^{⋆}D_{1}) into pair of functions (ρ, σ) with mean zero, such that −∇ρ + (∇σ)^{⋆} = F, verifies the estimate

∥∇^{∗}D_{1}^{−1}F ∥_{L^{2}(S)} ≲ ∥F ∥_{L^{2}(S)}.

(v) The operator ^{⋆}D_{2} is invertible as an operator defined on the quotient of H^{1}-vector fields by the kernel of ^{⋆}D_{2}. Its inverse ^{⋆}D_{2}^{−1} takes S-tangent 2-forms Z which is in L^{2} space into S tangent 1-forms F (orthogonal to the kernel of D_{2}), such that ⋆D_{2}F = Z, verifies the estimate

∥∇ ·^{⋆} D_{2}^{−1}Z∥_{L^{2}(S)} ≲ ∥Z∥_{L^{2}(S)}.

As a consequence of (i)-(v), let D^{−1} be one of the operators D_{1}^{−1}, D_{2}^{−1}, ^{⋆}D_{1}^{−1} or ^{⋆}D_{2}^{−1}. By dual argument ,we have the following estimate for appropriate ^{4} tensor fields F,

∥D^{−1} div F ∥_{L^{2}(S)} ≲ ∥F ∥_{L^{2}(S)}.

4By “ appropriate” , we mean the tensor F such that div F is in the space where D^{−1} is well-defined.

We record the following basic properties of Λ^{a} for a ∈ ℝ in the following, Proposition 4.3. For any a, b ∈ ℝ and k ∈ ℕ there hold

Λ^{0} = Id, Λ^{a} · Λ^{b} = Λ^{a+b}, Λ^{2k} = (s^{−2}Id − Δ_{γ})^{k}

Moreover for any S-tangent tensor field F there hold for any a < 0

s^{a}∥Λ^{a}F ∥_{L^{2}(S)} ≲ ∥F ∥_{L^{2}(S)},

and for any b ≥ a ≥ 0

s^{a}∥Λ^{a}F ∥_{L^{2}(S)} ≲ s^{b}∥Λ^{b}F ∥_{L^{2}(S)},

∥Λ^{a}F ∥_{L^{2}(S)} ≲ ∥Λ^{b}F ∥L^{2}(S)^{\frac{a}{b}}∥F ∥_{L^{2}(S)}^{1− \frac{a}{b}}.

For the proof of the above proposition, refer to [15, Sec.7] and [26, Chap.1] . Proposition 4.4. For any tensors F and G on S there holds with 0 ≤ a < 1,

∥Λ^{a}(F · G)∥_{L^{2}(S)} ≤ ∥ΛF ∥_{L^{2}(S)}∥Λ^{a}G∥_{L^{2}(S)} + ∥Λ^{a}F ∥_{L^{2}(S)}∥ΛG∥_{L^{2}(S)}. We refer the reader to [15, Sec.9 ] for the proof. In order to prove Proposition 4.2, let us first assume the following result which will

be proved later. Proposition 4.5. The following estimate holds for all \frac{1}{2} < a < 1,

\underline{K}_{a} := ∥Λ^{−a}(K − s^{−2})∥_{L_{t}^{∞}L_{x}^{2}} ≲ Δ_{0}  (4.18)

where Λ^{−a} = (s^{−2}I − Δ)^{− \frac{a}{2}} . Proof of Proposition 4.2. In order to prove (i) and (iii), we need the following Böchner identity which holds true on any 2-surface S = S_{s} and for any scalar function f:

∫_{S} |∇^{2}f|^{2} = ∫_{S} |Δf|^{2} − ∫_{S} K|∇f|^{2},

it implies that

∫_{S} |∇^{2}f|^{2} + s^{−2} ∫_{S}|∇f|^{2} = ∫_{S} |Δf|^{2} − ∫_{S} (K − s^{−2}) |∇f|^{2}
= ∫_{S} |Δf|^{2} − ∫_{S} Λ^{−a} (K − s^{−2}) Λ^{a}|∇f|^{2}
≲ ∫_{S} |Δf|^{2} + ∥Λ^{−a} (K − s^{−2})∥_{L^{2}(S)} ∥Λ^{a}|∇f|^{2}∥_{L^{2}(S)}  (4.19)

To estimate the term ∥Λ^{a}|∇f|^{2}∥_{L^{2}(S)}, by using Proposition 4.4 we have

∥Λ^{a}|∇f|^{2}∥_{L^{2}(S)} ≲ ∥Λ^{a}∇f∥_{L^{2}(S)}∥Λ∇f∥_{L^{2}(S)} ≲ s^{1−a}∥Λ∇f∥_{L^{2}(S)}^{2} ≲ ∥Λ∇f∥_{L^{2}(S)}^{2}
≲ (∥∇^{2}f∥_{L^{2}(S)} + s^{−1}∥∇f∥_{L^{2}(S)})^{2}

as long as a ≤ 1. Then return to (4.19), we have

∥∇^{2}f∥_{L^{2}(S)}^{2} + ∥s^{−1}∇f∥_{L^{2}(S)}^{2} ≲ ∥Δf∥_{L^{2}(S)}^{2} + Δ_{0} (∥∇^{2}f∥_{L^{2}(S)} + ∥s^{−1}∇f∥_{L^{2}(S)})^{2}

which immediately implies ∥∇^{2}f∥_{L^{2}(S)} + ∥s^{−1}∇f∥_{L^{2}(S)} ≲ ∥Δf∥_{L^{2}(S)}. Thus we get (iii). Now we prove (i). Recall Proposition 2.5 (i), we have

∫_{S} (|∇F |^{2} + s^{−2}|F |^{2}) =2 ∫_{S} (| div F |^{2} + |curlF |^{2}) − ∫_{S} (K − s^{−2}) |F |^{2}
=2 ∫_{S} (| div F |^{2} + |curlF |^{2}) − ∫_{S} Λ^{−a} (K − s^{−2}) Λ^{a}(|F |^{2})
≲∥D_{1}F ∥_{L^{2}(S)}^{2} + ∥Λ^{−a} (K − s^{−2}) ∥_{L^{2}(S)} · ∥Λ^{a}(|F |^{2})∥_{L^{2}(S)}.

It follows from Proposition 4.5 that

∫_{S} (|∇F |^{2} + s^{−2}|F |^{2}) ≲∥D_{1}F ∥_{L^{2}(S)}^{2} + Δ_{0} · s^{1−a}∥ΛF ∥_{L^{2}(S)}^{2}
≲∥D_{1}F ∥_{L^{2}(S)}^{2} + Δ_{0} (∥∇F ∥_{L^{2}(S)}^{2} + ∥s^{−1}F ∥_{L^{2}(S)}^{2}) .

Since Δ_{0} < 1/2 is sufficiently small, we conclude ∥∇F ∥_{L^{2}(S)}^{2}+∥s^{−1}F ∥_{L^{2}(S)}^{2} ≲ ∥D_{1}F ∥_{L^{2}(S)}. Proposition 4.2 (ii) can be proved in the same way by using Proposition 2.5 (ii). (iii) follows immediately from Proposition 2.5 (iii). (iv) can be derived by using (4.18), Proposition 2.5 (iv) and a contradiction argument.

Recall that the Gaussian curvature K of S_{s} is given by the formula ( see [2]).

K = −\frac{1}{4}trχtr\underline{χ} + \frac{1}{2}χ̂ · \underline{χ̂} − ρ.

With the help of the definitions trχ = \frac{2}{s} + V and tr\underline{χ} = −\frac{2}{s} + \underline{V} we write it as

K − \frac{1}{s^{2}} = −\frac{1}{2s}\underline{V} + \frac{1}{2s}V − \frac{1}{4}V · \underline{V} − ρ̌.

Since the term \frac{1}{2s}V is better than \frac{1}{2s} \underline{V}, we may neglect it and also neglect signs and constants, then rewrite the above identity into the following form

K − \frac{1}{s^{2}} = \frac{1}{s}\underline{V} + V · \underline{V} + ρ̌.  (4.20)

Thus, according to (BA1) and (BA2),

∥K − s^{−2}∥_{L^{2}(H)} ≲ ∥s^{−1}\underline{V} ∥_{L^{2}(H)} + ∥V · \underline{V} ∥_{L^{2}(H)} + ∥ρ̌∥_{L^{2}(H)}
≲ Δ_{0} + Δ_{0}^{2} + R_{0}.

This proves the following

Proposition 4.6. The Gauss curvature K of (S_{s}, γ) verifies the following estimate

∥K − s^{−2}∥_{L^{2}(H)} ≲ Δ_{0}.

It remains to show Proposition 4.5.

Proof of Proposition 4.5. First for a ≥ 1/2, we have by L^{2} boundedness property of Λ^{−a}

∥Λ^{−a}(s^{−1}\underline{V})∥_{L^{2}(S)} ≲ s^{−1+a}∥\underline{V} ∥_{L^{2}(S)} ≲ Δ_{0}.  (4.21)

The last step follows from (BA2). Next we show

∥Λ^{−a}(V · \underline{V})∥_{L^{2}(S)} ≲ Δ_{0}^{2}.  (4.22)

Set

J_{a}^{2}(s) = ∫_{S_{s}} Λ^{−2a}(V · \underline{V}) · (V · \underline{V}),

we deduce by Hölder’s inequality and (BA2)

J_{a}^{2}(s) ≲ s^{\frac{1}{2}} ∥Λ^{−2a}(V · \underline{V}) · V ∥_{L^{2}(S_{s})}∥s^{− \frac{1}{2}} \underline{V} ∥_{L^{2}(S_{s})}
≲ Δ_{0}s^{\frac{1}{2}} ∥Λ^{−2a}(V · \underline{V})∥_{L^{2}(S_{s})}∥V ∥_{L^{∞}(S_{s})}.

According to (BA1), ∥V ∥_{L^{∞}(H)} ≲ Δ_{0}, we infer from L^{2} boundness property of Λ^{−a} that for any 1 ≥ s > 0,

J_{a}^{2}(s) ≲ Δ_{0}^{2}s^{\frac{1}{2} +a}∥Λ^{−a}(V · \underline{V})∥_{L^{2}(S_{s})} = Δ_{0}^{2}J_{a},

whence J_{a}(s) ≲ Δ_{0}.

So far we have the following estimate for 1 ≥ s > 0,

∥Λ^{−a} (K − s^{−2} − ρ̌)∥_{L^{2}(S_{s})} ≲ Δ_{0},  (4.23)

it remains to estimate the curvature term ∥Λ^{−a}ρ̌∥_{L_{t}^{∞}L_{x}^{2}} which is a bit involved.

We first set W (s) = −∥Λ^{−a}ρ̌∥_{L^{2}(S_{s})}^{2} + ∥Λ^{−a}ρ̌∥_{L^{2}(S_{1})}^{2}, then

W (s) = − ∫_{S_{s}} Λ^{−a}ρ̌ · Λ^{−a}ρ̌dA_{s} + ∫_{S_{1}} Λ^{−a}ρ̌ · Λ^{−a}ρ̌dA_{1}

≈ − ∫_{|ω|=1} Λ^{−a}ρ̌ · Λ^{−a}ρ̌s^{2}dsdω + ∫_{|ω|=1} Λ^{−a}ρ̌ · Λ^{−a}ρ̌(1, ω)dω

= 2 ∫_{|ω|=1} ∫_{s}^{1} ∇_{L}(Λ^{−a}ρ̌) · Λ^{−a}ρ̌ · s^{2}dsdω + 2 ∫_{|ω|=1} ∫_{s}^{1} sΛ^{−a}ρ̌ · Λ^{−a}ρ̌dsdω

= 2 ∫_{H} ∇_{L}(Λ^{−a}ρ̌) · Λ^{−a}ρ̌ + 2 ∫_{H} s^{−1}Λ^{−a}ρ̌ · Λ^{−a}ρ̌

= 2 ∫_{H} Λ^{−2a}(∇_{L}ρ̌) · ρ̌ + 2 ∫_{H}[∇_{L}, Λ^{−a}]_{g}ρ̌ · Λ^{−a}ρ̌ + 2(1 − a) ∫_{H} s^{−1}Λ^{−a}ρ̌ · Λ^{−a}ρ̌

≲ ∥Λ^{−2a}∇_{L}ρ̌∥_{L_{t}^{2}L_{x}^{2}} ∥ρ̌∥_{L_{t}^{2}L_{x}^{2}} + ∥[∇_{L}, Λ^{−a}]_{g}ρ̌∥_{L_{t}^{1}L_{x}^{2}}∥Λ^{−a}ρ̌∥_{L_{t}^{∞}L_{x}^{2}} + ∥s^{−1/2}Λ^{−a}ρ̌∥_{L^{2}(H)}^{2} (4.24)

where [Λ^{−a}, ∇_{L}]_{g}f := s^{a}C_{a} ∫_{0}^{∞} τ^{a/2−1}e^{−τ}[U(τ), ∇_{L}]fdτ, with C_{a} = \frac{1}{Γ(a/2)}. Since according to Proposition 4.3 with a > 1/2,

∥s^{− \frac{1}{2}} Λ^{−a}ρ̌∥_{L_{t}^{2}L_{x}^{2}} ≲ ∥ρ̌∥_{L_{t}^{2}L_{x}^{2}} ≲ R_{0},

we then conclude the third term is bounded by R_{0}^{2}. ∥Λ^{−a}ρ̌∥_{L^{2}(S_{1})} can be bounded by (4.24) which is obtained by using a cut-off function θ(t) with θ(1) = 1 and supported in [1/2, 1], then proceeding in the same manner as we estimated ∥F (1)∥_{L_{ω}^{2}} in the proof of (3.39). Thus, from (4.24)

∥Λ^{−a}ρ̌∥_{L_{t}^{∞}L_{x}^{2}}^{2} ≲ ∥Λ^{−2a}(∇_{L}ρ̌)∥_{L_{t}^{2}L_{x}^{2}}∥ρ̌∥_{L_{t}^{2}L_{x}^{2}} + ∥[∇_{L}, Λ^{−a}]ρ̌∥_{L_{t}^{1}L_{x}^{2}} · ∥Λ^{−a}ρ̌∥_{L_{t}^{∞}L_{x}^{2}} + Δ_{0}^{2} (4.25)

In order to verify that (4.25) is bounded by Δ_{0}^{2}, we need to establish the following estimates

∥[∇_{L}, Λ^{−a}]ρ̌∥_{L_{t}^{1}L_{x}^{2}} ≲ Δ_{0}  (4.26)

and

∥Λ^{−2a}∇_{L}ρ̌∥_{L_{t}^{2}L_{x}^{2}} ≲ Δ_{0}.  (4.27)

Proof of (4.27). From the transport equation (3.21) i.e.

L(ρ̌) + \frac{3}{2}trχ · ρ̌ = div β − ζ · β + \frac{1}{2}χ̂ · (∇ ˆ⊗ζ + \frac{1}{2}\underline{V} · χ̂ − \frac{2}{s}χ̂ − ζ ˆ⊗ζ) ,

we infer

∥Λ^{−2a}∇_{L}ρ̌∥_{L_{t}^{2}L_{x}^{2}} ≲ ∥Λ^{−2a} div β∥_{L_{t}^{2}L_{x}^{2}} + ∥Λ^{−2a}(A · R̄)∥_{L_{t}^{2}L_{x}^{2}} + ∥t^{−1}Λ^{−2a}ρ̌∥_{L_{t}^{2}L_{x}^{2}} (4.28)

where R̄ = R + ∇A + \frac{1}{s}A + A · \underline{A} + A · A. Combining Hölder inequality, (4.8), (BA1) and (BA2), we can easily check

∥R̄∥_{L^{2}(H)} ≲ R_{0} + Δ_{0}.  (4.29)

According to Proposition 4.3, ∥s^{−1}Λ^{−2a}ρ̌∥_{L^{2}(H)} ≲ ∥s^{2a−1}ρ̌∥_{L^{2}(H)} ≲ ∥ρ̌∥_{L^{2}(H)} ≲ R_{0}, since a ≥ 1/2. It suffices to estimate the other two terms on the right hand side of (4.28). We first show

∥Λ^{−2a} div β∥_{L_{t}^{2}L_{x}^{2}} ≲ ∥β∥_{L_{t}^{2}L_{x}^{2}},  (4.30)

by proving the following lemma

Lemma 4.3. For a 1-forms F and a ≥ \frac{1}{2}, we have

∥Λ^{−2a} div F ∥_{L_{t}^{2}L_{x}^{2}} ≲ ∥F ∥_{L_{t}^{2}L_{x}^{2}}  (4.31)

Proof. Note that for scalar functions f and a > 1/2,

∥∇Λ^{−2a}f∥_{L^{2}(S)}^{2} = ∫_{S} ΔΛ^{−2a}f · Λ^{−2a}f = ∫_{S} ΔΛ^{−4a}f · f
≲ ∥(Δ − s^{−2}Id)Λ^{−4a}f∥_{L^{2}(S)}∥f∥_{L^{2}(S)} + ∥s^{−1}Λ^{−2a}f∥_{L_{x}^{2}}^{2} ≲ ∥f∥_{L^{2}(S)}^{2}

gives the desired result.

It remains to estimate the term ∥Λ^{−2a}(A · R̄)∥_{L_{t}^{2}L_{x}^{2}} in (4.28). Note

∥Λ^{−2a}(A · R̄)∥_{L_{t}^{2}L_{x}^{2}}^{2} = ∫ Λ^{−4a}(A · R̄) · A · R̄
≲ ∥Λ^{−4a}(A · R̄)∥_{L_{t}^{2}L_{x}^{4}} ∥A∥_{L_{t}^{∞}L_{x}^{4}}∥R̄∥_{L_{t}^{2}L_{x}^{2}}

Combine (3.39) and (BA1), we infer

∥A∥_{L_{t}^{∞}L_{x}^{4}} ≲ N_{1}(A) ≲ Δ_{0}.

Since ∥R̄∥_{L_{t}^{2}L_{x}^{2}} ≲ Δ_{0} + R_{0}, by Sobolev embedding (3.6), we derive

∥Λ^{−2a}(A · R̄)∥_{L_{t}^{2}L_{x}^{2}}^{2} ≲ (Δ_{0}^{2} + R_{0})∥Λ^{−4a+ \frac{1}{2} +}(A · R̄)∥_{L_{t}^{2}L_{x}^{2}}
≲ (Δ_{0}^{2} + R_{0})∥s^{2a− \frac{1}{2} −}Λ^{−2a}(A · R̄)∥_{L_{t}^{2}L_{x}^{2}}.

As a consequence, ∥Λ^{−2a}(A · R̄)∥_{L_{t}^{2}L_{x}^{2}} ≲ Δ_{0}^{2} + R_{0}. Return to (4.28), we conclude

∥Λ^{−2a}∇_{L}ρ̌∥_{L_{t}^{2}L_{x}^{2}} ≲ Δ_{0}^{2} + R_{0}.

Thus we justified (4.27).

Proof of (4.26). To prove (4.26), We will employ the identity (3.28), i.e. for arbitrary scalars f on S_{s}

LΔf − ΔLf = −2χ · ∇^{2}f + (2χ̂ · ζ + ∇trχ + trχ · ζ)∇f
= −\frac{2}{s}Δf + ∇ (A · ∇f) + (∇A + A · A + s^{−1}A) · ∇f.  (4.32)

By Duhamel’s Principle, we have

s^{−a}[Λ^{−a}, ∇_{L}]_{g}f = C_{a} ∫_{0}^{∞} τ^{\frac{a}{2} −1}e^{−τ}[U(τ), ∇_{L}]fdτ
= C_{a} ∫_{0}^{∞} dττ^{\frac{a}{2} −1}e^{−τ} ∫_{0}^{τ} s^{2}U(τ − τ′)([Δ, ∇_{L}]U(τ′)f
−\frac{2}{s}ΔU(τ′)f)dτ′.  (4.33)

Set

ϕ_{1} = A · ∇U(τ′)f and ϕ_{2} = (∇A + A · A + s^{−1}A) · ∇U(τ′)f.

Then

s^{−a}[Λ^{−a}, ∇_{L}]_{g}f = Φ_{1} + Φ_{2}  (4.34)

where

Φ_{1} = C_{a} ∫_{0}^{∞} s^{2}τ^{\frac{a}{2} −1}e^{−τ}dτ ∫_{0}^{τ} U(τ − τ′)∇ϕ_{1}(τ′)dτ′,
Φ_{2} = C_{a} ∫_{0}^{∞} s^{2}τ^{\frac{a}{2} −1}e^{−τ}dτ ∫_{0}^{τ} U(τ − τ′)ϕ_{2}(τ′)dτ′

and C_{a} is some universal constant. Then we infer

∥Φ_{1}∥_{L_{t}^{1}L_{x}^{2}} ≲ ∫_{0}^{∞} τ^{\frac{a}{2} −1}e^{−τ}dτ ∫_{0}^{τ} ∥s^{2}U(τ − τ′)∇ϕ_{1}(τ′)∥_{L_{t}^{1}L_{x}^{2}}dτ′
≲ ∫_{0}^{∞} τ^{\frac{a}{2} −1}e^{−τ}dτ ∫_{0}^{τ} |τ − τ′|^{− \frac{1}{2}} ∥sϕ_{1}(τ′)∥_{L_{t}^{1}L_{x}^{2}}dτ′.  (4.35)

Therefore

∥sϕ_{1}∥_{L_{t}^{1}L_{x}^{2}} ≲ ∥s^{−1+ \frac{2}{p}} A∥_{L_{t}^{2}L \frac{2p}{p−2} ; x} ∥s^{2− \frac{2}{p}} ∇U(τ′)f∥_{L_{t}^{2}L_{x}^{p}}.  (4.36)

By Sobolev inequality (3.37), we first infer for p > 2 that

∥s^{−1+ \frac{2}{p}} A∥_{L \frac{2p}{p−2} ; x} ≲ ∥∇A∥L_{x}^{2}^{\frac{2}{p}}∥s^{−1}A∥_{L_{x}^{2}}^{1− \frac{2}{p}} + ∥s^{−1}A∥_{L_{x}^{2}},

Then by (BA1), ∥s^{−1+ \frac{2}{p}} A∥_{L_{t}^{2}L \frac{2p}{p−2} ; x} ≲ Δ_{0}.

On the other hand, the Böchner identity for scalar functions on S_{s} implies:

∥∇^{2}U(τ′)f∥_{L^{2}(S_{s})} ≲ ∥ΔU(τ′)f∥_{L^{2}(S_{s})} + I_{a} · ∥s^{−1}∇U(τ′)f∥_{L^{2}(S_{s})}  (4.37)

where I_{a} = 1 + \underline{K}a^{\frac{1}{1−a}} + \underline{K}a^{\frac{1}{2}} , with \underline{K}_{a} = ∥Λ^{−a}(K − s^{−2})∥_{L^{2}(S_{s})}. From (3.37), we infer

∥s^{2− \frac{2}{p}} ∇U(τ)f∥_{L^{p}(S_{s})} ≲ ∥s^{2}∇^{2}U(τ)f∥_{L^{2}(S_{s})}^{1− \frac{2}{p}}·∥s∇U(τ)f∥L^{2}(S_{s})^{\frac{2}{p}}+∥s∇U(τ)f∥_{L^{2}(S_{s})}. (4.38)

We use (4.37) to get ∥s^{2− \frac{2}{p}} ∇U(τ′)f∥_{L^{p}(S_{s})}

≲ (∥s^{2}ΔU(τ′)f∥_{L^{2}(S_{s})} + I_{a} · ∥s∇U(τ′)f∥_{L^{2}(S_{s})})^{1− \frac{2}{p}} ∥s∇U(τ′)f∥L^{2}(S_{s})^{\frac{2}{p}} + ∥s∇U(τ′)f∥_{L^{2}(S_{s})}
≲ (\frac{1}{τ′} ∥f∥_{L^{2}(S_{s})} + I_{a} · ∥s∇U(τ′)f∥_{L^{2}(S_{s})})^{1− \frac{2}{p}} ∥s∇U(τ′)f∥L^{2}(S_{s})^{\frac{2}{p}} + ∥s∇U(τ′)f∥_{L^{2}(S_{s})}
≲ (τ′^{−1+ \frac{2}{p}} + (I_{a}^{−2} · τ′)^{− \frac{1}{2} + \frac{1}{p}} ) τ′^{− \frac{1}{p}} ∥f∥_{L^{2}(S_{s})} + τ′^{− \frac{1}{2}} ∥f∥_{L^{2}(S_{s})}.  (4.39)

Taking L_{t}^{2} norm yields

∥t^{2− \frac{2}{p}} ∇U(τ′)f∥_{L_{t}^{2}L_{x}^{p}} ≲∥f∥_{L_{t}^{2}L_{x}^{2}} (τ′^{−1+ \frac{2}{p}} + (I_{a}^{−2} · τ′)^{− \frac{1}{2} + \frac{1}{p}} ) τ′^{− \frac{1}{p}} + τ′^{− \frac{1}{2}} ∥f∥_{L_{t}^{2}L_{x}^{2}} (4.40)

Thus

∥tϕ_{1}∥_{L_{t}^{2}L_{x}^{2}} ≲ (Δ_{0}τ′^{−1+ \frac{1}{p}} + τ′^{− \frac{1}{2}} Δ_{0}I_{a}^{1− \frac{2}{p}} ) ∥f∥_{L_{t}^{2}L_{x}^{2}}.

We conclude

∥Φ_{1}∥_{L_{t}^{1}L_{x}^{2}} ≲ ∥f∥_{L^{2}(H)}(Δ_{0} + Δ_{0}I_{a}^{1− \frac{2}{P}}).  (4.41)

As to Φ_{2}, we estimate as follows

∥Φ_{2}∥_{L_{t}^{1}L_{x}^{2}} ≲ ∫_{0}^{∞} s^{2}τ^{\frac{a}{2} −1}e^{−τ}dτ ∫_{0}^{τ} ∥U(τ − τ′)ϕ_{2}(τ′)∥_{L_{t}^{1}L_{x}^{2}}dτ′
≲ ∫_{0}^{∞} τ^{\frac{a}{2} −1}e^{−τ}dτ ∫_{0}^{τ}(1 + |τ − τ′|)^{− \frac{1}{r} + \frac{1}{2}} ∥s^{2− \frac{2}{p}} ϕ_{2}(τ′)∥_{L_{t}^{1}L_{x}^{r}}dτ′

where p = \frac{2r}{2−r} > 2, and the term about ϕ_{2} can be estimated by using Hölder inequality and (4.39)

∥s^{2− \frac{2}{p}} ϕ_{2}(τ′)∥_{L_{t}^{1}L_{x}^{r}} ≲ ∥∇A + A · A + s^{−1}A∥_{L_{t}^{2}L_{x}^{2}}∥s^{2− \frac{2}{p}} ∇U(τ′)f∥_{L_{t}^{2}L_{x}^{p}}
≲ Δ_{0}∥f∥_{L_{t}^{2}L_{x}^{2}}((τ′^{−1+ \frac{2}{p}} + (I_{a}^{−2} · τ′)^{− \frac{1}{2} + \frac{1}{p}})τ′^{− \frac{1}{p}} + τ^{′− \frac{1}{2}} ).

Therefore we get

∥Φ_{2}∥_{L_{t}^{1}L_{x}^{2}} ≲ ∥f∥_{L_{t}^{2}L_{x}^{2}}(Δ_{0} + Δ_{0}I_{a}^{1− \frac{2}{p}}).  (4.42)

Recall f = ρ̌, according to (4.9), ∥f∥_{L^{2}(H)} ≲ Δ_{0}^{2} + R_{0}. Hence, combine (4.34), (4.41) with (4.42), we conclude

∥s^{−a}[∇_{L}, Λ^{−a}]_{g}ρ̌∥_{L_{t}^{1}L_{x}^{2}} ≲ R_{0}Δ_{0}I_{a}^{1− \frac{2}{p}} + R_{0}Δ_{0}.  (4.43)

Then combine (4.23), (4.24), (4.27) with (4.43), we obtain

∥Λ^{−a} (K − s^{−2})∥_{L^{2}(S_{s})} ≲ Δ_{0} + R_{0}Δ_{0}I_{a}^{1− \frac{2}{p}} .  (4.44)

Due to the definition of \underline{K}_{a} and I_{a}, we let \frac{1}{1−a}(1 − \frac{2}{p}) < 1, i.e. a < \frac{2}{p}. Then (4.44) implies (4.26).

Therefore we complete the proof of Proposition 4.5.

4.4 Commutation relation between ∇_{L} and Hodge operators D^{−1}

We consider now commutators between L and D_{1}^{−1}, D_{2}^{−1}, ^{⋆}D_{1}^{−1}. Recall that we denote by R(D) the L^{2}(S) range of D and denote by ker(D) its L^{2}(S) kernel. The operator P is defined to be the projection operator from L^{2}(S) to R(D). For (ρ, σ) ∈ R(D_{1}), we consider first

[L, D_{1}^{−1}](ρ, σ) := ∇_{L}D_{1}^{−1}(ρ, σ) − D_{1}^{−1}(Lρ, Lσ).

Let F̂ = D_{1}^{−1}(ρ, σ) and F = D_{1}^{−1}(Lρ, Lσ), then

D_{1}^{−1}(L( div F̂, curl F̂)) = D_{1}^{−1}(Lρ, Lσ)
D_{1}^{−1}( div F, curl F) = D_{1}^{−1}(Lρ, Lσ).

Therefore

D_{1}^{−1}( div (∇_{L}F̂ − F), curl (∇_{L}F̂ − F)) = D_{1}^{−1}([ div , L]F̂, [ curl , L]F̂).

Hence we obtain

∇_{L}F̂ − F = D_{1}^{−1}([ div , L]F̂, [ curl , L]F̂) = D_{1}^{−1}([D_{1}, L]F̂) = D_{1}^{−1}[D_{1}, L]D_{1}^{−1}(ρ, σ),

whence [L, D_{1}^{−1}] = D_{1}^{−1}[D_{1}, L]D_{1}^{−1}. Proceeding in precisely the same manner for the other Hodge operators, we have the following

Proposition 4.7. Let D^{−1} be any of the operators D_{1}, D_{2}^{−1}, ^{⋆}D_{1}^{−1}. Then,

[L, D^{−1}] = D^{−1}[D, L]D^{−1}.

4.5 Estimates of second derivatives

In addition to Proposition 4.2, we shall also estimate ∥∇^{2}D^{−1}F ∥_{L^{2}(H)}. We start with considering D = D_{1}. Let F be tensor fields verifying D_{1}F = (ρ, σ). In view of the formula (2.25), the Hodge operators D_{1} and ^{⋆}D_{1} satisfies the identities,

(−Δ + K)F = ^{⋆}D_{1}(D_{1}F) = ^{⋆}D_{1}(ρ, σ).

Recall the Böchner identity for 1−forms F on S = S_{s},

∫_{S} |∇^{2}F |^{2} = ∫_{s} |ΔF |^{2} − ∫_{S} K(2|∇F |^{2} − | div F |^{2}) + ∫_{S} K^{2}|F |^{2}.  (4.45)

Using Proposition 4.5 and Proposition 4.4, with a > 1/2, we obtain that

| ∫_{S}(K − \frac{1}{s^{2}} )(2|∇F |^{2} − | div F |^{2})| ≲ Δ_{0} · ∥Λ^{a}(2|∇F |^{2} − | div F |^{2})∥_{L^{2}(S)}
≲ Δ_{0}(∥∇^{2}F ∥_{L^{2}(S)}^{2} + ∥s^{−1}∇F ∥_{L^{2}(S)}^{2}).

Then we derive from (4.45) that

∥∇^{2}F ∥_{L^{2}(S)}^{2} + ∥s^{−1}∇F ∥_{L^{2}(S)}^{2} ≲ ∥^{⋆}D_{1}(ρ, σ)∥_{L^{2}(S)}^{2} + Δ_{0}(∥∇^{2}F ∥_{L^{2}(S)}^{2} + ∥s^{−1}∇F ∥_{L^{2}(S)}^{2})

+ ∥F ∥_{L^{∞}(S)}^{2}∥K − s^{−2}∥_{L^{2}(S)}^{2} + ∥s^{−2}F ∥_{L^{2}(S)}^{2} + ∥s^{−1}∇F ∥_{L^{2}(S)}^{2}.

which combined with (3.39) implies

∥∇^{2}F ∥_{L^{2}(H)}^{2} + ∥s^{−1}∇F ∥_{L^{2}(H)}^{2} ≲ ∥^{⋆}D_{1}(ρ, σ)∥_{L^{2}(H)}^{2} + Δ_{0}^{2}(∥s^{−1}∇_{L}F ∥_{L^{2}(H)}^{2}
+ ∥∇∇_{L}F ∥_{L^{2}(H)}^{2}) + ∥s^{−1}F ∥_{H_{x}^{1}}^{2}.

Since D_{1}F = (ρ, σ), we conclude

∥∇^{2}D_{1}^{−1}(ρ, σ)∥_{L^{2}(H)} ≲ ∥^{⋆}D_{1}(ρ, σ)∥_{L^{2}(H)} + Δ_{0}(∥s^{−1}∇_{L}F ∥_{L^{2}(H)}
+ ∥∇∇_{L}F ∥_{L^{2}(H)}) + ∥s^{−2}F ∥_{L^{2}(H)} + ∥s^{−1}∇F ∥_{L^{2}(H)}

as desired.

Proceeding in the same manner, we can derive the similar estimates for D_{2}^{−1} and ⋆D_{1}^{−1} as follows.

Proposition 4.8. Let D denote either D_{1} or D_{2} with ^{⋆}D and D^{−1} the corresponding adjoints and inverses. The following estimate holds true,

∥∇^{2}D^{−1}F ∥_{L^{2}(H)} ≲ ∥^{⋆}DF ∥_{L^{2}(H)} + Δ_{0} (∥s^{−1}∇_{L}D^{−1}F ∥_{L^{2}(H)} + ∥∇∇_{L}D^{−1}F ∥_{L^{2}(H)})
+ ∥s^{−2}F ∥_{L^{2}(H)} + ∥s^{−1}∇F ∥_{L^{2}(H)}

where for F = (ρ, σ) being pairs of scalar functions, D is understood to be D_{1}; for F being S-tangent one forms, D is D_{2}.

We will use Proposition 4.8 to prove

Lemma 4.4. For appropriate tensor fields P, the following inequalities hold true

N_{2}(D^{−1}P) ≲ N_{1}(P),  (4.46)
N_{1}(∇D^{−1}P) ≲ N_{1}(P).  (4.47)

Proof of Lemma 4.4. By the estimate of ∥∇^{2}D^{−1}P ∥_{L^{2}(H)} in Proposition 4.8 and the definition of N_{2}(·), we have

N_{2}(D^{−1}P) = ∥s^{−2}D^{−1}P ∥_{L^{2}(H)} + ∥s^{−1}∇ · D^{−1}P ∥_{L^{2}(H)} + ∥∇^{2}D^{−1}P ∥_{L^{2}(H)}
+ ∥s^{−1}∇_{L}D^{−1}P ∥_{L^{2}(H)} + ∥∇∇_{L}D^{−1}P ∥_{L^{2}(H)}
≲ ∥s^{−1}P ∥_{L^{2}(H)} + ∥∇P ∥_{L^{2}(H)} + ∥s^{−1}∇_{L}D^{−1}P ∥_{L^{2}(H)}
+ ∥∇[∇_{L}, D^{−1}]P ∥_{L^{2}(H)} + ∥∇_{L}P ∥_{L^{2}(H)}
≲ N_{1}(P) + ∥∇[∇_{L}, D^{−1}]P ∥_{L^{2}(H)} + ∥s^{−1}[∇_{L}, D^{−1}]P ∥_{L^{2}(H)}.

Then we only need to establish the following estimates

∥∇[∇_{L}, D^{−1}]P ∥_{L^{2}(H)} ≲ N_{1}(P)  (4.48)
∥s^{−1}[∇_{L}, D^{−1}]P ∥_{L^{2}(H)} ≲ N_{1}(P).  (4.49)

In view of (3.25), we have the symbolic commutation formula:

∇[∇_{L}, D^{−1}]P = ∇((A + s^{−1}) · ∇D^{−1}P + (A + s^{−1}) · A · D^{−1}P ) + ∇ · D^{−1}(β · D^{−1}P),

Then by applying Proposition 4.2 and Hölder’s inequality followed by Sobolev inequality to the above equation, (4.48) can be obtained as follows

∥∇[∇_{L}, D^{−1}]P ∥_{L^{2}(H)}

≲ ∥P ∥_{L^{2}(H)} + (∥β∥_{L^{2}(H)} + ∥A∥_{L_{t}^{∞}L_{x}^{4}} + ∥A · (\frac{1}{s} + A)∥_{L^{2}(H)})N_{2}(D^{−1}P)

≲ ∥P ∥_{L^{2}(H)} + Δ_{0}N_{2}(D^{−1}P).

Similarly we can get (4.49). Hence

N_{2}(D^{−1}P) ≲ N_{1}(P) + Δ_{0} · N_{2}(D^{−1}P) with 0 < R_{0} ≤ Δ_{0} < 1/2.

Then N_{2}(D^{−1}P) ≲ N_{1}(P). Therefore (4.46) is proved.

We still need to show (4.47) to complete the proof of Lemma 4.4. In view of the definition of N_{1}(·), we have by (4.46)

N_{1}(∇ · D^{−1}P) ≲ ∥∇^{2}D^{−1}P ∥_{L^{2}(H)} + ∥s^{−1}∇ · D^{−1}P ∥_{L^{2}(H)} + ∥∇_{L}∇ · D^{−1}P ∥_{L^{2}(H)}
≲ N_{1}(P) + ∥∇_{L}∇ · D^{−1}P ∥_{L^{2}(H)}.

It remains to show

∥∇_{L}∇ · D^{−1}P ∥_{L^{2}(H)} ≲ N_{1}(P).

Indeed, commuting ∇_{L} with ∇ and using (4.46), (3.39), Hölder’s inequality and Sobolev inequality, we infer

∥∇_{L}∇ · D^{−1}P ∥_{L^{2}(H)} ≤ ∥∇ · ∇_{L}D^{−1}P ∥_{L^{2}(H)} + ∥[∇_{L}, ∇] · D^{−1}P ∥_{L^{2}(H)} ≲ ∥∇ · ∇_{L}D^{−1}P ∥_{L^{2}(H)} + ∥s^{−1}∇D^{−1}P ∥_{L_{t}^{2}L_{x}^{2}} + ∥A · ∇D^{−1}P ∥_{L_{t}^{2}L_{x}^{2}}

+ ∥((A + s^{−1}) · A + β) · D^{−1}P ∥_{L^{2}(H)}

≲ (1 + ∥β∥_{L_{t}^{2}L_{x}^{2}} + ∥(A + s^{−1}) · A∥_{L_{t}^{2}L_{x}^{2}})N_{2}(D^{−1}P) + ∥A∥_{L_{t}^{∞}L_{x}^{4}}∥∇ · D^{−1}P ∥_{L_{t}^{2}L_{x}^{4}}

≲ (1 + Δ_{0})N_{2}(D^{−1}P) ≲ N_{1}(P)

Thus we get (4.47) and finish the proof of Lemma 4.4.

Transport-Product Estimates in Besov Norms

In this chapter, we will provide a series of estimates in Besov norms which are of fundamental importance for the later applications.

We first record the following embedding estimates which can be easily proved by using Sharp Bernstein inequality.

Proposition 5.1. For scalar functions f on S = S_{s} the following sharp Sobolev inequality holds,

∥f∥_{L^{∞}(S)} ≲ ∥f∥_{B_{2,1}^{1}(S)}

and

∥f∥_{B_{2,1}^{1}(S)} ≲ ∥s^{−1}f∥_{L^{2}(S)} + ∥∇f∥_{B^{0}}.

We also can easily check the following estimate,

Proposition 5.2. Let f be a scalar and U an arbitrary tensor on a fixed S = S_{s}. The following bilinear estimate hold true,

∥f · U∥_{B_{2,1}^{0}(S)} ≲ (∥∇f∥_{L^{2}(S)} + ∥f∥_{L^{∞}(S)})∥U∥_{B_{2,1}^{0}(S)}.

5.1 Product Estimates

The main purpose of this section is to provide the transport-product estimates. Denote by S the 2-D leaves S_{s}, in the following three theorems we will always assume the bootstrap assumptions (BA1), (BA2) and the weakly spherical condition.

Theorem 5.1. Assume that the S-tangent tensors W, F and G satisfy the transport equation ∇_{L}W + \frac{1}{s}W = ∇_{L}F · G along H and lim_{s→0} s|W | = 0. Then

∥W ∥_{B^{0}} ≲ N_{1}(F) · (N_{1}(G) + ∥G∥_{L_{ω}^{∞}L_{t}^{2}})

Theorem 5.2. Assume that the S-tangent tensors W, F and G satisfy the transport equation ∇_{L}W + \frac{1}{s}W = F · G and lim_{s→0} s|W | = 0 along H. There holds the estimate

∥W ∥_{B^{0}} ≲ ∥F ∥_{P^{0}}(N_{1}(G) + ∥G∥_{L_{ω}^{∞}L_{t}^{2}}).

Theorem 5.3. For any pair of S-tangent tensors G and W such that W satisfies a transport equation of the form ∇_{L}W + \frac{1}{s}W = F along H , there holds

∥G · W ∥_{P^{0}} ≲ (lim_{s→0} ∥W ∥_{B_{2,1}^{0}(S)} + ∥F ∥_{P^{0}})(N_{1}(G) + ∥G∥_{L_{ω}^{∞}L_{t}^{2}})

Remark 5.1. When the S-tangent tensors F, G and W satisfy the equations of the type ∇_{L}W + \frac{k}{s}W = F · G, or ∇_{L}F · G, or F, with k = 1, 2, ⋯, the same estimates in Theorems 5.1–5.3 still hold, and can be obtained in the same way as for the case k = 1.

Before proving the above results, let us modify [14, Lemma 4.13] as follows

Proposition 5.3. (Reduction to scalar estimates) Assume that the conclusions of Theorems 5.1–5.3 have been verified for the scalar transport equations, i.e. W is a scalar function. Then they also hold true in the tensor case. Moreover, in the case of Theorem 5.2 and Theorem 5.3, we can reduce the corresponding estimates to a fully scalar case, i.e. F, G and W are scalar functions.

Indeed, without loss of generality, we can assume W to be a 1-form and multiply the transport equation for W in Theorem 5.1 and Theorem 5.2 by the vector fields X which are chosen to satisfy (4.14). In this way, we obtain a scalar equation

∇_{L}(X · W) + \frac{1}{s}(X · W) = ∇_{L}F · (Gˆ⊗X), or F · (Gˆ⊗X),

where ˆ⊗ denotes either a tensor product or a contraction. Note N_{1}[Gˆ⊗X]+∥Gˆ⊗X∥_{L_{ω}^{∞}L_{t}^{2}} is invariant with respect to a tensor multiplication by a vector field X verifying the requirement in (4.14). In particular for Theorem 5.2, we have already reduce it to the case ∇_{L}w + \frac{1}{s}w = F · G where w is a scalar function, we can further reduce it to the case that F and G are both scalar functions.

For Theorem 5.3, we can also assume F, W and G are all scalar functions. It can be easily proved by combining Proposition 5.2 with the product estimate (5.3) which will be stated soon.

Based on the above reduction argument, Theorems 5.1–5.3 can be derived from the following product estimates.

Proposition 5.4. For any S-tangent tensors F and G of same type there hold

∥t^{−1} ∫_{0}^{t} s∇_{L}F · Gds∥_{B^{0}} ≲ N_{1}(F) (N_{1}(G) + ∥G∥_{L_{ω}^{∞}L_{t}^{2}}) .  (5.1)

For any scalar function F and G, we have

∥t^{−1} ∫_{0}^{t} sF · Gds∥_{B^{0}} ≲ (N_{1}(G) + ∥G∥_{L_{ω}^{∞}L_{t}^{2}}) ∥F ∥_{P^{0}},  (5.2)

For F and G being scalar functions,

∥t^{−1}G · ∫_{0}^{t} sF ds∥_{P^{0}} ≲ (N_{1}(G) + ∥G∥_{L_{ω}^{∞}L_{t}^{2}})∥F ∥_{P^{0}}  (5.3)

The proofs of (5.1)-(5.3) in the case when 1/2 < t < 2 have been given in [14]. However, by using Proposition 4.1 we are able to simplify the proofs. In the following we will prove (5.2). The proof of (5.1) is deferred to Section 5.1.2 since it is more intricate. The proof of (5.3) can be found in [14, Proposition 1.3].

Proof of (5.2). Using Proposition 4.1, it suffices to prove (5.2) with B^{0} and P^{0} norms replaced by B̃^{0} and P̃^{0} norms respectively. Thus we can use the classical LP theory.

Write E_{k} ∫_{0}^{t} sf · gds = A_{k} + B_{k}, where

A_{k} = E_{k} ∫_{0}^{t} sE_{<k}f · gds and B_{k} = E_{k} ∫_{0}^{t} sE_{≥k}f · gds.

With the help of Proposition 2.3 (iv) and (v), we infer

∥A_{k}∥_{L_{ω}^{2}} = 2^{−k} ∥Ẽ_{k}∂ ∫_{0}^{t} sE_{<k}f · gds∥_{L_{ω}^{2}}

≲ 2^{−k} ∥∫_{0}^{t} s∂E_{<k}f · gds∥_{L_{ω}^{2}} + 2^{−k} ∥∫_{0}^{t} sE_{<k}f · ∂gds∥_{L_{ω}^{2}}

≲ ∑_{a<k} 2^{a−k}∥tE_{a}f∥_{L_{t}^{2}L_{ω}^{2}}(∥g∥_{L_{ω}^{∞}L_{t}^{2}} + ∥∂g∥_{L_{ω}^{2}L_{t}^{2}}).

Since for scalar functions g, it is clear that ∥∂g∥_{L_{t}^{2}L_{ω}^{2}} ≲ ∥∇g∥_{L_{t}^{2}L_{x}^{2}}, by using Proposition 4.1 we have,

∑_{k>0} ∥A_{k}∥_{L_{t}^{∞}L_{ω}^{2}} ≲ ∥f∥_{P^{0}}(∥g∥_{L_{ω}^{∞}L_{t}^{2}} + ∥∇g∥_{L_{t}^{2}L_{x}^{2}}).

Similarly, for the term B_{k} we obtain

∥B_{k}∥_{L_{ω}^{2}} ≲ ∑_{b>k} 2^{−b} ∥E_{k} ∫_{0}^{t} s∂(Ẽ_{b}f · g)ds∥_{L_{ω}^{2}} + ∑_{b>k} 2^{−b} ∥E_{k} ∫_{0}^{t} sẼ_{b}f · ∂gds∥_{L_{ω}^{2}}

≲ ∑_{b>k} 2^{k−b} ∥sẼ_{b}f · g∥_{L_{ω}^{2}L_{t}^{1}} + ∑_{b>k} 2^{k−b}∥sẼ_{b}f∥_{L_{ω}^{2}L_{t}^{2}}∥∂g∥_{L_{ω}^{2}L_{t}^{2}}

≲ ∑_{b>k} 2^{k−b}∥sẼ_{b}f∥_{L_{ω}^{2}L_{t}^{2}} (∥g∥_{L_{ω}^{∞}L_{t}^{2}} + ∥∂g∥_{L_{ω}^{2}L_{t}^{2}}) .

Therefore using Proposition 4.1, we obtain

∑_{k>0} ∥B_{k}∥_{L_{t}^{∞}L_{ω}^{2}} ≲ ∥f∥_{P^{0}}(∥g∥_{L_{ω}^{∞}L_{t}^{2}} + ∥∇g∥_{L_{t}^{2}L_{x}^{2}}).

Thus we complete the proof of (5.2).

5.1.1 Trace Theorem

Theorem 5.4. Let F be an S-tangent tensor which admits a decomposition of the form ∇F = ∇_{L}P + E with tensors P and E of the same type. If ^{1} |s^{0−}F | and |s^{1−}∇F | are finite relative to the angular variable ω when s → 0, then

∥F ∥_{L_{ω}^{∞}L_{t}^{2}} ≤ N_{1}(F) + N_{1}(P) + ∥E∥_{P^{0}}.

Proof. Let us set G(t) = ∫_{0}^{t} |F |^{2}ds, then ∇_{L}G = |F |^{2}. Since |s^{0−}F | is bounded when s → 0, we obtain lim_{s→0} |G| = 0.

Due to Proposition 5.1, it suffices to estimate the sum of ∥∇G∥_{B^{0}} and ∥t^{−1}G∥_{L_{t}^{∞}L_{x}^{2}} instead of ∥G∥_{L^{∞}(H)}. First, it is easy to see

∥t^{−1}G∥_{L_{t}^{∞}L_{x}^{2}} ≲ ∥F ∥_{L_{ω}^{∞}L_{t}^{2}} · ∥t^{−1}F ∥_{L_{t}^{2}L_{x}^{2}}.

To estimate B^{0} norm of ∇G, using Lemma 4.2, it is identical to estimate ∑_{i=1}^{l} ∥X_{i} · ∇G∥_{B^{0}}, where X_{i} verifies (4.14). ^{2}

In view of (3.26) i.e. [∇_{L}, ∇]G = −χ · ∇G, ∇G satisfies the transport equation

∇_{L}∇G + \frac{1}{s}∇G = 2F · ∇F − \frac{1}{2}V · ∇G − χ̂ · ∇G.

Using the decomposition ∇F = ∇_{L}P + E and the fact that ∇_{L}X_{i} = 0, we obtain

∥X_{i}·∇G−lim_{t→0} X_{i}·∇G∥_{B^{0}} ≲ ∥\frac{1}{t} ∫_{0}^{t} sF · (∇_{L}P + E) · X_{i}ds∥_{B^{0}}+∥\frac{1}{t} ∫_{0}^{t} sA · ∇G · X_{i}ds∥_{B^{0}} .

(5.4)

Now we determine lim_{s→0} s|X_{i} · ∇G|. Due to (4.14), it is equal to lim_{s→0} s|∇G| up to a universal positive constant. By the weakly spherical condition, we only need to consider \frac{∂G}{∂ω_{i}} under the transport coordinate (s, ω_{1}, ω_{2}), where i = 1, 2. Note that

\frac{∂G}{∂ω_{i}} = 2 ∫_{0}^{t}⟨∇_{\frac{∂}{∂ω_{i}}} F, F ⟩ds,

1We will frequently use a+ to represent a number q > a, and a− to represent a number q < a.

2X_{i} · ∇G = X_{i}(G).

where i = 1, 2, we infer

|\frac{∂G}{∂ω_{i}} | ≲ | ∫_{0}^{t}⟨∇_{\frac{∂}{∂ω_{i}}} F, F ⟩ds| ≲ sup_{0<s≤t} |s^{0−}F | · |s∇_{\frac{∂}{∂ω_{i}}} F | |∫_{0}^{t} s^{−1+}ds| .  (5.5)

From the condition that |s∇F | is finite relative to the angular variable ω when s → 0, let t → 0, we conclude that |\frac{∂G}{∂ω_{i}} | → 0.

Thus

lim_{t→0} tX_{i} · ∇G = 0 as s → 0,

where i = 1, ⋯, l.

Therefore the second term on the left hand side of (5.4) vanishes. From Lemma 4.2, ∥∇G∥_{B^{0}} ≈ ∑_{i=1}^{l} ∥X_{i} · ∇G∥_{B^{0}}, hence we can rewrite (5.4) as follows

∥∇G∥_{B^{0}} ≲ ∑_{i=1}^{l}\{∥\frac{1}{t} ∫_{0}^{t} sF · (∇_{L}P + E) · X_{i}ds∥_{B^{0}} + ∥\frac{1}{t} ∫_{0}^{t} sA · ∇G · X_{i}ds∥_{B^{0}}\}.

With the help of Proposition 5.4 and Lemma 4.2, we have

∑_{i=1}^{l} ∥\frac{1}{t} ∫_{0}^{t} sX_{i} · F · ∇_{L}P ds∥_{B^{0}} ≲ N_{1}(P) (N_{1}(F) + ∥F ∥_{L_{ω}^{∞}L_{t}^{2}})
∑_{i=1}^{l} ∥\frac{1}{t} ∫_{0}^{t} sF · E · X_{i}ds∥_{B^{0}} ≲ (N_{1}(F) + ∥F ∥_{L_{ω}^{∞}L_{t}^{2}}) ∥E∥_{P^{0}}
∑_{i=1}^{l} ∥\frac{1}{t} ∫_{0}^{t} sA · ∇G · X_{i}ds∥_{B^{0}} ≲ (N_{1}(A) + ∥A∥_{L_{ω}^{∞}L_{t}^{2}}) ∥∇G∥_{P^{0}}.

These inequalities, together with (BA1) and the fact ∥∇G∥_{P^{0}} ≲ ∥∇G∥_{B^{0}} yield

∥∇G∥_{B^{0}} ≲ (N_{1}(F) + ∥F ∥_{L_{ω}^{∞}L_{t}^{2}})(N_{1}(P) + ∥E∥_{P^{0}}).

Combining the above inequality with the estimate for ∥t^{−1}G∥_{L_{t}^{∞}L_{x}^{2}}, we get the desired result.

Remark 5.2. In Theorem 5.4, we require F to satisfy certain initial conditions. Note that we only apply this theorem to χ̂ and ζ to derive their L_{ω}^{∞}L_{t}^{2} estimates. For F = χ̂ and ζ, there hold |s^{0−}F | → 0 and |s∇F | → 0 when s → 0, in view of Proposition 3.1 and Proposition 3.2. Thus the initial conditions in Theorem 5.4 are satisfied. In Chapter 8 where the calculations for ∥χ̂∥_{L_{ω}^{∞}L_{t}^{2}} and ∥ζ∥_{L_{ω}^{∞}L_{t}^{2}} are carried on, we will not mention the initial condition any more.

The rest of Section 5.1 is devoted to proving (5.1).

5.1.2 Proof of Sharp Trace Estimate

In this subsection, we will sketch the main steps of the proof for the sharp trace estimate (5.1). Throughout this subsection, we denote by \underline{K} the term K − s^{−2}.

Step 1. From Proposition 5.3 and Proposition 4.1 it suffices to prove

∥t^{−1} ∫_{0}^{t} s∇_{L}F · G∥B̃^{0} ≲ N_{1}(F) · N_{1}(G)  (5.6)

for arbitrary S tangent tensor fields F and G of same order.

The advantage of doing this is that the classical LP projection E_{k} commute with ∫_{0}^{t}. This fact allows us to circumvent the difficult commutation estimate of the operator [P_{k}, ∫_{0}^{t}] in [14]. Unfortunately we can not rely on the classical LP projection completely to prove (5.6). The reason is that the H^{1} component of tensor field F relative to a coordinate system is not equivalent to the L^{2} norm of the covariant derivative ∇F, because terms of the form ∥F · Γ∥_{L^{2}(H)} involving the Christoffel symbol Γ require the estimate of ∥Γ∥_{L_{t}^{2}L_{x}^{∞}}. However our metric assumption is ∥Γ∥_{L_{x}^{2}L_{t}^{∞}} < ∞, or ∥Γ∥_{B_{2,1}^{0}} < ∞ at most, under which we can not bound the L^{2} norm of F · Γ on S_{s} uniformly by N_{1}(F). The method of proof here combines the classical LP projections which are applied to ∫_{0}^{t} ∇_{L}F · G and the geometric LP projections which are applied to decompose the two tensor fields ∇_{L}F and G.

Step 2. Denoting P_{n}(·) by (·)_{n}, for fixed k > 0, observe that it suffices to consider the terms

∑_{m,n>0} ∫_{0}^{t} E_{k}(s(∇_{L}F)_{n} · G_{m})ds

after dropping the low frequency terms. According to the relation of m and n, we divide it into two parts

∑_{m,n>0} ∫_{0}^{t} E_{k}(s(∇_{L}F)_{n} · G_{m})ds = A_{k} + B_{k}

where

A_{k} = A_{k}^{1} + A_{k}^{2} + A_{k}^{3}

= ∑_{k<n<m} ∫_{0}^{t} E_{k}(s(∇_{L}F)_{n} · G_{m})ds + ∑_{0<n<k<m} ∫_{0}^{t} E_{k}(s(∇_{L}F)_{n} · G_{m})ds

+ ∑_{0<n<m<k} ∫_{0}^{t} E_{k}(s(∇_{L}F)_{n} · G_{m})ds

and

B_{k} = B_{k}^{1} + B_{k}^{2} + B_{k}^{3}

= ∑_{k<m<n} ∫_{0}^{t} E_{k}(s(∇_{L}F)_{n} · G_{m})ds + ∑_{0<m<k<n} ∫_{0}^{t} E_{k}(s(∇_{L}F)_{n} · G_{m})ds

+ ∑_{0<m<n<k} ∫_{0}^{t} E_{k}(s(∇_{L}F)_{n} · G_{m})ds.

We then take L_{t}^{∞}L_{x}^{2} of the above terms.

1. We take L_{t}^{∞}L_{x}^{2} norms of the terms in A_{k}, i.e. A_{k}^{1}, A_{k}^{2} and A_{k}^{3}. For each term of A_{k}^{1}, consider ∥ ∫_{0}^{t} E_{k}(s(∇_{L}F)_{n} · G_{m})ds∥_{L_{t}^{∞}L_{x}^{2}} with k < n < m. We employ the sharp Bernstein inequality followed by the finite band property; for each term of A_{k}^{2}, since n < k < m, we first apply the property for Laplacian Δ in Proposition 2.2 to G_{m}, i.e. G_{m} = 2^{−2m}s^{2}ΔG_{m}, then take L_{t}^{∞}L_{x}^{2} norms followed by Leibniz’s rule and the finite band property; for each term of A_{k}^{3}, due to 0 < n < m < k we apply the derivative finite band property of Proposition 2.3 to E_{k} first, i.e. for scalar functions f, E_{k}f = 2^{−k}∂Ẽ_{k}f with Ẽ_{k} a slight modification of the LP projection E_{k}. Summing over n, m, k, we can get ∑_{k} ∥A_{k}∥_{L_{t}^{∞}L_{x}^{2}} ≲ N_{1}(F)N_{1}(G). We do not need to perform integration by parts since n < m.

2: To take L_{t}^{∞}L_{x}^{2} norms of the terms in B_{k}, since the term of (∇_{L}F)_{n} has higher frequency, we will perform integration by parts ^{3} with the help of

(∇_{L}F)_{n} · G_{m} = ∇_{L}(F_{n} · G_{m}) + [P_{n}, ∇_{L}]F · G_{m} − F_{n} · (∇_{L}G)_{m} + F_{n} · [P_{m}, ∇_{L}]G. (5.7)

Then it is necessary to give the estimates of the commutator [P_{n}, ∇_{L}]F and its covariant derivative.

We will follow the same way as the proof for [14, Theorem 4.9 (78)]. However, since we are considering null hypersurfaces H which emanate from a point in Einstein vacuum spacetimes, in each step we have to take the weight of t i.e. t^{a} for some a ∈ ℝ into account.

Analogously to [14, Lemma 5.3] we have the following estimates, the proof of which is given in Appendix 9.2.

Lemma 5.1. (Commutator Esimates.) For any smooth S_{s} tangent tensor fields F and all q < 2 sufficiently close to q = 2,

∥t^{\frac{1}{2} − \frac{1}{q}} [P_{k}, ∇_{L}]F ∥_{L_{t}^{q}L_{x}^{2}} + 2^{−k}∥t^{\frac{3}{2} − \frac{1}{q}} ∇[P_{k}, ∇_{L}]F ∥_{L_{t}^{q}L_{x}^{2}} ≲ 2^{− \frac{k}{2} +}N_{1}(F)  (5.8)

3The reason of doing this can be easily seen by proving (5.1) for scalar functions f and g. We refer the reader to [14] for its detail.

for q = 1,

∥t^{−1/2}[P_{k}, ∇_{L}]F ∥_{L_{t}^{q}L_{x}^{2}} + 2^{−k}∥t^{1/2}∇[P_{k}, ∇_{L}]F ∥_{L_{t}^{q}L_{x}^{2}} ≲ 2^{−k+}N_{1}(F).  (5.9)

5.1.3 Preliminary Estimates

In order to prove (5.1), we recall the notion of an N_{1}–envelope of tensor fields introduced in [14, p.31].

Definition 5.1. For a given S-tangent tensorfield F and a sufficiently small ϵ > 0 we define its N_{1}-envelope (of order ϵ) to be any sequence of positive real numbers N_{1}[F_{k}] satisfying the following properties:

N_{1}[F_{k}] ≲ 2^{ϵ|k−k′|}N_{1}[F_{k′}], for any k, k′,
∑_{k} N_{1}[F_{k}]^{2} ≈ N_{1}(F)^{2}.

The existence of an envelope follows from the construction

N̄_{1}[F_{k}] = ∥t^{−1}F_{k}∥_{L_{t}^{2}L_{x}^{2}} + ∥∇F_{k}∥_{L_{t}^{2}L_{x}^{2}} + ∥(∇_{L}F)_{k}∥_{L_{t}^{2}L_{x}^{2}}.  (5.10)

Then ∑_{k} N̄_{1}[F_{k}]^{2} ≈ N_{1}(F)^{2} since P_{k}F verify Bessel inequality and ∑_{k} 2^{2k}t^{−2}∥P_{k}F ∥^{2} ≲ ∥∇F ∥_{L_{x}^{2}}^{2}. Therefore the sequences

N_{1}[F_{k}] = ∑_{k′} 2^{−ϵ|k−k′|}N̄_{1}[F′_{k}]

are desired envelopes.

We record the following property of an envelope.

Lemma 5.2. Let {N_{1}[F_{k}]} be an envelope (of order ϵ) for a tensor field F, then for any 0 < α ≤ ϵ the sequence {N_{1}[F_{k}] + (2^{k}t^{−1})^{−α}N_{1}(F)} is also an envelope. Moreover, for any α > ϵ the sequence {N_{1}[F_{k}] + (2^{k}t^{−1})^{−α}N_{1}(F)} is dominated by an ϵ- N_{1}-envelope for F.

As a corollary of Lemma 5.1, we obtain the following results which will be used frequently.

Lemma 5.3. For any smooth S-tangent tensor field F and for exponent 2 ≤ q ≤ ∞, we have the following dyadic Gagliardo-Nirenberg inequality

∥t^{− \frac{1}{2} − \frac{1}{q}} F_{k}∥_{L_{t}^{q}L_{x}^{2}} ≲ 2^{− \frac{1}{2} k− \frac{1}{q} k}N_{1}[F_{k}].  (5.11)

Proof. The result is trivial when q = 2. So we only need to consider the case q > 2. It is easy to get the following estimate

∥t^{− \frac{1}{2} − \frac{1}{q}} F_{k}∥_{L_{t}^{q}L_{x}^{2}} ≲ ∥t^{−1}F_{k}∥L_{t}^{2}L_{x}^{2}^{\frac{2}{q}}∥t^{− \frac{1}{2}} F_{k}∥L_{t}^{∞}L_{x}^{2}^{\frac{q−2}{q}}.

Moreover we have by integrating along an arbitrary null geodesic from s = t to s = 1,

∥t^{− \frac{1}{2}} F_{k}∥_{L_{x}^{2}L_{t}^{∞}}^{2} ≲ ∥(∇_{L}F)_{k}∥_{L_{t}^{2}L_{x}^{2}}∥t^{−1}F_{k}∥_{L_{t}^{2}L_{x}^{2}} + ∥t^{−1}[P_{k}, ∇_{L}]F · F_{k}∥_{L_{t}^{1}L_{x}^{1}}
+ ∥t^{−1}F_{k}∥_{L_{t}^{2}L_{x}^{2}}^{2} + ∥F_{k}(1)∥_{L_{x}^{2}}^{2}.  (5.12)

Note the term ∥F_{k}(1)∥_{L_{x}^{2}} can be obtained by multiplying |F |^{2} with a test function 0 ≤ θ ≤ 1 supported on [\frac{1}{2}, 1] verifying θ(1) = 1 and sup_{t≤s≤1} |θ′| ≤ \frac{1}{1−t}. Note also that

|F_{k}(1, ω)|^{2} = ∫_{t}^{1} \frac{d}{ds}(sθ(s)|F_{k}(s, ω)|^{2})ds, with 0 < t < 1/2

in the same way as we estimated ∥F (1)∥_{L_{ω}^{2}} in the proof of (3.39) (also refer to [12, Corollary 4.7]), we can see that ∥F_{k}(1)∥_{L_{x}^{2}}^{2} is controlled by

∥(∇_{L}F)_{k}∥_{L_{t}^{2}L_{x}^{2}}∥t^{−1}F_{k}∥_{L_{t}^{2}L_{x}^{2}} + ∥t^{−1}[P_{k}, ∇_{L}]F · F_{k}∥_{L_{t}^{1}L_{x}^{1}} + ∥t^{−1}F_{k}∥_{L_{t}^{2}L_{x}^{2}}^{2}.

For the commutator in (5.12), we choose an exponent r < 2 sufficiently close to 2 and obtain the following estimate with \frac{1}{r} + \frac{1}{r′} = 1

∥t^{−1}[P_{k}, ∇_{L}F] · F_{k}∥_{L_{t}^{1}L_{x}^{1}} ≲ ∥t^{\frac{1}{2} − \frac{1}{r}} [P_{k}, ∇_{L}]F ∥_{L_{t}^{r}L_{x}^{2}}∥t^{− \frac{1}{r′} − \frac{1}{2}} F_{k}∥_{L_{t}^{r′}L_{x}^{2}}
≲ 2^{− \frac{k}{2} +}N_{1}(F)∥t^{− \frac{1}{r′} − \frac{1}{2}} F_{k}∥_{L_{t}^{r′}L_{x}^{2}},

where we employed Lemma 5.1 to derive the last inequality.

Combining the above estimates we obtain

∥t^{− \frac{1}{2} − \frac{1}{q}} F_{k}∥_{L_{t}^{q}L_{x}^{2}}^{q} ≲ ∥t^{−1}F_{k}∥_{L_{t}^{2}L_{x}^{2}}^{2}(∥(∇_{L}F)_{k}∥_{L_{t}^{2}L_{x}^{2}}∥t^{−1}F_{k}∥_{L_{t}^{2}L_{x}^{2}}
+ 2^{− \frac{k}{2} +}N_{1}(F)∥t^{− \frac{1}{r′} − \frac{1}{2}} F_{k}∥_{L_{t}^{r′}L_{x}^{2}} + ∥t^{−1}F_{k}∥_{L_{t}^{2}L_{x}^{2}}^{2})^{\frac{q}{2} −1}  (5.13)

for any r′ > 2 sufficiently close to 2. Let q = r′ in (5.13), and plug in the estimate of ∥t^{− \frac{1}{2} − \frac{1}{r′}} F_{k}∥_{L_{t}^{r′}L_{x}^{2}} into (5.13). Since q > 2, we get for any 0 ≤ α < \frac{1}{4} − \frac{1}{2q},

∥t^{− \frac{1}{2} − \frac{1}{q}} F_{k}∥_{L_{t}^{q}L_{x}^{2}} ≲ 2^{−k( \frac{1}{2} + \frac{1}{q})} (N_{1}[F_{k}] + 2^{−αk}N_{1}(F))

as desired.

Remark 5.3. It is clear that (5.12) implies ∥t^{−1/2}F_{k}∥_{L_{x}^{2}L_{t}^{∞}} ≲ 2^{−k/2}N_{1}[F_{k}].

We will also need the following lemma.

Lemma 5.4. 1. For all q with 1 ≤ q < 2 sufficiently close to q = 2 and any smooth S-tangent tensor fields F,

∥t^{\frac{1}{2} − \frac{1}{q} +}∇_{L}F_{k}∥_{L_{t}^{q}L_{x}^{2}} ≲ N_{1}[F_{k}].  (5.14)

2. For any smooth S-tangent tensor field F and any 2 ≤ q < 4, we have the following

inequality

∥t^{1− \frac{1}{q}} ∇F_{k}∥_{L_{t}^{q}L_{x}^{4}} ≲ 2^{k(1− \frac{1}{q})}N_{1}[F_{k}].  (5.15)

3. For any exponent 2 ≤ q < ∞,

∥t^{\frac{1}{2} − \frac{1}{q}} F_{k}∥_{L_{t}^{q}L_{x}^{∞}} ≲ 2^{k( \frac{1}{2} − \frac{1}{q})}N_{1}[F_{k}].  (5.16)

Proof. We first write ∇_{L}F_{k} = P_{k}∇_{L}F + [P_{k}, ∇_{L}]F. According to the commutator estimate for ∥t^{\frac{1}{2} − \frac{1}{q}} [P_{k}, ∇_{L}]F ∥_{L_{t}^{q}L_{x}^{2}} in Lemma 5.1, we get

∥t^{\frac{1}{2} − \frac{1}{q} +}∇_{L}F_{k}∥_{L_{t}^{q}L_{x}^{2}} ≲ ∥t^{\frac{1}{2} − \frac{1}{q} +}P_{k}∇_{L}F ∥_{L_{t}^{q}L_{x}^{2}} + 2^{− \frac{k}{2} +}N_{1}(F),

which implies (5.14). By Gagliardo-Nirenberg inequality,

∥t^{1− \frac{1}{q}} ∇F_{k}∥_{L_{t}^{q}L_{x}^{4}}^{q} ≲ ∥t^{q−1}∥∇^{2}F_{k}∥L_{x}^{2}^{\frac{q}{2}}∥∇F_{k}∥L_{x}^{2}^{\frac{q}{2}} + ∥∇F_{k}∥_{L_{x}^{2}}^{q}t^{q/2−1}∥_{L_{t}^{1}} .  (5.17)

On the other hand, according to Lemma 9.1 (iii), we have

∥∇^{2}F_{k}∥_{L_{x}^{2}} ≲ ∥ΔF_{k}∥_{L_{x}^{2}} + (t^{\frac{1}{p−1}} ∥\underline{K}∥L_{x}^{2}^{\frac{p}{p−1}} + ∥\underline{K}∥_{L_{x}^{2}})∥∇F_{k}∥_{L_{x}^{2}}

for any 2 ≤ p < ∞. Choose the exponent p so that 2 = \frac{qp}{2(p−1)}, the existence of which is guaranteed by the condition 2 ≤ q < 4. Thus by the estimate

∥t^{\frac{1}{2} − \frac{1}{q}} ∇F_{k}∥_{L_{t}^{q}L_{x}^{2}} ≲ 2^{\frac{k}{2} (1− \frac{2}{q})}N_{1}[F_{k}],

which can be obtained in the same way as to Proposition 9.2 (i), we infer from the finite band property for Δ and (5.11) that

∥t^{1− \frac{1}{q}} ∇F_{k}∥_{L_{t}^{q}L_{x}^{4}} ≲ ∥t^{\frac{3}{2} − \frac{1}{q}} ΔF_{k}∥L_{t}^{q}L_{x}^{2}^{\frac{1}{2}}∥t^{\frac{1}{2} − \frac{1}{q}} ∇F_{k}∥L_{t}^{q}L_{x}^{2}^{\frac{1}{2}}

+ ∥\underline{K}∥L_{t}^{2}L_{x}^{2}^{\frac{2}{q}}∥t^{1− \frac{1}{q}} ∇F_{k}∥_{L_{t}^{∞}L_{x}^{2}} + ∥t^{\frac{1}{2} − \frac{1}{q}} ∇F_{k}∥_{L_{t}^{q}L_{x}^{2}}

≲ 2^{k(1− \frac{1}{q})}N_{1}[F_{k}], which gives (5.15). Observe that from Lemma 9.1 (i), we have the following dyadic L^{∞} inequality (also refer to [15, Theorem 10.1])

∥F_{k}∥_{L_{x}^{∞}} ≲ (2^{k}t^{−1})(1 + 2^{\frac{k}{p}} + (2^{k}t^{−1})^{− \frac{1}{p}} ∥\underline{K}∥L_{x}^{2}^{\frac{1}{p}} + (2^{k}t^{−1})^{− \frac{1}{p−1}} ∥\underline{K}∥L_{x}^{2}^{\frac{1}{p−1}} )∥F_{k}∥_{L_{x}^{2}},

Noting ∥\underline{K}∥_{L_{t}^{2}L_{x}^{2}} ≲ Δ_{0}, take L_{t}^{q} norm, with \frac{1}{q_{1}} = \frac{1}{q} − \frac{1}{2p} and \frac{1}{q_{2}} = \frac{1}{q} − \frac{1}{2(p−1)}, we infer

∥t^{\frac{1}{2} − \frac{1}{q}} F_{k}∥_{L_{t}^{q}L_{x}^{∞}} ≲ 2^{k}∥t^{− \frac{1}{q} − \frac{1}{2}} F_{k}∥_{L_{t}^{q}L_{x}^{2}} + 2^{(1− \frac{1}{p})k}∥\underline{K}∥L_{t}^{2}L_{x}^{2}^{\frac{1}{p}}∥t^{− \frac{1}{2} + \frac{1}{p} − \frac{1}{q}} F_{k}∥_{L_{t}^{q_{1}}L_{x}^{2}}

+ 2^{(1− \frac{1}{p−1} )k}∥\underline{K}∥L_{t}^{2}L_{x}^{2}^{\frac{1}{p−1}}∥t^{− \frac{1}{2} + \frac{1}{p−1} − \frac{1}{q}} F_{k}∥_{L_{t}^{q_{2}}L_{x}^{2}}

≲ 2^{k( \frac{1}{2} − \frac{1}{q})}N_{1}[F_{k}] + (2^{k( \frac{1}{2} − \frac{1}{q} − \frac{1}{2p})}∥\underline{K}∥L_{t}^{2}L_{x}^{2}^{\frac{1}{p}}

+ 2^{k( \frac{1}{2} − \frac{1}{q} − \frac{1}{2(p−1)} )}∥\underline{K}∥L_{t}^{2}L_{x}^{2}^{\frac{1}{p−1}})N_{1}[F_{k}].

Thus we finish the proof of (5.16).

5.1.4 Estimates of B_{k}^{3}

Assuming Lemma 5.1 and using the estimates in the previous subsection, we will estimate B_{k} and A_{k} in the following three subsections. First we estimate B_{k} by performing integration by parts. Recall that

B_{k} = B_{k}^{1} + B_{k}^{2} + B_{k}^{3}

= ∑_{k<m<n} ∫_{0}^{t} E_{k}(s(∇_{L}F)_{n} · G_{m})ds + ∑_{0<m<k<n} ∫_{0}^{t} E_{k}(s(∇_{L}F)_{n} · G_{m})ds

+ ∑_{0<m<n<k} ∫_{0}^{t} E_{k}(s(∇_{L}F)_{n} · G_{m})ds,

in which the hardest term is B_{k}^{3} since m < n < k. This subsection is devoted to the estimation of B_{k}^{3} only. The estimation of B_{k}^{1} and B_{k}^{2} will be given in the next subsection.

In view of (5.7), B_{k}^{3} can be written as follows

B_{k}^{3} = ∑_{m<n<k} ∫_{0}^{t} E_{k}s((∇_{L}F)_{n} · G_{m})ds = ∑_{m<n<k} ∫_{0}^{t} E_{k}s(∇_{L}F_{n} · G_{m} + [P_{n}, ∇_{L}]F · G_{m})ds

= ∑_{m<n<k} ∫_{0}^{t} {sE_{k}(∇_{L}(F_{n} · G_{m}) − F_{n}∇_{L}G_{m}) + sE_{k}([P_{n}, ∇_{L}]F · G_{m})}ds

= ∑_{m<n<k} ∫_{0}^{t} {E_{k}(s∇_{L}(F_{n} · G_{m})) − E_{k}(sF_{n} · (∇_{L}G)_{m})

− sE_{k}(F_{n} · [∇_{L}, P_{m}]G) + sE_{k}([P_{n}, ∇_{L}]F · G_{m})}ds.

Note that

∥∫_{0}^{t} E_{k}s∇_{L}(F_{n} · G_{m})ds∥_{L_{t}^{∞}L_{ω}^{2}}

≲ ∥tE_{k}(F_{n} · G_{m})(t) − lim_{s→0} E_{k}(s(F_{n} · G_{m}))∥_{L_{t}^{∞}L_{ω}^{2}} + ∥∫_{0}^{t} E_{k}(F_{n} · G_{m})ds∥_{L_{t}^{∞}L_{ω}^{2}}

≲ ∥E_{k}(F_{n} · G_{m})∥_{L_{t}^{∞}L_{x}^{2}} + ∥ ∫_{0}^{t} E_{k}(F_{n} · G_{m})ds∥_{L_{t}^{∞}L_{ω}^{2}},

where the term ∥ lim_{s→0} E_{k}(s(F_{n} ·G_{m}))∥_{L_{ω}^{2}} is bounded by ∥E_{k}(F_{n} ·G_{m})∥_{L_{t}^{∞}L_{x}^{2}}. Summing the above inequality over m < n < k, the resulting right hand side can be bounded by E_{1} + ⋯ + E_{5} which are defined in the following. Therefore we only need to show the following inequalities

E_{1} := ∑_{m<n<k} ∥E_{k}(F_{n} · G_{m})∥_{L_{t}^{∞}L_{x}^{2}} ≲ N_{1}(F)N_{1}(G)  (5.18)
E_{2} := ∑_{m<n<k} ∥ ∫_{0}^{t} E_{k}(F_{n} · G_{m})ds∥_{L_{t}^{∞}L_{ω}^{2}} ≲ N_{1}(F)N_{1}(G)  (5.19)
E_{3} := ∑_{m<n<k} ∥ ∫_{0}^{t} E_{k}(sF_{n} · (∇_{L}G)_{m})ds∥_{L_{t}^{∞}L_{ω}^{2}} ≲ N_{1}(F)N_{1}(G)  (5.20)
E_{4} := ∑_{m<n<k} ∥ ∫_{0}^{t} sE_{k}(F_{n} · [∇_{L}, P_{m}]G)ds∥_{L_{t}^{∞}L_{ω}^{2}} ≲ N_{1}(F)N_{1}(G)  (5.21)
E_{5} := ∑_{m<n<k} ∥ ∫_{0}^{t} sE_{k}([P_{n}, ∇_{L}]F · G_{m})ds∥_{L_{t}^{∞}L_{ω}^{2}} ≲ N_{1}(F)N_{1}(G).  (5.22)

By the finite band property, Lemma 5.4 and Bernstein inequality we first have

E_{3} ≲ ∑_{m<n<k} ∥E_{k}(F_{n} · (∇_{L}G)_{m})∥_{L_{t}^{1}L_{x}^{2}} ≲ ∑_{m<n<k} 2^{−k}∥t∇(F_{n}(∇_{L}G)_{m})∥_{L_{t}^{1}L_{x}^{2}}

≲ ∑_{m<n<k} 2^{−k}(∥t∇(F_{n}) · (∇_{L}G)_{m}∥_{L_{t}^{1}L_{x}^{2}} + ∥F_{n} · t∇(∇_{L}G)_{m}∥_{L_{t}^{1}L_{x}^{2}})

≲ ∑_{m<n<k} (2^{−k}∥t^{1/2}∇F_{n}∥_{L_{t}^{2}L_{x}^{4}}∥t^{1/2}(∇_{L}G)_{m}∥_{L_{t}^{2}L_{x}^{4}} + 2^{m−k}∥F_{n}∥_{L_{t}^{2}L_{x}^{∞}}∥(∇_{L}G)_{m}∥_{L_{t}^{2}L_{x}^{2}})

≲ ∑_{m<n<k} (2^{−k}2^{\frac{n+m}{2}} N_{1}[F_{n}]∥(∇_{L}G)_{m}∥_{L_{t}^{2}L_{x}^{2}} + 2^{m−k}N_{1}[F_{n}]∥(∇_{L}G)_{m}∥_{L_{t}^{2}L_{x}^{2}})

≲ N_{1}(F)∥∇_{L}G∥_{L_{t}^{2}L_{x}^{2}} and

E_{2} ≲ ∑_{m<n<k} 2^{−k}∥∇(F_{n} · G_{m})∥_{L_{t}^{1}L_{x}^{2}}

≲ ∑_{m<n<k} 2^{−k} (∥t^{\frac{1}{2}} ∇F_{n}∥_{L_{t}^{2}L_{x}^{4}}∥t^{− \frac{1}{2}} G_{m}∥_{L_{t}^{2}L_{x}^{4}} + ∥t^{− \frac{1}{2}} F_{n}∥_{L_{t}^{2}L_{x}^{4}}∥t^{\frac{1}{2}} ∇G_{m}∥_{L_{t}^{2}L_{x}^{4}})

≲ ∑_{m<n<k} 2^{−k+ \frac{n+m}{2}} (N_{1}[F_{n}]∥t^{−1}G_{m}∥_{L_{t}^{2}L_{x}^{2}} + ∥t^{−1}F_{n}∥_{L_{t}^{2}L_{x}^{2}}N_{1}[G_{m}])

≲ N_{1}(F)N_{1}(G).

To show (5.18), noticing that by Proposition 4.1 we have

E_{1} ≲ ∑_{m<n} ∥P_{k}(F_{n} · G_{m})∥_{L_{x}^{2}} ≤ E_{1}^{(1)} + E_{1}^{(2)} + E_{1}^{(3)},  (5.23)

where

E_{1}^{(1)} := ∑_{m<n<k} ∥P_{k}(F_{n} · G_{m})∥_{L_{t}^{∞}L_{x}^{2}},

E_{1}^{(2)} := ∑_{m<k<n} ∥P_{k}(F_{n} · G_{m})∥_{L_{t}^{∞}L_{x}^{2}},

E_{1}^{(3)} := ∑_{k<m<n} ∥P_{k}(F_{n} · G_{m})∥_{L_{t}^{∞}L_{x}^{2}}.

We can assume m > 0, k > 0 since lower frequency terms can be dropped. We start with E_{1}^{(1)}, By the finite band property, the weak Bernstein inequality and Lemma 5.3, we have

E_{1}^{(1)} ≲ ∑_{m<n<k} 2^{−2k}(∥t^{2}P_{k}(ΔF_{n} · G_{m})∥_{L_{t}^{∞}L_{x}^{2}} + ∥t^{2}P_{k}(F_{n} · ΔG_{m})∥_{L_{t}^{∞}L_{x}^{2}}
+ ∥t^{2}P_{k}(∇F_{n} · ∇G_{m})∥_{L_{t}^{∞}L_{x}^{2}})
≲ ∑_{m<n<k} ((2^{−2(k−n)} + 2^{−2(k−m)})∥F_{n} · G_{m}∥_{L_{t}^{∞}L_{x}^{2}} + 2^{−k+ \frac{n+m}{2}} N_{1}[F_{n}]N_{1}[G_{m}]) (5.24)

Indeed, the derivation of the last term of (5.24) follows from the sharp Bernstein inequality and Lemma 5.3 as follows

2^{−2k}∥t^{2}P_{k}(∇F_{n} · ∇G_{m})∥_{L_{t}^{∞}L_{x}^{2}} ≲ 2^{−k}∥t∇F_{n} · ∇G_{m}∥_{L_{t}^{∞}L_{x}^{1}}

≲ 2^{−k}∥t^{\frac{1}{2}} ∇F_{n}∥_{L_{t}^{∞}L_{x}^{2}}∥t^{\frac{1}{2}} ∇G_{m}∥_{L_{t}^{∞}L_{x}^{2}} ≲ 2^{−k+n+m}∥t^{− \frac{1}{2}} F_{n}∥_{L_{t}^{∞}L_{x}^{2}}∥t^{− \frac{1}{2}} G_{m}∥_{L_{t}^{∞}L_{x}^{2}} ≲ 2^{−k+ \frac{n+m}{2}} N_{1}[F_{n}]N_{1}[G_{m}].

Then we conclude E_{1}^{(1)} ≲ N_{1}(F)N_{1}(G). As to E_{1}^{(2)}, we proceed by weak Bernstein inequality and Lemma 5.3

E_{1}^{(2)} ≲ ∑_{m<k<n} ∥t^{− \frac{1}{4}} F_{n}∥_{L_{t}^{∞}L \frac{8}{3} ; x} ∥t^{\frac{1}{4}} G_{m}∥_{L_{t}^{∞}L_{x}^{8}} ≲ ∑_{m<k<n} 2^{\frac{1}{4} n+ \frac{3}{4} m}∥t^{− \frac{1}{2}} F_{n}∥_{L_{t}^{∞}L_{x}^{2}}∥t^{− \frac{1}{2}} G_{m}∥_{L_{t}^{∞}L_{x}^{2}}

≲ ∑_{m<k<n} 2^{− \frac{1}{4} (n−m)}N_{1}[F_{n}] · N_{1}[G_{m}] ≲ ∑_{m<k<n} 2^{− \frac{1}{8} (n−m)}2^{− \frac{1}{8} (k−m)}N_{1}[F_{n}] · N_{1}[G_{m}]

≲ N_{1}(F)N_{1}(G).

In the same manner we derive

E_{1}^{(3)} ≲ ∑_{k<m<n} 2^{k}∥t^{−1}F_{n} · G_{m}∥_{L_{t}^{∞}L_{x}^{1}} ≲ ∑_{k<m<n} 2^{k}∥t^{−1/2}F_{n}∥_{L_{t}^{∞}L_{x}^{2}}∥t^{−1/2}G_{m}∥_{L_{t}^{∞}L_{x}^{2}}

≲ ∑_{k<m<n} 2^{k− \frac{n+m}{2}} N_{1}[F_{n}]N_{1}[G_{m}] ≲ N_{1}(F)N_{1}(G).

Therefore we complete the proof of (5.18).

As to (5.21), it follows from finite band property that

E_{4} ≲ ∑_{m<n<k} ∥E_{k}(F_{n} · [∇_{L}, P_{m}]G)∥_{L_{t}^{1}L_{x}^{2}} ≲ ∑_{m<n<k} 2^{−k}∥t∇(F_{n} · [∇_{L}, P_{m}]G)∥_{L_{t}^{1}L_{x}^{2}}

≲ ∑_{m<n<k} 2^{−k}∥t∇F_{n} · [∇_{L}, P_{m}]G∥_{L_{t}^{1}L_{x}^{2}} + ∑_{m<n<k} 2^{−k}∥tF_{n} · ∇[∇_{L}, P_{m}]G∥_{L_{t}^{1}L_{x}^{2}}

≲ ∑_{m<n<k} 2^{−k}∥t^{\frac{1}{q}} ∇F_{n}∥_{L_{t}^{q′}L_{x}^{4}}∥t^{1− \frac{1}{q}} [∇_{L}, P_{m}]G∥_{L_{t}^{q}L_{x}^{4}}

+ ∑_{m<n<k} 2^{−k}∥t^{\frac{1}{q} − \frac{1}{2}} F_{n}∥_{L_{t}^{q′}L_{x}^{∞}}∥t^{\frac{3}{2} − \frac{1}{q}} ∇[∇_{L}, P_{m}]G∥_{L_{t}^{q}L_{x}^{2}}.

Since we have

∥t^{1− \frac{1}{q}} [∇_{L}, P_{m}]G∥_{L_{t}^{q}L_{x}^{4}} ≲ ∥t^{\frac{3}{2} − \frac{1}{q}} ∇[∇_{L}, P_{m}]G∥L_{t}^{q}L_{x}^{2}^{\frac{1}{2}}∥t^{\frac{1}{2} − \frac{1}{q}} [∇_{L}, P_{m}]G∥L_{t}^{q}L_{x}^{2}^{\frac{1}{2}}
+ ∥t^{1/2−1/q}[∇_{L}, P_{m}]G∥_{L_{t}^{q}L_{x}^{2}},

by Lemma 5.1, we get ∥t^{1− \frac{1}{q}} [∇_{L}, P_{m}]G∥_{L_{t}^{q}L_{x}^{4}} ≲ 2^{(0+)m}N_{1}(G). With the help of Lemma 5.4, we can get

∥t^{\frac{1}{q}} ∇F_{k}∥_{L_{t}^{q′}L_{x}^{4}} ≲ 2^{\frac{1}{q} n}N_{1}[F_{k}] and ∥t^{\frac{1}{q} − \frac{1}{2}} F_{k}∥_{L_{t}^{q′}L_{x}^{∞}} ≲ 2^{( \frac{1}{q} − \frac{1}{2} )k}N_{1}[F_{k}].  (5.25)

We then infer that as 1 < q < 2 and q sufficiently close to 2,

E_{4} ≲ ∑_{m<n<k} 2^{\frac{1}{q} n+(0+)m−k}N_{1}[F_{n}]N_{1}(G) + ∑_{m<n<k} 2^{( \frac{1}{q} − \frac{1}{2} )n+( \frac{1}{2} +)m−k}N_{1}[F_{n}]N_{1}(G)
≲ N_{1}(F)N_{1}(G)

At last we are going to show (5.22), by Lemma 5.1, following the same argument as for (5.21), we have

E_{5} ≲ ∑_{m<n<k} ∥E_{k}([P_{n}, ∇_{L}]F · G_{m})∥_{L_{t}^{1}L_{x}^{2}} ≲ ∑_{m<n<k} 2^{−k}∥t∇([P_{n}, ∇_{L}]F · G_{m})∥_{L_{t}^{1}L_{x}^{2}}

≲ ∑_{m<n<k} 2^{−k}(∥t^{\frac{3}{2} − \frac{1}{q}} ∇[P_{n}, ∇_{L}]F ∥_{L_{t}^{q}L_{x}^{2}}∥∥t^{\frac{1}{q} − \frac{1}{2}} G_{m}∥_{L_{t}^{q′}L_{x}^{∞}}

+ ∥t^{1− \frac{1}{q}} [P_{n}, ∇_{L}]F ∥_{L_{t}^{q}L_{x}^{4}}∥t^{\frac{1}{q}} ∇G_{m}∥_{L_{t}^{q′}L_{x}^{4}})

≲ ∑_{m<n<k} (2^{( \frac{1}{q} − \frac{1}{2} )m+( \frac{1}{2} +)n−k}N_{1}(F)N_{1}[G_{m}] + 2^{\frac{1}{q} m+(0+)n−k}N_{1}(F)N_{1}[G_{m}])

≲ N_{1}(F)N_{1}(G),

which implies (5.22). Then we complete the estimation on ∑_{k>0} ∥B_{k}^{3}∥_{L_{t}^{∞}L_{x}^{2}}.

5.1.5 Estimates of B_{k}^{1} and B_{k}^{2}

We are going to estimate B_{k}^{1} and B_{k}^{2}. As to ∑_{k>0} ∥B_{k}^{1}∥_{L_{t}^{∞}L_{x}^{2}}, we can proceed in the same manner as for B_{k}^{3} and write

∑_{k>0} ∥B_{k}^{1}∥_{L_{t}^{∞}L_{x}^{2}} ≲ I + II + III + IV + V,

where

I := ∑_{k<m<n} ∥∫_{0}^{t} E_{k}(sF_{n}(∇_{L}G)_{m})ds∥_{L_{t}^{∞}L_{ω}^{2}} ,  (5.26)
II := ∑_{k<m<n} ∥∫_{0}^{t} E_{k}(F_{n} · G_{m})ds∥_{L_{t}^{∞}L_{ω}^{2}} ,  (5.27)
III := ∑_{k<m<n} ∥E_{k}(F_{n} · G_{m})∥_{L_{t}^{∞}L_{x}^{2}},  (5.28)
IV := ∑_{k<m<n} ∥∫_{0}^{t} sE_{k}(F_{n} · [∇_{L}, P_{m}]G)ds∥_{L_{t}^{∞}L_{ω}^{2}} ,  (5.29)
V := ∑_{k<m<n} ∥∫_{0}^{t} sE_{k}([P_{n}, ∇_{L}]F · G_{m})ds∥_{L_{t}^{∞}L_{ω}^{2}} .  (5.30)

By the sharp Bernstein inequality and finite band property, we first have

I ≲ ∑_{k<m<n} 2^{k}∥t^{−1}F_{n}(∇_{L}G)_{m}∥_{L_{t}^{1}L_{x}^{1}} ≲ ∑_{k<m<n} 2^{k}∥(∇_{L}G)_{m}∥_{L_{t}^{2}L_{x}^{2}}∥t^{−1}F_{n}∥_{L_{t}^{2}L_{x}^{2}}

≲ ∑_{k<m<n} 2^{k−n}N_{1}[F_{n}]∥(∇_{L}G)_{m}∥_{L_{t}^{2}L_{x}^{2}} ≲ N_{1}(F)N_{1}(G),

and

II ≲ ∑_{k<m<n} 2^{k}∥t^{−2}F_{n} · G_{m}∥_{L_{t}^{1}L_{x}^{1}} ≲ ∑_{k<m<n} 2^{k−n}∥∇F_{n}∥_{L_{t}^{2}L_{x}^{2}}∥t^{−1}G_{m}∥_{L_{t}^{2}L_{x}^{2}} ≲ N_{1}(F)N_{1}(G).

As to the boundary term (5.28), it follows from sharp Bernstein inequality and Lemma 5.3 that

III ≲ ∑_{k<m<n} 2^{k}∥t^{− \frac{1}{2}} F_{n}∥_{L_{t}^{∞}L_{x}^{2}}∥t^{− \frac{1}{2}} G_{m}∥_{L_{t}^{∞}L_{x}^{2}}

≲ ∑_{k<m<n} 2^{k− \frac{m+n}{2}} N_{1}[F_{n}]N_{1}[G_{m}] ≲ N_{1}(F)N_{1}(G).

The inequality (5.29) can be proved by Lemma 5.1 and Lemma 5.3 as follows

IV ≲ ∑_{k<m<n} ∥E_{k}(F_{n} · [∇_{L}, P_{m}]G)∥_{L_{t}^{1}L_{x}^{2}} ≲ ∑_{k<m<n} 2^{k}∥t^{−1}F_{n} · [∇_{L}, P_{m}]G∥_{L_{t}^{1}L_{x}^{1}}

≲ ∑_{k<m<n} 2^{k}∥t^{\frac{1}{q} − \frac{3}{2}} F_{n}∥_{L_{t}^{q′}L_{x}^{2}}∥t^{\frac{1}{2} − \frac{1}{q}} [∇_{L}, P_{m}]G∥_{L_{t}^{q}L_{x}^{2}}

≲ ∑_{k<m<n} 2^{k}2^{−n( \frac{1}{2} + \frac{1}{q′})}2^{(− \frac{1}{2} +)m}N_{1}[F_{n}]N_{1}(G)

≲ N_{1}(F)N_{1}(G),

where 1 < q < 2 and \frac{1}{q′} + \frac{1}{q} = 1.

As to the inequality (5.30), by sharp Bernstein inequality, Lemma 5.1 and Lemma 5.3, we have

V ≲ ∑_{k<n<m} ∥E_{k}([P_{n}, ∇_{L}]F · G_{m})∥_{L_{t}^{1}L_{x}^{2}} ≲ ∑_{k<n<m} 2^{k}∥t^{−1}[P_{n}, ∇_{L}]F · G_{m}∥_{L_{t}^{1}L_{x}^{1}}

≲ ∑_{k<n<m} 2^{k}∥t^{\frac{1}{2} − \frac{1}{q}} [P_{n}, ∇_{L}]F ∥_{L_{t}^{q}L_{x}^{2}}∥t^{− \frac{1}{2} − \frac{1}{q′}} G_{m}∥_{L_{t}^{q′}L_{x}^{2}}

≲ ∑_{k<n<m} 2^{k−( \frac{1}{2} −)n−m( \frac{1}{2} + \frac{1}{q′})}N_{1}(F)N_{1}[G_{m}]

≲ N_{1}(F)N_{1}(G).

We then conclude that ∑_{k>0} ∥B_{k}^{1}∥_{L_{t}^{∞}L_{x}^{2}} ≲ N_{1}(F)N_{1}(G).

At last we are going to estimate B_{k}^{2}. We simply change (5.26)-(5.30) to the case when m < k < n, then we show the following estimates hold true:

Ĩ:= ∑_{m<k<n} ∥E_{k}(F_{n} · G_{m})∥_{L_{t}^{∞}L_{x}^{2}} ≲ N_{1}(F)N_{1}(G)  (5.31)
ĨI := ∑_{m<k<n} ∥∫_{0}^{t} E_{k}(F_{n} · G_{m})ds∥_{L_{t}^{∞}L_{ω}^{2}} ≲ N_{1}(F)N_{1}(G)  (5.32)
ĨII := ∑_{m<k<n} ∥∫_{0}^{t} E_{k}(sF_{n} · (∇_{L}G)_{m})ds∥_{L_{t}^{∞}L_{ω}^{2}} ≲ N_{1}(F)N_{1}(G)  (5.33)
ĨV := ∑_{m<k<n} ∥∫_{0}^{t} sE_{k}(F_{n} · [∇_{L}, P_{m}]G)ds∥_{L_{t}^{∞}L_{ω}^{2}} ≲ N_{1}(F)N_{1}(G)  (5.34)
Ṽ := ∑_{m<k<n} ∥∫_{0}^{t} sE_{k}([P_{n}, ∇_{L}]F · G_{m})ds∥_{L_{t}^{∞}L_{ω}^{2}} ≲ N_{1}(F)N_{1}(G).  (5.35)

The estimate (5.31) can be proved as E_{1}. By the finite band property, sharp Bernstein inequality, weak Bernstein inequality and Lemma 5.4, we have

ĨI ≲ ∑_{m<k<n} ∥E_{k}(F_{n} · G_{m})∥_{L_{t}^{1}L_{ω}^{2}} ≲ ∑_{m<k<n} 2^{−2n}∥tE_{k}(ΔF_{n} · G_{m})∥_{L_{t}^{1}L_{x}^{2}}

≲ ∑_{m<k<n} 2^{−2n}∥tE_{k}( div (∇F_{n} · G_{m}) − ∇F_{n} · ∇G_{m})∥_{L_{t}^{1}L_{x}^{2}}

≲ ∑_{m<k<n} (2^{−2n+k}∥∇F_{n} · G_{m}∥_{L_{t}^{1}L_{x}^{2}} + 2^{−2n+k}∥∇F_{n}∥_{L_{t}^{2}L_{x}^{2}}∥∇G_{m}∥_{L_{t}^{2}L_{x}^{2}})

≲ ∑_{m<k<n} 2^{−2n+k} (∥∇F_{n}∥_{L_{t}^{2}L_{x}^{2}}N_{1}[G_{m}] + ∥∇F_{n}∥_{L_{t}^{2}L_{x}^{2}}∥∇G_{m}∥_{L_{t}^{2}L_{x}^{2}})

≲ N_{1}(F)N_{1}(G)

4 and

ĨII ≲ ∑_{m<k<n} ∥E_{k}(F_{n}(∇_{L}G)_{m})ds∥_{L_{t}^{1}L_{x}^{2}} ≲ ∑_{m<k<n} 2^{−2n}∥E_{k}(t^{2}ΔF_{n}(∇_{L}G)_{m})∥_{L_{t}^{1}L_{x}^{2}}

≲ ∑_{m<k<n} 2^{−2n} (∥E_{k}(t div (t∇F_{n} · (∇_{L}G)_{m}))∥_{L_{t}^{1}L_{x}^{2}} + ∥E_{k}(t^{2}(∇F_{n}) · ∇(∇_{L}G)_{m})∥_{L_{t}^{1}L_{x}^{2}})

≲ ∑_{m<k<n} 2^{−2n+k} (∥t∇F_{n} · (∇_{L}G)_{m}∥_{L_{t}^{1}L_{x}^{2}} + ∥t∇F_{n} · ∇(∇_{L}G)_{m}∥_{L_{t}^{1}L_{x}^{1}})

≲ ∑_{m<k<n} 2^{−2n+k} (∥t^{\frac{1}{2}} ∇F_{n}∥_{L_{t}^{2}L_{x}^{4}}∥t^{\frac{1}{2}} (∇_{L}G)_{m}∥_{L_{t}^{2}L_{x}^{4}} + ∥∇F_{n}∥_{L_{t}^{2}L_{x}^{2}}∥t∇(∇_{L}G)_{m}∥_{L_{t}^{2}L_{x}^{2}})

≲ ∑_{m<k<n} 2^{−2n+k+ \frac{n+m}{2}} N_{1}[F_{n}]∥(∇_{L}G)_{m}∥_{L_{t}^{2}L_{x}^{2}}

+ ∑_{m<k<n} 2^{−2n+k+m}∥∇F_{n}∥_{L_{t}^{2}L_{x}^{2}}∥(∇_{L}G)_{m}∥_{L_{t}^{2}L_{x}^{2}}

≲ N_{1}(F)N_{1}(G),

which give (5.32) and (5.33),

As to (5.34), with q < 2 sufficiently close to 2 we have from finite band property, Bernstein inequality, (5.25) and Lemma 5.3 that

ĨV ≲ ∑_{m<k<n} ∥E_{k}(F_{n} · [∇_{L}, P_{m}]G)∥_{L_{t}^{1}L_{x}^{2}} = ∑_{m<k<n} 2^{−2n}∥E_{k}(t^{2}ΔF_{n} · [∇_{L}, P_{m}]G)∥_{L_{t}^{1}L_{x}^{2}}

≲ ∑_{m<k<n} 2^{−2n}∥E_{k}(t^{2} div ∇F_{n} · [∇_{L}, P_{m}]G)∥_{L_{t}^{1}L_{x}^{2}}

≲ ∑_{m<k<n} 2^{−2n} (2^{k}∥t∇F_{n} · [∇_{L}, P_{m}]G∥_{L_{t}^{1}L_{x}^{2}} + 2^{k}∥t∇F_{n} · ∇[∇_{L}, P_{m}]G∥_{L_{t}^{1}L_{x}^{1}})

≲ ∑_{m<k<n} 2^{−2n+k}(∥t^{\frac{1}{q}} ∇F_{n}∥_{L_{t}^{q′}L_{x}^{4}}∥t^{1− \frac{1}{q}} [∇_{L}, P_{m}]G∥_{L_{t}^{q}L_{x}^{4}}

+ ∥t^{\frac{1}{q} − \frac{1}{2}} ∇F_{n}∥_{L_{t}^{q′}L_{x}^{2}}∥t^{\frac{3}{2} − \frac{1}{q}} ∇[∇_{L}, P_{m}]G∥_{L_{t}^{q}L_{x}^{2}})

≲ ∑_{m<k<n} 2^{−2n+k+ \frac{1}{q} n+(0+)m}N_{1}[F_{n}]N_{1}(G)

+ ∑_{m<k<n} 2^{−2n+k+2n( \frac{1}{q} − \frac{1}{2} )+m(1− \frac{1}{2} +)}N_{1}(F)N_{1}(G) ≲ N_{1}(F)N_{1}(G).

4The inequality ∥E_{k} div F ∥_{L_{x}^{2}} ≲ 2^{k}∥t^{−1}F ∥_{L_{x}^{2}} for any 1-forms F can be derived by dual argument.

Similarly, as to (5.35) we have

Ṽ ≲ ∑_{m<k<n} ∥E_{k}([P_{n}, ∇_{L}]F · G_{m})∥_{L_{t}^{1}L_{x}^{2}} ≲ ∑_{m<k<n} 2^{\frac{1}{2} k}∥t^{− \frac{1}{2}} [P_{n}, ∇_{L}]F · G_{m}∥_{L_{t}^{1}L \frac{4}{3} ; x}

≲ ∑_{m<k<n} 2^{\frac{1}{2} k}∥t^{− \frac{1}{2}} [P_{n}, ∇_{L}]F ∥_{L_{t}^{1}L_{x}^{2}}∥G∥_{L_{t}^{∞}L_{x}^{4}}

≲ ∑_{m<k<n} 2^{\frac{1}{2} k−(n−)}N_{1}(G)N_{1}(F) ≲ N_{1}(G)N_{1}(F).

This completes the estimate for B_{k}^{2}.

5.1.6 Estiamtes of A_{k}

The estimation for A_{k} follows from Bernstein inequality, the finite band property and Lemma 5.4. We will provide the estimation of A_{k}^{1} first. Recall for k > 0 fixed, A_{k}^{1} = ∑_{k<n<m} ∫_{0}^{t} E_{k}(s(∇_{L}F)_{n} · G_{m})ds, then

A_{k}^{1} ≲ ∑_{k<n<m} ∥E_{k}((∇_{L}F)_{n} · G_{m})∥_{L_{t}^{1}L_{x}^{2}} ≲ ∑_{k<n<m} 2^{k}∥(∇_{L}F)_{n}∥_{L_{t}^{2}L_{x}^{2}}∥t^{−1}G_{m}∥_{L_{t}^{2}L_{x}^{2}}

≲ ∑_{k<n<m} 2^{k−m}∥(∇_{L}F)_{n}∥_{L_{t}^{2}L_{x}^{2}}∥∇G_{m}∥_{L_{t}^{2}L_{x}^{2}},

which implies ∑_{k>0} A_{k}^{1} ≲ N_{1}(F)N_{1}(G).

The term A_{k}^{2} can be estimated in the following way

A_{k}^{2} ≲ ∑_{0<n<k<m} 2^{−2m}∥s^{2}E_{k}(∇_{L}F)_{n}ΔG_{m}∥_{L_{t}^{1}L_{x}^{2}}

≲ ∑_{0<n<k<m} 2^{−2m}∥s^{2}{E_{k} div ((∇_{L}F)_{n}∇G_{m}) − E_{k}(∇(∇_{L}F)_{n} · ∇G_{m})}∥_{L_{t}^{1}L_{x}^{2}}

≲ ∑_{0<n<k<m} 2^{−2m+k} {∥s(∇_{L}F)_{n} · ∇G_{m}∥_{L_{t}^{1}L_{x}^{2}} + ∥s∇(∇_{L}F)_{n}∥_{L_{t}^{2}L_{x}^{2}}∥∇G_{m}∥_{L_{t}^{2}L_{x}^{2}}}

≲ ∑_{0<n<k<m} 2^{−2m+k} \{∥s^{\frac{1}{2}} (∇_{L}F)_{n}∥_{L_{t}^{2}L_{x}^{4}}∥s^{\frac{1}{2}} ∇G_{m}∥_{L_{t}^{2}L_{x}^{4}} + 2^{n}∥(∇_{L}F)_{n}∥_{L_{t}^{2}L_{x}^{2}}∥∇G_{m}∥_{L_{t}^{2}L_{x}^{2}}\}

≲ ∑_{0<n<k<m} \{2^{− \frac{3}{2} m+k+ \frac{n}{2}} ∥(∇_{L}F)_{n}∥_{L_{t}^{2}L_{x}^{2}}N_{1}[G_{m}] + 2^{−2m+k+n}∥(∇_{L}F)_{n}∥_{L_{t}^{2}L_{x}^{2}}N_{1}[G_{m}]\}

which yields the inequality ∑_{k>0} A_{k}^{2} ≲ N_{1}(F)N_{1}(G).

At last we will estimate A_{k}^{3} = ∑_{0<n<m<k} ∥ ∫_{0}^{t}(E_{k}(s∇_{L}F)_{n} · G_{m})ds∥_{L_{t}^{∞}L_{x}^{2}},

A_{k}^{3} ≲ ∑_{0<n<m<k} 2^{−k}∥t∇((∇_{L}F)_{n} · G_{m})∥_{L_{t}^{1}L_{x}^{2}}

≲ ∑_{0<n<m<k} 2^{−k}∥t∇(∇_{L}F)_{n}∥_{L_{t}^{2}L_{x}^{2}}∥G_{m}∥_{L_{t}^{2}L_{x}^{∞}} + ∥(∇_{L}F)_{n} · t∇G_{m}∥_{L_{t}^{1}L_{x}^{2}}

≲ ∑_{0<n<m<k} \{2^{−k+n}∥(∇_{L}F)_{n}∥_{L_{t}^{2}L_{x}^{2}}N_{1}[G_{m}] + 2^{−k}∥t^{\frac{1}{2}} (∇_{L}F)_{n}∥_{L_{t}^{2}L_{x}^{4}}∥t^{\frac{1}{2}} ∇G_{m}∥_{L_{t}^{2}L_{x}^{4}}\}

≲ ∑_{0<n<m<k} (2^{−k+n} + 2^{−k+ \frac{1}{2} n+ \frac{1}{2} m}) N_{1}[F_{n}]N_{1}[G_{m}],

which implies ∑_{k>0} A_{k}^{3} ≲ N_{1}(F)N_{1}(G). Thus we conclude the proof of (5.6). ^{5}

5.2 Non-Sharp Product Estimates

The following non-sharp product estimates will be used in Chapter 7.

Lemma 5.5. For any S_{s} tangent tensors F and G,

∥F · G∥_{P^{0}} ≲ N_{1}(F)(∥G∥_{L_{t}^{a}L_{x}^{2}} + ∥∇G∥_{L_{t}^{2}L_{x}^{2}}), with a > 4  (5.36)
∥F · G∥_{P^{0}} ≲ N_{2}(t^{1/2}F)∥G∥_{P^{0}},  (5.37)
∥F · G∥_{P^{0}} ≲ N_{1}(t^{\frac{1}{2}} F)(∥∇G∥_{L_{t}^{2}L_{x}^{2}} + ∥G∥_{L_{ω}^{∞}L_{t}^{2}}).  (5.38)

In particular, for tensors F and G being of the same order, i.e. F · G are scalars. there holds

∥F · G∥_{P^{0}} ≲ N_{1}(t^{1/2}F)∥∇G∥_{L_{t}^{2}L_{x}^{2}}.  (5.39)

Remark 5.4. Observe that (5.38) is a consequence of (5.39), we will only prove (5.36), (5.37) and (5.39).

Proof. In what follows, we will frequently use Proposition 2.2 (iii) and Lemma 4.4. We will employ the inequalities which have been established in Section 5.1.2 (refer to (5.15), Remark 5.3 and Definition 5.1)

∥t^{1− \frac{1}{b}} ∇F_{m}∥_{L_{t}^{b}L_{x}^{4}} ≲ 2^{(1− \frac{1}{b})m}N_{1}[F_{m}], with 4 > b ≥ 2  (5.40)
∥t^{− \frac{1}{2}} F_{m}∥_{L_{x}^{2}L_{t}^{∞}} ≲ 2^{− \frac{m}{2}} N_{1}[F_{m}],  (5.41)
∥N_{1}[F_{m}]∥_{l^{2}} ≈ N_{1}(F).  (5.42)

We first prove (5.36).

5In the above calculation, we dropped the lower frequency term.

We begin with decomposing F ·G by geometric LP projections P_{k} such that ∑_{k} P_{k}^{2} = Id. We may drop the low frequency terms and get ^{6}

P_{k}(F · G) = ∑_{k>n,m>0} P_{k}(F_{m} · G_{n}) + ∑_{k<n,m} P_{k}(F_{m} · G_{n})

+ ( ∑_{n<k<m} + ∑_{m<k<n})P_{k}(F_{m} · G_{n}) = A_{[1]}^{k} + A_{[2]}^{k} + A_{[3]}^{k}.

By finite band property,

∥A_{[1]}^{k}∥_{L_{x}^{2}} ≤ ∑_{k>n,m>0} ∥P_{k}(F_{m} · G_{n})∥_{L_{x}^{2}} = ∑_{k>n,m>0} 2^{−2k}∥P_{k}Δ(F_{m} · G_{n})∥_{L_{x}^{2}}
≲ ∑_{k>n,m>0} 2^{−2k}∥P_{k}(ΔF_{m} · G_{n} + ΔG_{n} · F_{m} + ∇F_{m} · ∇G_{n})∥_{L_{x}^{2}}.  (5.43)

Consider the case k > n > m, using the weak Bernstein inequality of Proposition 2.2 (v) followed with the finite band property of Proposition 2.2 (iv), taking L_{t}^{2} norm, we get (5.43) is bounded by

∑_{k>n>m>0}(2^{−2k+k/2+2m} + 2^{−2k+k/2+2n})∥F_{m}∥_{L_{t}^{∞}L_{x}^{4}}∥t^{− \frac{1}{2}} G_{n}∥_{L_{t}^{2}L_{x}^{2}}
+ ∑_{k>n>m>0} 2^{−2k+k/2}∥t^{1− \frac{1}{b}} ∇F_{m}∥_{L_{t}^{b}L_{x}^{4}}∥t^{\frac{1}{b} + \frac{1}{2}} ∇G_{n}∥_{L_{t}^{b′}L_{x}^{2}}  (5.44)

∥F_{m}∥_{L_{t}^{∞}L_{x}^{4}} can be treated by using (3.39) and Proposition 2.2 (i) in the following way,

∥F_{m}∥_{L_{t}^{∞}L_{x}^{4}} ≲ ∥F ∥_{L_{t}^{∞}L_{x}^{4}} ≲ N_{1}(F).

Hence,

(5.44) ≲ ∑_{k>n>m}(2^{−3k/2+2m−n} + 2^{−3k/2+n})N_{1}(F)∥t^{1/2}∇G∥_{L_{t}^{2}L_{x}^{2}}  (5.45)
+ ∑_{k>n>m} 2^{− \frac{3k}{2}} ∥t^{1− \frac{1}{b}} ∇F_{m}∥_{L_{t}^{b}L_{x}^{4}}∥t^{\frac{1}{2} + \frac{1}{b}} ∇G_{n}∥_{L_{t}^{b′}L_{x}^{2}}  (5.46)

where \frac{1}{b} + \frac{1}{b′} = \frac{1}{2}. The term in (5.45) can be bounded by N_{1}(F)∥t^{1/2}∇G∥_{L_{t}^{2}L_{x}^{2}}. As to (5.46), we will employ the following two inequalities. Set Π_{n}^{λ} = ∥∇G_{n}∥_{L_{t}^{2}L_{x}^{2}}^{λ}∥G∥_{L_{t}^{a}L_{x}^{2}}^{1−λ}, where a > 2 and 0 ≤ λ ≤ 1 are some numbers to be chosen. By interpolation,

∥t^{1−λ}∇G_{n}∥_{L_{t}^{b′}L_{x}^{2}} ≲ ∥∇G_{n}∥_{L_{t}^{2}L_{x}^{2}}^{λ}∥∇G_{n}∥_{L_{t}^{a}L_{x}^{2}}^{1−λ} ≲ 2^{(1−λ)n}Π_{n}^{λ},  (5.47)

where \frac{1−λ}{a} + \frac{λ}{2} = \frac{1}{b′}. Similarly, we have

∥t^{−λ}G_{n}∥_{L_{t}^{b′}L_{x}^{2}} ≲ 2^{−λn}Π_{n}^{λ}.  (5.48)

6By k > n, m, we mean k > n and k > m. We also ignored the terms with k = n, k = m, n = m.

Combine \frac{1−λ}{a} + \frac{λ}{2} = \frac{1}{b′} with \frac{1}{b} + \frac{1}{b′} = \frac{1}{2}, we get

\frac{1 − λ}{a} + \frac{λ}{2} = \frac{1}{2} − \frac{1}{b}.  (5.49)

Due to the condition 2 ≤ b < 4 as required in (5.40), we infer from (5.49) that \frac{1}{a} < \frac{\frac{1}{4} − \frac{λ}{2}}{1−λ}. This implies that a > 4. Now let λ and b verify the condition

\frac{1}{2} − λ − \frac{1}{b} < 0.  (5.50)

By (5.49), it is identical to

\frac{1}{a} < \frac{λ}{2 − 2λ}.  (5.51)

In what follows, we choose a > 4 and 0 < λ < 1 verifying (5.51).

Using (5.40) followed by Young’s inequality, (5.46) can be bounded by

∑_{k>n>m>0} 2^{(1− \frac{1}{b})m+(1−λ)n− \frac{3}{2} k}N_{1}[F_{m}]Π_{n}^{λ} ≲ N_{1}(F)(∥∇G∥_{L_{t}^{2}L_{x}^{2}} + ∥G∥_{L_{t}^{a}L_{x}^{2}}).

In the case k > m > n, we use the weak Bernstein inequality and the finite band property followed with (5.40), (5.47) and (5.48) to obtain

∥P_{k}(F_{m} · G_{n})∥_{L_{t}^{2}L_{x}^{2}} ≲ 2^{−2k}∥t^{2}P_{k}Δ(F_{m} · G_{n})∥_{L_{t}^{2}L_{x}^{2}}

≲ 2^{− \frac{3}{2} k}∥t^{\frac{3}{2}} ∇F_{m} · ∇G_{n}∥_{L_{t}^{2}L_{x}^{4/3}} + (2^{−2k+2m} + 2^{−2k+2n})∥P_{k}(F_{m} · G_{n})∥_{L_{t}^{2}L_{x}^{2}} ≲ 2^{− \frac{3}{2} k}∥t^{\frac{3}{2}} ∇F_{m}∥_{L_{t}^{b}L_{x}^{4}}∥∇G_{n}∥_{L_{t}^{b′}L_{x}^{2}} + 2^{−2k+2m}∥t^{− \frac{1}{b}} F_{m}∥_{L_{t}^{b}L_{x}^{2}}∥t^{\frac{1}{b}} G_{n}∥_{L_{t}^{b′}L_{x}^{4}}

+ 2^{− \frac{3}{2} k+2n}∥t^{− \frac{1}{2} − \frac{1}{b}} F_{m}∥_{L_{t}^{b}L_{x}^{2}}∥t^{\frac{1}{b}} G_{n}∥_{L_{t}^{b′}L_{x}^{4}}

≲ (2^{− \frac{3}{2} k}2^{(1− \frac{1}{b})m}2^{(1−λ)n} + 2^{−2k+2m− \frac{m}{b} + \frac{n}{2} −nλ} + 2^{− \frac{3k}{2} − \frac{m}{2} − \frac{m}{b} −λn+n \frac{5}{2}} )N_{1}[F_{m}]Π_{n}^{λ}.

Thus, due to \frac{1}{2} − λ − \frac{1}{b} < 0,

∑_{k>m>n} ∥P_{k}(F_{m} · G_{n})∥_{L_{t}^{2}L_{x}^{2}} ≲ N_{1}(F)(∥∇G∥_{L_{t}^{2}L_{x}^{2}} + ∥G∥_{L_{t}^{a}L_{x}^{2}}),

Therefore we conclude

∑_{k>0} ∥A_{[1]}^{k}∥_{L_{t}^{2}L_{x}^{2}} ≲ N_{1}(F)(∥∇G∥_{L_{t}^{2}L_{x}^{2}} + ∥G∥_{L_{t}^{a}L_{x}^{2}}).

As to A_{[2]}^{k}, when k < m < n, using Hölder’s inequality followed with Proposition 2.2 (i), the finite band property of Proposition 2.2 (iv) and (3.39), we obtain

∥P_{k}(F_{m} · G_{n})∥_{L_{t}^{2}L_{x}^{2}} ≲ 2^{\frac{k}{2}} ∥t^{−1/2}F_{m} · G_{n}∥_{L_{t}^{2}L_{x}^{4/3}} ≲ 2^{\frac{1}{2} k}∥F_{m}∥_{L_{t}^{∞}L_{x}^{4}}∥t^{−1/2}G_{n}∥_{L_{t}^{2}L_{x}^{2}} (5.52)
≲ 2^{\frac{k}{2} −n}N_{1}(F)∥t^{1/2}∇G∥_{L_{t}^{2}L_{x}^{2}}.

When k < n < m, using the weak Bernstein inequality and (5.41), we have

∥P_{k}(F_{m} · G_{n})∥_{L_{t}^{2}L_{x}^{2}} ≲ 2^{\frac{k}{2}} ∥t^{−1/2}F_{m}∥_{L_{t}^{∞}L_{x}^{2}}∥G_{n}∥_{L_{t}^{2}L_{x}^{4}} ≲ 2^{\frac{k−m−n}{2}} N_{1}[F_{m}]∥t^{1/2}∇G∥_{L_{t}^{2}L_{x}^{2}}.

Summing over k < m < n or k < n < m for the respective case, we conclude

∑_{k>0} ∥A_{[2]}^{k}∥_{L_{t}^{2}L_{x}^{2}} ≲ N_{1}(F)∥∇G∥_{L_{t}^{2}L_{x}^{2}}.

Now we estimate A_{[3]}^{k}. We first consider the case n < k < m,

∥P_{k}(F_{m} · G_{n})∥_{L_{t}^{2}L_{x}^{2}} ≲ 2^{−2m}∥t^{2}P_{k}(ΔF_{m} · G_{n})∥_{L_{t}^{2}L_{x}^{2}}
≲ 2^{−2m}(∥t^{2}P_{k}∇(∇F_{m} · G_{n})∥_{L_{t}^{2}L_{x}^{2}} + ∥t^{2}P_{k}(∇F_{m} · ∇G_{n})∥_{L_{t}^{2}L_{x}^{2}})
≲ 2^{−2m}(2^{k}∥t∇F_{m} · G_{n}∥_{L_{t}^{2}L_{x}^{2}} + 2^{\frac{k}{2}} ∥t^{\frac{3}{2}} ∇F_{m} · ∇G_{n}∥_{L_{t}^{2}L_{x}^{4/3}})

≲ 2^{−2m+k}∥t^{1− \frac{1}{b}} ∇F_{m}∥_{L_{t}^{b}L_{x}^{4}}∥t^{\frac{1}{b}} G_{n}∥_{L_{t}^{b′}L_{x}^{4}} + 2^{−2m+ \frac{k}{2}} ∥t^{1− \frac{1}{b}} ∇F_{m}∥_{L_{t}^{b}L_{x}^{4}}∥t^{\frac{1}{b} + \frac{1}{2}} ∇G_{n}∥_{L_{t}^{b′}L_{x}^{2}}

≲ (2^{−2m+k+(1− \frac{1}{b})m+ \frac{n}{2} −λn} + 2^{−2m+ \frac{k}{2} +(1− \frac{1}{b})m+(1−λ)n})N_{1}[F_{m}]Π_{n}^{λ}.

Summing over n < k < m implies

∑_{n>k>m} ∥P_{k}(F_{m} · G_{n})∥_{L_{t}^{2}L_{x}^{2}} ≲ N_{1}(F)(∥∇G∥_{L_{t}^{2}L_{x}^{2}} + ∥G∥_{L_{t}^{a}L_{x}^{2}}).

It remains to consider the last case n > k > m with 1 < p < 2, \frac{1}{p∗} + \frac{1}{2} = \frac{1}{p}. Note that by the finite band property and the weak Bernstein inequality, we can obtain

∥P_{k}(F_{m} · G_{n})∥_{L_{x}^{2}} ≲ 2^{k( \frac{2}{p} −1)}t^{1− \frac{2}{p}} ∥F_{m} · G_{n}∥_{L_{x}^{p}} ≲ 2^{k( \frac{2}{p} −1)}t^{1− \frac{2}{p}} ∥F_{m}∥_{L_{x}^{p∗}}∥G_{n}∥_{L_{x}^{2}}
≲ 2^{k( \frac{2}{p} −1)}2^{m(1− \frac{2}{p∗})−n}∥F_{m}∥_{L_{x}^{2}}∥∇G∥_{L_{x}^{2}}.

By using (5.41) and (5.42), we infer

∑_{n>k>m} ∥P_{k}(F_{m} · G_{n})∥_{L_{t}^{2}L_{x}^{2}} ≲ ∑_{n>k>m} 2^{k( \frac{2}{p} −1)}2^{m(2− \frac{2}{p})−n− \frac{m}{2}} N_{1}[F_{m}]∥t^{1/2}∇G∥_{L_{t}^{2}L_{x}^{2}}
≲ N_{1}(F)∥∇G∥_{L_{t}^{2}L_{x}^{2}}.

Therefore ∑_{k>0} ∥A_{[3]}^{k}∥_{L_{t}^{2}L_{x}^{2}} ≲ N_{1}(F)(∥∇G∥_{L_{t}^{2}L_{x}^{2}} + ∥G∥_{L_{t}^{a}L_{x}^{2}}).

Combining all the above estimates we conclude

∥F · G∥_{P^{0}} ≲ N_{1}(F)(∥G∥_{L_{t}^{a}L_{x}^{2}} + ∥∇G∥_{L_{t}^{2}L_{x}^{2}}).

The proof of (5.36) is therefore complete.

Next, we prove (5.37). It suffices to show it for tensor fields F and G being of the same order. Note

P_{k}(F · G) = ∑_{k<m} P_{k}(F · G_{m}) + ∑_{k>m} P_{k}(F · G_{m}),

we drop the low frequency terms and consider the terms ∑_{k>m>0} P_{k}(F · G_{m}) and ∑_{0<k<m} P_{k}(F · G_{m}). First by the finite band property, followed by Bernstein inequality and (3.39), we get for the case k ≥ m > 0

∥P_{k}(F · G_{m})∥_{L_{t}^{2}L_{x}^{2}} ≲ 2^{−k}{∥t∇F · G_{m}∥_{L_{t}^{2}L_{x}^{2}} + ∥tF · ∇G_{m}∥_{L_{t}^{2}L_{x}^{2}}}
≲ 2^{−k}\{∥t^{1/2}∇F ∥_{L_{t}^{∞}L_{x}^{4}}∥t^{\frac{1}{2}} G_{m}∥_{L_{t}^{2}L_{x}^{4}} + ∥F ∥_{L_{t}^{∞}L_{x}^{∞}}∥t∇G_{m}∥_{L_{t}^{2}L_{x}^{2}}\}
≲ 2^{−k+ \frac{m}{2}} N_{2}(t^{1/2}F)∥G_{m}∥_{L^{2}(H)} + 2^{−k+m}N_{2}(t^{1/2}F)∥G_{m}∥_{L^{2}(H)}.

Summing over k ≥ m > 0 yields

∑_{k≥m>0} ∥P_{k}(F · G_{m})∥_{L_{t}^{2}L_{x}^{2}} ≲ ∑_{k≥m>0} 2^{−k+m}N_{2}(t^{1/2}F)∥G_{m}∥_{L_{t}^{2}L_{x}^{2}} ≲ N_{2}(t^{1/2}F)∥G∥_{P^{0}}.

Then we consider the case 0 < k < m by using sharp Bernstein inequality, finite band property and (3.39),

∥P_{k}(F · G_{m})∥_{L_{t}^{2}L_{x}^{2}} ≲ 2^{−2m}∥t^{2}P_{k}(F · ΔG_{m})∥_{L_{t}^{2}L_{x}^{2}}

≲ 2^{−2m}∥t^{2}P_{k} div (F · ∇G_{m})∥_{L_{t}^{2}L_{x}^{2}} + 2^{−2m}∥t^{2}P_{k}(∇F · ∇G_{m})∥_{L_{t}^{2}L_{x}^{2}}

≲ 2^{−2m+k}∥tF · ∇G_{m}∥_{L_{t}^{2}L_{x}^{2}} + 2^{−2m+k}∥t∇F · ∇G_{m}∥_{L_{t}^{2}L_{x}^{1}}
≲ 2^{−2m+k+m}(∥F ∥_{L_{t}^{∞}L_{x}^{∞}}∥G_{m}∥_{L_{t}^{2}L_{x}^{2}} + ∥∇F ∥_{L_{t}^{∞}L_{x}^{2}}∥G_{m}∥_{L_{t}^{2}L_{x}^{2}}).

Summing over 0 < k < m gives ∑_{0<k<m} ∥P_{k}(F · G_{m})∥_{L_{t}^{2}L_{x}^{2}} ≲ N_{2}(t^{1/2}F)∥G∥_{P^{0}}. Thus we complete the proof of (5.37).

Set C_{mnk} = ∥P_{m}(F_{n}·G_{k})∥_{L_{t}^{2}L_{x}^{2}}. To prove (5.39), it suffices to estimate ∑_{m,n,k>0} C_{mnk}. When k > m, by using sharp Bernstein inequality followed by (5.41) and finite band property, we have

C_{mnk} ≲ 2^{m}∥t^{−1}F_{n} · G_{k}∥_{L_{t}^{2}L_{x}^{1}} ≲ 2^{m−k}∥F_{n}∥_{L_{t}^{∞}L_{x}^{2}}∥∇G∥_{L_{t}^{2}L_{x}^{2}}
≲ 2^{m−k− \frac{n}{2}} N_{1}[t^{1/2}F_{n}]∥∇G∥_{L_{t}^{2}L_{x}^{2}},

hence ∑_{n>k>m} C_{mnk} ≲ N_{1}(t^{1/2}F)∥∇G∥_{L_{t}^{2}L_{x}^{2}}. Similarly, we can obtain

∑_{k>n>m} C_{mnk} ≲ N_{1}(t^{1/2}F)∥∇G∥_{L_{t}^{2}L_{x}^{2}}.

For the case k > m > n, we use finite band property for Laplacian followed by (2.7) and Hölder inequality to get

C_{mnk} ≲ 2^{−2k}∥t^{2}P_{m}(F_{n} · ΔP_{k}G)∥_{L_{t}^{2}L_{x}^{2}}
≲ 2^{−2k}∥t^{2}P_{m} div (F_{n} · ∇P_{k}G)∥_{L_{t}^{2}L_{x}^{2}} + 2^{−2k}∥t^{2}P_{m}(∇F_{n} · ∇P_{k}G)∥_{L_{t}^{2}L_{x}^{2}}
≲ 2^{−2k+ \frac{3}{2} m}∥t^{1/2}F_{n} · ∇P_{k}G∥_{L_{t}^{2}L_{x}^{4/3}} + 2^{−2k+m}∥t∇F_{n} · ∇P_{k}G∥_{L_{t}^{2}L_{x}^{1}}
≲ 2^{−2k+ \frac{3}{2} m}∥t^{1/2}F_{n}∥_{L_{t}^{∞}L_{x}^{4}}∥∇G∥_{L_{t}^{2}L_{x}^{2}} + 2^{−2k+m}∥t∇F_{n}∥_{L_{t}^{∞}L_{x}^{2}}∥∇G∥_{L_{t}^{2}L_{x}^{2}}
≲ 2^{−2k+ \frac{3}{2} m}N_{1}(t^{1/2}F)∥∇G∥_{L_{t}^{2}L_{x}^{2}},

which implies

∑_{k>m>n} C_{mnk} ≲ N_{1}(t^{1/2}F)∥∇G∥_{L_{t}^{2}L_{x}^{2}}.

When m > k > n, by finite band property,

C_{mnk} ≲ 2^{−2m}∥t^{2}P_{m}Δ(F_{n} · G_{k})∥_{L_{t}^{2}L_{x}^{2}}
≲ 2^{−2m}(∥t^{2}P_{m}(ΔF_{n} · G_{k})∥_{L_{t}^{2}L_{x}^{2}} + ∥t^{2}P_{m}(F_{n} · ΔP_{k}G)∥_{L_{t}^{2}L_{x}^{2}}
+ ∥t^{2}P_{m}(∇F_{n} · ∇P_{k}G)∥_{L_{t}^{2}L_{x}^{2}}).  (5.53)

For the first term of (5.53), we employ finite band property and sharp Bernstein inequality to obtain

2^{−2m}∥t^{2}P_{m}(ΔF_{n} · G_{k})∥_{L_{t}^{2}L_{x}^{2}} ≲ 2^{−2m+m+2n}∥F_{n}∥_{L_{t}^{∞}L_{x}^{2}}∥t^{−1}P_{k}G∥_{L_{t}^{2}L_{x}^{2}}
≲ 2^{−m+ \frac{3n}{2} −k}N_{1}(t^{1/2}F)∥∇G∥_{L_{t}^{2}L_{x}^{2}}.

For the second term of (5.53), using finite band property and weak Bernstein inequality, we can get the following estimate

2^{−2m}∥t^{2}P_{m}(F_{n} · ΔP_{k}G)∥_{L_{t}^{2}L_{x}^{2}} ≲ 2^{−2m+k+ \frac{m}{2}} ∥t^{1/2}F_{n}∥_{L_{t}^{∞}L_{x}^{4}}∥∇G∥_{L_{t}^{2}L_{x}^{2}} ≲ 2^{− \frac{3m}{2} +k}N_{1}(t^{\frac{1}{2}} F)∥∇G∥_{L_{t}^{2}L_{x}^{2}}.

For the last term of (5.53) we obtain by using sharp Bernstein inequality

2^{−2m}∥t^{2}P_{m}(∇F_{n} · ∇P_{k}G)∥_{L_{t}^{2}L_{x}^{2}} ≲ 2^{−2m+m}∥t∇F_{n}∥_{L_{t}^{∞}L_{x}^{2}}∥∇P_{k}G∥_{L_{t}^{2}L_{x}^{2}}
≲ 2^{−m+ \frac{n}{2}} N_{1}(t^{1/2}F)∥∇G∥_{L_{t}^{2}L_{x}^{2}}.

Combining the above estimates for (5.53), we conclude that

∑_{m>k>n} C_{mnk} ≲ N_{1}(t^{1/2}F)∥∇G∥_{L_{t}^{2}L_{x}^{2}}.

Now we consider the case when n > m > k.

C_{mnk} ≲ 2^{−2n}∥t^{2}P_{m}(ΔF_{n} · G_{k})∥_{L_{t}^{2}L_{x}^{2}}
≲ 2^{−2n}(∥t^{2}P_{m} div (∇F_{n} · G_{k})∥_{L_{t}^{2}L_{x}^{2}} + ∥t^{2}P_{m}(∇F_{n} · ∇P_{k}G)∥_{L_{t}^{2}L_{x}^{2}}).  (5.54)

For the first term in (5.54), we use (2.7) to obtain

2^{−2n+ \frac{2}{p} m}∥t∇F_{n}∥_{L_{t}^{∞}L_{x}^{2}}∥t^{1− \frac{2}{p}} P_{k}G∥_{L_{t}^{2}L_{x}^{p∗}}
≲ 2^{−2n+ \frac{2}{p} m+ \frac{n}{2} − \frac{2}{p∗} k}N_{1}[t^{1/2}F_{n}]∥∇G∥_{L_{t}^{2}L_{x}^{2}},

where \frac{1}{p} = \frac{1}{2} + \frac{1}{p∗} and 2 > p > 4/3.

As to the second term in (5.54), we employ sharp Bernstein inequality to derive

2^{−2n}∥t^{2}P_{m}(∇F_{n} · ∇P_{k}G)∥_{L_{t}^{2}L_{x}^{2}} ≲ 2^{−2n+m}∥t∇F_{n}∥_{L_{t}^{∞}L_{x}^{2}}∥∇P_{k}G∥_{L_{t}^{2}L_{x}^{2}}
≲ 2^{− \frac{3}{2} n+m}N_{1}(t^{1/2}F)∥∇G∥_{L_{t}^{2}L_{x}^{2}}.

Combining the above two estimates for (5.54), we obtain

∑_{n>m>k} C_{mnk} ≲ N_{1}(t^{1/2}F)∥∇G∥_{L_{t}^{2}L_{x}^{2}}.

When m > n > k, we derive

C_{mnk} ≲ 2^{−2m}∥t^{2}P_{m}Δ(F_{n} · G_{k})∥_{L_{t}^{2}L_{x}^{2}}
≲ 2^{−2m}(∥t^{2}P_{m}(ΔF_{n} · G_{k})∥_{L_{t}^{2}L_{x}^{2}} + ∥t^{2}P_{m}(F_{n} · ΔP_{k}G)∥_{L_{t}^{2}L_{x}^{2}}
+ ∥t^{2}P_{m}(∇F_{n} · ∇P_{k}G)∥_{L_{t}^{2}L_{x}^{2}}).  (5.55)

As to the first term of (5.55), with \frac{1}{p∗} + \frac{1}{p} = \frac{1}{2} and p∗ < 4, we obtain

2^{−2m+2n}∥F_{n}∥_{L_{t}^{∞}L_{x}^{p∗}}∥P_{k}G∥_{L_{t}^{2}L_{x}^{p}} ≲ 2^{−2m+2n+ \frac{n}{2} (1− \frac{4}{p∗})−k \frac{2}{p∗}} N_{1}(t^{1/2}F)∥∇G∥_{L_{t}^{2}L_{x}^{2}}. For the second term of (5.55), we derive by using finite band property and sharp Bernstein inequality that

2^{−2m}∥t^{2}P_{m}(F_{n} · ΔP_{k}G)∥_{L_{t}^{2}L_{x}^{2}} ≲ 2^{−m}∥tF_{n} · P_{k}G∥_{L_{t}^{2}L_{x}^{1}}
≲ 2^{−2m+m+k}∥tF_{n}∥_{L_{t}^{∞}L_{x}^{2}}∥∇G∥_{L_{t}^{2}L_{x}^{2}}
≲ 2^{−m− \frac{n}{2} +k}N_{1}(t^{1/2}F)∥∇G∥_{L_{t}^{2}L_{x}^{2}}.

For the last term, by sharp Bernstein inequality, we have

2^{−2m+m}∥t∇F_{n}∥_{L_{t}^{∞}L_{x}^{2}}∥∇P_{k}G∥_{L_{t}^{2}L_{x}^{2}} ≲ 2^{−m+ \frac{n}{2}} N_{1}(t^{1/2}F)∥∇G∥_{L_{t}^{2}L_{x}^{2}}.

Summing the above three estimates over m > n > k, we conclude

∑_{m>n>k} C_{mnk} ≲ N_{1}(t^{1/2}F)∥∇G∥_{L_{t}^{2}L_{x}^{2}}.

Thus we complete the proof of (5.39).

Elliptic P^{σ} Estimates of Hodge operators on H

In this chapter we provide P^{σ} estimates for 0–order Hodge operators. Throughout this chapter, we denote \underline{K}_{γ_{0}} = ∥Λ^{−γ_{0}}(K − s^{−2})∥_{L_{t}^{∞}L_{x}^{2}} and \underline{K} = K − s^{−2}, with a constant 1 > γ_{0} > 1/2 and K being the Gaussian curvature on S_{s}.

6.1 Preliminary Estimates

In this section we provide some preliminary estimates which are important for the next section, where the main theorem of this chapter is presented and proved. These preliminary estimates are also frequently used in Chapter 7.

We begin with a basic estimate in the norm of L^{2}(S).

Proposition 6.1. For any S-tangent tensor F on H, and 1 < p ≤ 2,

∥D^{−1}F ∥_{L^{2}(S)} ≲ ∥s^{2− \frac{2}{p}} F ∥_{L^{p}(S)},

where D represents one of the operators D_{1}, D_{2}, ^{⋆}D_{1} and ^{⋆}D_{2}.

Proof. We first note that

⟨D^{−1}F, ψ⟩ = ⟨F,^{⋆}D^{−1}ψ⟩ = ⟨s^{\frac{2}{p} −2}^{⋆}D^{−1}ψ, s^{2− \frac{2}{p}} F ⟩ ≲ ∥s^{\frac{2}{p} −2}^{⋆}D^{−1}ψ∥_{L^{p′}(S)}∥s^{2− \frac{2}{p}} F ∥_{L^{p}(S)}.

Since the Gagliardo-Nirenberg inequality implies

∥s^{\frac{2}{p} −2}^{⋆}D^{−1}ψ∥_{L^{p′}(S)} ≲ ∥∇^{⋆}D^{−1}ψ∥_{L^{2}(S)}^{1− \frac{2}{p′}}∥s^{−1}^{⋆}D^{−1}ψ∥L^{2}(S)^{\frac{2}{p′}} + ∥s^{−1}D^{−1}ψ∥_{L^{2}(S)}

then we infer from Proposition 4.2 that

∥s^{\frac{2}{p} −2}^{⋆}D^{−1}ψ∥_{L^{p′}(S)} ≲ ∥ψ∥_{L^{2}(S)}.

Consequently, we have for any appropriate ψ, the inequality

⟨D^{−1}F, ψ⟩ ≲ ∥ψ∥_{L^{2}(S)}∥s^{2− \frac{2}{p}} F ∥_{L^{p}(S)}

as desired.

Lemma 6.1. Let D denote one of the Hodge operators D_{1}, D_{2}, ^{⋆}D_{1} and ^{⋆}D_{2}, let D^{−1} denote the inverse of D, for P_{k}F with P_{k} the geometric Littlewood Paley decomposition associated to the heat equation (3.33),

∥D^{−1}P_{k}F ∥_{L_{x}^{2}}, ∥P_{k}D^{−1}F ∥_{L_{x}^{2}} ≲ 2^{−k}∥sF ∥_{L_{x}^{2}} and ∥P_{k}D^{−1}F ∥_{L_{x}^{2}} ≲ 2^{−(2− \frac{2}{p})k}∥s^{2− \frac{2}{p}} F ∥_{L_{x}^{p}}.

Proof of Lemma 6.1. Let ^{⋆}D denote the L^{2} adjoint of D. We prove the first inequality by duality, for appropriate tensor fields g

∥P_{k}^{⋆}D^{−1}g∥_{L_{x}^{2}} ≲ 2^{−2k}∥s^{2}P_{k}Δ^{⋆}D^{−1}g∥_{L^{2}}
= 2^{−2k}∥s^{2}P_{k} div ∇^{⋆}D^{−1}g∥_{L_{x}^{2}}
≲ 2^{−k}∥s∇^{⋆}D^{−1}g∥_{L_{x}^{2}} ≲ 2^{−k}∥sg∥_{L_{x}^{2}},

where we relied on the estimates

∥P_{k} div g∥_{L_{x}^{2}} ≲ 2^{k}∥s^{−1}g∥_{L_{x}^{2}}, ∥∇^{⋆}D^{−1}g∥_{L_{x}^{2}} ≲ ∥g∥_{L_{x}^{2}}.

The second inequality in Lemma 6.1 follows from the finite band property and the elliptic estimate, ∥P_{k}D^{−1}F ∥_{L_{x}^{2}} ≲ 2^{−k}∥s∇D^{−1}F ∥_{L_{x}^{2}} ≲ 2^{−k}∥sF ∥_{L_{x}^{2}}. Now we show the last inequality in Lemma 6.1. By duality, it suffices to show for appropriate tensor fields G,

∥^{⋆}D^{−1}P_{k}G∥_{L_{x}^{p′}} ≲ 2^{−(2− \frac{2}{p})k}∥s^{2− \frac{2}{p}} G∥_{L_{x}^{2}},

where \frac{1}{p′} + \frac{1}{p} = 1. It can be obtained by using Sobolev inequality and ∥^{⋆}D^{−1}P_{k}G∥_{L_{x}^{2}} ≲ 2^{−k}s∥G∥_{L_{x}^{2}},

∥^{⋆}D^{−1}P_{k}G∥_{L_{x}^{p′}} ≲ ∥∇^{⋆}D^{−1}P_{k}G∥_{L_{x}^{2}}^{1− \frac{2}{p′}} ∥^{⋆}D^{−1}P_{k}G∥L_{x}^{2}^{\frac{2}{p′}} + s^{−1+ \frac{2}{p′}} ∥D^{−1}P_{k}G∥_{L_{x}^{2}}
≲ s^{\frac{2}{p′}} 2^{− \frac{2}{p′} k}∥G∥_{L_{x}^{2}}.

The following proposition follows from the last estimate in Lemma 6.1 immediately.

Proposition 6.2. Let D^{−1} denote either D_{1}^{−1}, ^{⋆}D_{1}^{−1}, D_{2}^{−1}, then for appropriate Stangent tensor fields F on H and any 1 < p ≤ 2,

∥D^{−1}F ∥_{B_{2,1}^{0}(S)} ≲ ∥s^{2− \frac{2}{p}} F ∥_{L^{p}(S)}.  (6.1)

Also along H,

∥D^{−1}F ∥_{B^{0}} ≲ ∥t^{2− \frac{2}{p}} F ∥_{L_{t}^{∞}L_{x}^{p}}  (6.2)

Moreover for 0 ≤ θ < \frac{1}{2} and \frac{2}{2−θ} < p ≤ 2,

∥D^{−1}F ∥_{P^{θ}} ≲ ∥t^{2− \frac{2}{p} −θ}F ∥_{L_{t}^{2}L_{x}^{p}}.  (6.3)

We record a simple lemma about Besov estimates as follows

Lemma 6.2. For arbitrary S_{s} tangent tensor fields F, ^{1}

∥F ∥_{B_{2,1}^{0}} ≲ ∥t∇F ∥_{L_{x}^{2}} + ∥F ∥_{L_{x}^{2}},
∥F ∥_{P^{0}} ≲ ∥t∇F ∥_{L^{2}(H)} + ∥F ∥_{L^{2}(H)},
∥F ∥_{B^{0}} ≲ ∥t∇F ∥_{L_{t}^{∞}L_{x}^{2}} + ∥F ∥_{L_{t}^{∞}L_{x}^{2}},

and

∥F ∥_{B^{0}} ≲ N_{1}(t^{1/2}F).

Indeed the estimates in the first line follows from the finite band property that ∥P_{k}F ∥_{L_{x}^{2}} ≲ 2^{−k}∥t∇F ∥_{L_{x}^{2}}. The inequality in the second line is an immediate consequence of (5.41).

The following lemma is used frequently in the next section.

Lemma 6.3. For any smooth m-covariant S-tangent tensor F and a constant 0 ≤ γ ≤ 1,

∥P_{k}∇Λ^{γ}F ∥_{L_{x}^{2}} ≲ 2^{(1+γ)k}s^{−1−γ}∥F ∥_{L_{x}^{2}} + 2^{k}s^{−1}∥\underline{K}∥_{L_{x}^{2}}^{γ}∥F ∥_{L_{x}^{2}}  (6.4)

and

∥Λ^{γ}∇P_{k}F ∥_{L_{x}^{2}} ≲ 2^{(1+γ)k}s^{−1−γ}∥F ∥_{L_{x}^{2}} + 2^{k}s^{−1}∥\underline{K}∥_{L_{x}^{2}}^{γ}∥F ∥_{L_{x}^{2}}.  (6.5)

Proof of Lemma 6.3. We now recall that the Böchner identity gives ^{2}

∥∇^{2}P_{k}G∥_{L_{x}^{2}} ≲ (2^{2k}s^{−2} + 2^{k}s^{−1}∥\underline{K}∥_{L_{x}^{2}})∥P_{k}G∥_{L_{x}^{2}}  (6.6)

By interpolation, combining (6.6) with the inequality,

∥∇P_{k}G∥_{L_{x}^{2}} ≲ 2^{k}s^{−1}∥P_{k}G∥_{L_{x}^{2}},

we can conclude (6.5) holds true for any 0 ≤ γ ≤ 1. With its aid, (6.4) can be obtained by duality.

1When using Lemma 6.2, we may ignore the lower order term.

2The actual estimate is slightly worse.

As a consequence of Lemma 6.3, we obtain the following result

Corollary 6.1. The following inequality holds for appropriate tensor fields F, for l > m and γ > 1/2

∥∇P_{l}∇P_{m}F ∥_{L_{x}^{2}} ≲ 2^{(1−γ)l}(2^{(1+γ)m}s^{−2} + 2^{m}s^{−2+γ}∥\underline{K}∥_{L_{x}^{2}}^{γ})∥P_{m}F ∥_{L_{x}^{2}}.  (6.7)

Proof of Corollary 6.1.

∥∇P_{l}∇P_{m}F ∥_{L_{x}^{2}} = ∥∇Λ^{−γ}P_{l}Λ^{γ}∇P_{m}F ∥_{L_{x}^{2}}
≲ ∥∇Λ^{−γ}P_{l}∥_{L_{x}^{2}→L_{x}^{2}} · ∥Λ^{γ}∇P_{m}F ∥_{L_{x}^{2}}  (6.8)

It is straightforward to get

∥∇Λ^{−γ}P_{l}F ∥_{L_{x}^{2}} ≲ 2^{l(1−γ)}s^{γ−1}∥F ∥_{L_{x}^{2}}.  (6.9)

Combining (6.8), (6.5) and (6.9), we get (6.7) as desired.

6.2 Hodge-Elliptic P^{σ} Estimates

The result in this section relies on Proposition 4.5 and Proposition 4.6, i.e.

\underline{K}_{γ_{0}}, ∥\underline{K}∥_{L^{2}(H)} ≲ Δ_{0} with γ_{0} > 1/2.  (6.10)

Now we state the main theorem of this chapter. ^{3}

Theorem 6.1. Hodge-Elliptic P^{σ} Estimate. Let D denote either D_{1}, D_{2} or their adjoint operators^{⋆}D_{1} and^{⋆}D_{2}. Let S-tangent tensor ξ be a solution of the Hodge system Dξ = F. Then for any \frac{1}{2} > σ ≥ 0,

∥∇ξ∥_{P^{σ}} ≲ ∥F ∥_{P^{σ}} + Δ_{0}∥D^{−1}F ∥_{L_{t}^{a}L_{x}^{2}}^{q}∥F ∥_{L^{2}(H)}^{1−q},  (6.11)

where 1 − σ > q > γ_{0} and a > 4.

Remark 6.1. In Theorem 6.1, a is derived from the formula \frac{q}{a} + \frac{1−q}{2} = \frac{1−γ}{2} as required by Hölder inequality, where 1 > q > γ > γ_{0} > 1/2.

Remark 6.2. The estimate (6.11) was stated in [12] where the second term on the right hand side is not included.

3This result and its proof are based on the unpublished notes of Klainerman and Rodnianski [9].

6.2.1 Sketch of the proof

We illustrate the main idea of the proof as follows:

Step 1.

According to the definition of ∥∇D^{−1}F ∥_{P^{0}}, we consider ∑_{k>0} ∥P_{k}∇D^{−1}F ∥_{L^{2}(H)}. Using the geometric Littlewood Paley decomposition, we have D^{−1}F = D^{−1} ∑_{l∈ℤ} P_{l}F. Dropping the low frequency terms, we will estimate the following

∑_{k,l>0} ∥P_{k}∇D^{−1}P_{l}F ∥_{L^{2}(H)}.

Note that when k < l, we can easily get

∥P_{k}∇D^{−1}P_{l}F ∥_{L^{2}(H)} ≲ 2^{k−l}∥P_{l}F ∥_{L^{2}(H)},

which implies

∑_{0<k<l} ∥P_{k}∇D^{−1}P_{l}F ∥_{L^{2}(H)} ≲ ∥F ∥_{P^{0}}.

However, for the case k > l > 0, this method fails. Thus in addition to decomposing D^{−1}F, we use geometric Littlewood Paley decomposition to decompose P_{k}∇D^{−1}P_{l}F into the form ∑_{m∈ℤ} P_{k}∇P_{m}D^{−1}P_{l}F. It suffices to consider the terms with m, l, k > 0. In Proposition 6.3, we will provide the estimate of ∥P_{k}∇P_{m}G∥_{L^{2}(S)} for appropriate tensor fields G. In Proposition 6.4, we will provide the estimate of ∥P_{m}D^{−1}P_{l}f∥_{L^{2}(S)} for appropriate tensor fields f. Let G = D^{−1}P_{l}F and f = F. With the help of the two propositions, we can estimate ∥P_{k}∇P_{m}D^{−1}P_{l}F ∥_{L_{x}^{2}}. Integrate the result in t, we then obtain the estimate of ∥∇D^{−1}F ∥_{P^{0}} by using (6.10).

Step 2. Below we will sketch the proofs of Proposition 6.3 and Proposition 6.4.

1). The idea of proving Proposition 6.3 is as follows:

When k ≤ m it is clear that ∥P_{k}∇P_{m}G∥_{L_{x}^{2}} ≲ 2^{k}s^{−1}∥P_{m}G∥_{L_{x}^{2}}. When k > m > 0, we employ the finite band property for Laplacian Δ first, which gives

∥P_{k}∇P_{m}G∥_{L_{x}^{2}} = 2^{−2m}s^{2}∥P_{k}∇ΔP_{m}G∥_{L_{x}^{2}}.

Then we commute ∇ with Δ according to the commutation formula in [15, P.7] and obtain

∥P_{k}∇P_{m}G∥_{L_{x}^{2}} ≲ 2^{−2k}s^{2}∥P_{k}∇ΔP_{m}G∥_{L_{x}^{2}} + 2^{−2k}s^{2}\{∥P_{k}∇(\underline{K}P_{m}G)∥_{L_{x}^{2}}
+ ∥P_{k}(\underline{K}∇P_{m}G)∥_{L_{x}^{2}}\} + 2^{−2k}∥P_{k}∇P_{m}G∥_{L_{x}^{2}}  (6.12)

where the last term is of lower order. The first term on the right hand side of (6.12) can be bounded by 2^{−2k+2m}2^{m}s^{−1}∥P_{m}G∥_{L_{x}^{2}}. Since k > m, the second term ∥P_{k}∇(\underline{K}P_{m}G)∥_{L^{2}(H)} of (6.12) is harder to estimate than the third one. In what follows we consider the second term only. Note that

P_{k}∇(\underline{K}P_{m}G) = P_{k}∇Λ^{γ} ∘ Λ^{−γ}(\underline{K}P_{m}G),

in Lemma 6.4, we will provide the estimate of

∥Λ^{−γ}(\underline{K}P_{m}G)∥_{L_{x}^{2}} with γ > 1/2  (6.13)

for tensor fields G with the help of (6.10). By Lemma 6.3, the estimate of ∥P_{k}∇Λ^{γ}F ∥_{L_{x}^{2}} is depending on the quantity of ∥\underline{K}∥_{L^{2}(S_{s})}. However, for the Gauss curvature, instead of the boundedness of ∥\underline{K}∥_{L_{t}^{∞}L_{x}^{2}} , we have ∥\underline{K}∥_{L^{2}(H)} ≲ Δ_{0} which is weaker. Thus we need to integrate both sides of (6.12) over t ∈ (0, 1] to get Theorem 6.1.

2). To prove Proposition 6.4, considering the case m > l > 0, we use the finite band property for Δ followed by using the relation Δ = ^{⋆}DD + K

∥P_{k}D^{−1}P_{l}f∥_{L_{x}^{2}} = 2^{−2k}s^{2}∥P_{k}ΔD^{−1}P_{l}f∥_{L_{x}^{2}}

≲ 2^{−2k}s^{2}∥P_{k}^{⋆}DP_{l}f∥_{L_{x}^{2}} + 2^{−2k}s^{2}∥P_{k}(\underline{K}D^{−1}P_{l}f)∥_{L_{x}^{2}} + 2^{−2k}∥P_{k}D^{−1}P_{l}f∥_{L_{x}^{2}},

where the last term is easy to estimate since it is of lower order. Clearly, the first term can be estimated by finite band property of ^{⋆}D. In Lemma 6.4 we will use the GLP projection again to estimate the second term. We can treat the case 0 < m ≤ l similarly.

6.2.2 Step 1

We will prove Theorem 6.1 by assuming the following two propositions.

Proposition 6.3. Let F be an m-covariant S-tangent tensor. Then for any γ > γ_{0} > \frac{1}{2}, where γ_{0} is determined by the condition that \underline{K}_{γ_{0}} < ∞, we have the following estimate

∥P_{k}∇P_{m}F ∥_{L^{2}(H)} ≲ 2^{min(m,k)}2^{−2|m−k|}∥t^{−1}P_{m}F ∥_{L_{t}^{2}L_{x}^{2}}

+ 2^{min(m,k)}2^{−(1−γ) max(m,k)}∥t^{−γ}P_{m}F ∥_{L_{t}^{2}L_{x}^{2}}

+ 2^{−|m−k|}∥\underline{K}∥_{L^{2}(H)}^{γ}∥P_{m}F ∥_{L \frac{2}{1−γ} ; t L_{x}^{2}}.

Proposition 6.4. Let f be an m-covariant S-tangent tensor and D be the corresponding elliptic Hodge operator. Then

∥P_{k}D^{−1}P_{l}f∥_{L^{2}(S)} ≲ 2^{− max(k,l)}2^{−|k−l|}s∥P_{l}f∥_{L^{2}(S)}
+ 2^{− max(k,l)}2^{−(1−γ) max(k,l)}s^{2−γ}∥P_{l}f∥_{L^{2}(S)},  (6.14)

where γ > γ_{0} > \frac{1}{2} and γ_{0} is determined by the condition that \underline{K}_{γ_{0}} < ∞.

Proof of Theorem 6.1. Now we prove Theorem 6.1 using Propositions 6.3 and Proposition 6.4. It suffices to consider the case when σ = 0. For the case when 1/2 > σ > 0, by taking q < 1 − σ, we can follow the same way.

Using Proposition 6.3 we have

∥P_{k}∇D^{−1}f∥_{L^{2}(H)} ≲ ∑_{m} ∥P_{k}∇P_{m}D^{−1}f∥_{L^{2}(H)}
≲ ∑_{m} 2^{min(m,k)}2^{−2|m−k|}∥t^{−1}P_{m}D^{−1}f∥_{L^{2}(H)}
+ ∑_{m} 2^{min(m,k)}2^{−(1−γ) max(m,k)}∥t^{−γ}P_{m}D^{−1}f∥_{L^{2}(H)}
+ ∑_{m} ∥K∥_{L^{2}(H)}^{γ}2^{−|m−k|}∥P_{m}D^{−1}f∥_{L \frac{2}{1−γ} ; t L_{x}^{2}} = A_{k}^{(1)} + A_{k}^{(2)} + A_{k}^{(3)}

Define

A_{kl}^{(1)} = ∑_{m} 2^{min(m,k)}2^{−2|m−k|}∥t^{−1}P_{m}D^{−1}P_{l}f∥_{L^{2}(H)},

A_{kl}^{(2)} = ∑_{m} 2^{min(m,k)}2^{−(1−γ) max(m,k)}∥t^{−γ}P_{m}D^{−1}P_{l}f∥_{L^{2}(H)}.

Our goal is to establish estimates of the form

∑_{k} A_{kl}^{(1)} ≲ ∥P_{l}f∥_{L^{2}(H)},  (6.15)
∑_{k} A_{kl}^{(2)} ≲ ∥P_{l}f∥_{L^{2}(H)}.  (6.16)

We estimate the term A_{kl}^{(1)} with the help of Proposition 6.4.

A_{kl}^{(1)} ≲ ∑_{m} 2^{min(m,k)}2^{−2|m−k|}2^{− max(m,l)}2^{−|m−l|}∥P_{l}f∥_{L^{2}(H)}
+ ∑_{m} 2^{min(m,k)}2^{−2|m−k|}2^{− max(m,l)}2^{−(1−γ) max(m,l)}∥t^{1−γ}P_{l}f∥_{L^{2}(H)}
≲ 2^{−|l−k|} ∑_{m} 2^{−2|m−k|}∥P_{l}f∥_{L^{2}(H)}
≲ 2^{−|l−k|}∥P_{l}f∥_{L^{2}(H)}.

The desired estimate (6.15) now follows after summing over k. Similarly,

A_{kl}^{(2)} ≲ ∑_{m} 2^{min(m,k)}2^{−(1−γ) max(m,k)}2^{−2|m−k|}2^{− max(m,l)}2^{−|m−l|}∥t^{1−γ}P_{l}f∥_{L^{2}(H)}
+ ∑_{m} 2^{min(m,k)}2^{−(1−γ) max(m,k)}2^{− max(m,l)}2^{−(1−γ) max(m,l)}∥t^{2−2γ}P_{l}f∥_{L^{2}(H)}
≲ 2^{−|k−l|} ∑_{m} 2^{−(1−γ)m}∥t^{1−γ}P_{l}f∥_{L^{2}(H)}
≲ 2^{−|k−l|}∥P_{l}f∥_{L^{2}(H)}

and the estimate (6.16) follows. We now estimate the term A_{k}^{(3)}. Note by interpolation we can easily obtain with \frac{q}{a} + \frac{1−q}{2} = \frac{1−γ}{2} ,

∥P_{m}D^{−1}f∥_{L \frac{2}{1−γ} ; t L_{x}^{2}} ≲ ∥P_{m}D^{−1}f∥_{L^{2}(H)} + ∥P_{m}D^{−1}f∥_{L_{t}^{a}L_{x}^{2}}^{q}∥P_{m}D^{−1}f∥_{L^{2}(H)}^{1−q},

then

∑_{k} A_{k}^{(3)} ≲ ∑_{m} ∥P_{m}D^{−1}f∥_{L^{2}(H)} + ∑_{m} 2^{−(1−q)m}∥D^{−1}f∥_{L_{t}^{a}L_{x}^{2}}^{q}∥tf∥_{L^{2}(H)}^{1−q}
≲ ∑_{m} 2^{−m}∥tf∥_{L^{2}(H)} + ∥D^{−1}f∥_{L_{t}^{a}L_{x}^{2}}^{q}∥tf∥_{L^{2}(H)}^{1−q}.

Thus,

∑_{k} A_{k}^{(3)} ≲ ∥tf∥_{L^{2}(H)} + ∥D^{−1}f∥_{L_{t}^{a}L_{x}^{2}}^{q}∥tf∥_{L^{2}(H)}^{1−q}.

Combining the estimates for A^{(1)}, A^{(2)}, A^{(3)} yields the desired conclusion.

6.2.3 Step 2

As discussed before, in order to verify the two propositions in Step 1, we need to establish the following lemma.

Lemma 6.4. Let S be a weakly regular surface with scalar curvature K satisfying the condition \underline{K}_{γ_{0}} < ∞ for some 1 > γ_{0} > \frac{1}{2}. Then for any smooth S-tangent tensor F and a constant γ > γ_{0},

∥Λ^{−γ}(\underline{K}P_{m}F )∥_{L^{2}(S)} ≲ \underline{K}_{γ_{0}}2^{m}s^{−1}∥P_{m}F ∥_{L^{2}(S)}  (6.17)

In addition, we have for arbitrary m, l, with an elliptic Hodge operator D acting on tensors,

∥P_{m}(\underline{K}D^{−1}P_{l}F )∥_{L^{2}(S)} ≲ \underline{K}_{γ_{0}}2^{γm}s^{−γ_{0}}∥P_{l}F ∥_{L^{2}(S)}.  (6.18)

Proof of Lemma 6.4. We have

∥Λ^{−γ}(\underline{K} · P_{l}F )∥_{L_{x}^{2}}^{2} ≲ ∑_{m} 2^{−2mγ}∥s^{γ}P_{m}(\underline{K} · P_{l}F )∥_{L_{x}^{2}}^{2}.

We now decompose

P_{m}(\underline{K} · P_{l}F ) = P_{m}(\underline{K}_{≤m} · P_{l}F ) + P_{m}(\underline{K}_{>m} · P_{l}F )

and estimate with p sufficiently close to 2 and \frac{1}{p∗} = \frac{1}{2} − \frac{1}{p},

∥P_{m}(\underline{K}_{≤m} · P_{l}F )∥_{L_{x}^{2}} ≲ ∑_{m′,l≤m} 2^{−m}s∥∇(\underline{K}_{m′} · P_{l}F )∥_{L_{x}^{2}} + ∑_{m′≤m≤l} ∥P_{m}(\underline{K}_{m′} · P_{l}F)∥_{L_{x}^{2}}
≲ \{ ∑_{m′≤l≤m} + ∑_{l≤m′≤m}\}2^{−m}s∥∇(\underline{K}_{m′} · P_{l}F)∥_{L_{x}^{2}}
+ ∑_{m′≤m≤l} 2^{m \frac{2}{p}} s^{− \frac{2}{p}} ∥\underline{K}_{m′}∥_{L_{x}^{p}}∥P_{l}F ∥_{L_{x}^{2}}
≲ ∑_{m′≤l≤m} 2^{−m}s\{∥∇\underline{K}_{m′}∥_{L_{x}^{4}}∥P_{l}F ∥_{L_{x}^{4}} + ∥\underline{K}_{m′} · ∇P_{l}F ∥_{L_{x}^{2}}\}
+ ∑_{l≤m′≤m} 2^{−m}s\{∥∇\underline{K}_{m′}∥_{L_{x}^{p}}∥P_{l}F ∥_{L_{x}^{p∗}} + ∥\underline{K}_{m′} · ∇P_{l}F ∥_{L_{x}^{2}}\}
+ ∑_{m′≤m≤l} 2^{\frac{2}{p} m}2^{m′(1− \frac{2}{p} +γ_{0})}s^{−1−γ_{0}}∥2^{−γ_{0}m′}s^{γ_{0}} \underline{K}_{m′}∥_{L_{x}^{2}}∥P_{l}F ∥_{L_{x}^{2}}
≲ 2^{l}\underline{K}_{γ_{0}} ∑_{m′≤m} 2^{m′γ_{1}}s^{−γ_{0}−1}∥P_{l}F ∥_{L_{x}^{2}}
≲ 2^{l}\underline{K}_{γ_{0}}2^{mγ_{1}}s^{−γ_{0}−1}∥P_{l}F ∥_{L_{x}^{2}}

with γ_{0} < γ_{1} < γ and γ_{0} − γ_{1} < \frac{2}{p} − 1. Thus,

∑_{m} 2^{−2mγ}∥s^{γ}P_{m}(\underline{K}_{≤m} · P_{l}F )∥_{L_{x}^{2}}^{2}

≲ 2^{2l}\underline{K}_{γ_{0}}^{2}∥s^{−1}P_{l}F ∥_{L_{x}^{2}}^{2} ∑_{m} 2^{−2m(γ−γ_{1})} ≲ 2^{2l}\underline{K}_{γ_{0}}^{2}∥s^{−1}P_{l}F ∥_{L_{x}^{2}}^{2}.

The other term is more difficult to treat. We start by expanding

∥P_{m}(\underline{K}_{>m} · P_{l}F )∥_{L_{x}^{2}} ≲ ∑_{m′>m,l} ∥P_{m}(\underline{K}_{m′} · P_{l}F )∥_{L_{x}^{2}} + ∑_{l≥m′>m} ∥P_{m}(\underline{K}_{m′} · P_{l}F )∥_{L_{x}^{2}}
= I_{1} + I_{2},

where the second term is needed only if l > m. Let p > 2 be sufficiently close to 2, we derive

I_{1} ≲ ∑_{m′>m,l} 2^{−2m′}∥s^{2}P_{m}(Δ\underline{K}_{m′} · P_{l}F )∥_{L_{x}^{2}}
≲ ∑_{m′>m,l} 2^{−2m′}s^{2}(∥P_{m} div (∇\underline{K}_{m′} · P_{l}F )∥_{L_{x}^{2}} + ∥P_{m}(∇\underline{K}_{m′} · ∇P_{l}F )∥_{L_{x}^{2}})
≲ ∑_{m′>m,l} 2^{−2m′}(2^{m}s∥∇\underline{K}_{m′} · P_{l}F ∥_{L_{x}^{2}} + s^{2}∥P_{m}∇(\underline{K}_{m′} · ∇P_{l}F )∥_{L_{x}^{2}})
+ ∑_{m′>m,l} 2^{−2m′}s^{2}∥P_{m}(\underline{K}_{m′} · ΔP_{l}F )∥_{L_{x}^{2}}
≲ ∑_{m′>m,l} 2^{−2m′}(2^{m}s∥∇\underline{K}_{m′} · P_{l}F ∥_{L_{x}^{2}} + 2^{m}s∥\underline{K}_{m′} · ∇P_{l}F ∥_{L_{x}^{2}}
+2^{2l}2^{\frac{2}{p} m}s^{− \frac{2}{p}} ∥\underline{K}_{m′}∥_{L_{x}^{2}}∥P_{l}F ∥_{L_{x}^{p}})

Hölder inequality, weak Bernstein inequality for tensors, the Gagliardo-Nirenberg, and the scalar Böchner inequalities imply that

∥∇\underline{K}_{m′} · P_{l}F ∥_{L_{x}^{2}} ≲ ∥∇\underline{K}_{m′}∥_{L_{x}^{3}}∥P_{l}F ∥_{L_{x}^{6}}
≲ 2^{\frac{2l}{3}} s^{− \frac{2}{3}} ∥∇^{2}\underline{K}_{m′}∥L_{x}^{2}^{\frac{1}{3}}∥∇\underline{K}_{m′}∥L_{x}^{2}^{\frac{2}{3}}∥P_{l}F ∥_{L_{x}^{2}}
≲ 2^{\frac{2l+4m′}{3}} s^{−2}∥\underline{K}_{m′}∥_{L_{x}^{2}}∥P_{l}F ∥_{L_{x}^{2}}

Similarly, Hölder inequality and the strong scalar Bernstein inequality (for the Gauss curvature K) give

∥\underline{K}_{m′} · ∇P_{l}F ∥_{L_{x}^{2}} ≲ ∥\underline{K}_{m′}∥_{L_{x}^{∞}}∥∇P_{l}F ∥_{L_{x}^{2}}
≲ 2^{m′+l}s^{−2}∥\underline{K}_{m′}∥_{L_{x}^{2}}∥P_{l}F ∥_{L_{x}^{2}}

Finally, by applying Hölder’s inequality and the weak Bernstein inequality, we obtain

I_{1} ≲ ∑_{m′>m,l} (2^{− \frac{2m′}{3}} 2^{m}2^{\frac{2l}{3}} + 2^{−m′}2^{m}2^{l} + 2^{−2m′}2^{2l}2^{m \frac{2}{p}} 2^{l(1− \frac{2}{p})})s^{−1}∥\underline{K}_{m′}∥_{L_{x}^{2}}∥P_{l}F ∥_{L_{x}^{2}}.

Then with γ_{0} < 2/3 and p > 2 sufficiently close to 2, we obtain

I_{1} ≲ 2^{mγ_{0}}2^{l}\underline{K}_{γ_{0}}s^{−1−γ_{0}}∥P_{l}F ∥_{L_{x}^{2}}.

Now we estimate I_{2}. Let \frac{1}{q} = \frac{1}{2} + \frac{1}{p} with p > 2 sufficiently close to 2, then

I_{2} ≲ ∑_{l≥m′>m} 2^{\frac{2}{p} m}s^{− \frac{2}{p}} ∥\underline{K}_{m′} · P_{l}F ∥_{L_{x}^{q}}
≲ ∑_{l≥m′>m} 2^{m \frac{2}{p} +m′(1− \frac{2}{p})}s^{−1}∥\underline{K}_{m′}∥_{L_{x}^{2}}∥P_{l}F ∥_{L_{x}^{2}}
≲ ∑_{l≥m′>m} 2^{( \frac{2}{p} −γ_{0})m+(1− \frac{2}{p} +γ_{0})m′−l}s^{−1}∥2^{−γ_{0}m′} \underline{K}_{m′}∥_{L_{x}^{2}}2^{l+γ_{0}m}∥P_{l}F ∥_{L_{x}^{2}}
≲ 2^{l+mγ_{0}}s^{−1−γ_{0}} \underline{K}_{γ_{0}}∥P_{l}F ∥_{L_{x}^{2}}

Finally, the desired result follows after summing over m.

To prove the estimate (6.18) we argue similarly. We have by using the weak Bernstein inequality followed by the Gagliardo-Nirenberg inequality.

∥P_{m}(\underline{K}_{≤m} · D^{−1}P_{l}F )∥_{L_{x}^{2}} ≲ ∑_{m′≤m} ∥P_{m}(\underline{K}_{m′} · D^{−1}P_{l}F )∥_{L_{x}^{2}}

≲ ∑_{m′≤m} 2^{−m}s\{∥∇\underline{K}_{m′}∥_{L_{x}^{p}}∥D^{−1}P_{l}F ∥_{L_{x}^{p∗}} + ∥\underline{K}_{m′}∥_{L_{x}^{∞}}∥∇D^{−1}P_{l}F ∥_{L_{x}^{2}}\}

≲ ∑_{m′≤m} 2^{−m}\{2^{m′(2− \frac{2}{p})−l(1− \frac{2}{p})} + 2^{m′}\}∥\underline{K}_{m′}∥_{L_{x}^{2}}∥P_{l}F ∥_{L_{x}^{2}}

≲ 2^{mγ}\underline{K}_{γ_{0}}s^{−γ_{0}}∥P_{l}F ∥_{L_{x}^{2}}

with 1+γ_{0}−γ < \frac{2}{p} and p > 2 sufficiently close to 2, where, to derive the third inequality, we have used the elliptic estimate ∥∇D^{−1}g∥_{L_{x}^{2}} ≲ ∥g∥_{L_{x}^{2}}, and the estimate

∥D^{−1}\underline{K}_{m′}∥_{L_{x}^{2}} ≲ 2^{−m}s∥\underline{K}_{m′}∥_{L_{x}^{2}}  (6.19)

which is given in Lemma 6.1. By using (6.19), we also estimate

∥P_{m}(\underline{K}_{>m} · D^{−1}P_{l}F )∥_{L_{x}^{2}} ≲ ∑_{m′>m} 2^{−2m′}s^{2}∥P_{m}(Δ\underline{K}_{m′} · D^{−1}P_{l}F )∥_{L_{x}^{2}}
≲ ∑_{m′>m} 2^{−2m′}s^{2}\{∥P_{m} div (∇\underline{K}_{m′} · D^{−1}P_{l}F )∥_{L_{x}^{2}}
+ ∥P_{m}(∇\underline{K}_{m′} · ∇D^{−1}P_{l}F )∥_{L_{x}^{2}}\}
≲ ∑_{m′>m} 2^{mγ}\{2^{m(1−γ)+m′(− \frac{2}{p} +γ_{0})−l(1− \frac{2}{p})}
+ 2^{( \frac{2}{p_{1}} −1−γ_{0})(m−m′)}2^{m(γ_{0}−γ)}\}∥2^{−γ_{0}m′} \underline{K}_{m′}∥_{L_{x}^{2}}∥P_{l}F ∥_{L_{x}^{2}}
≲ 2^{mγ}s^{−γ_{0}} \underline{K}_{γ_{0}}∥P_{l}F ∥_{L_{x}^{2}},

where p_{1} > 1 and sufficiently close to 1, and p > 2 verifying γ − γ_{0} > 1 − \frac{2}{p}.

Now we are ready to prove Proposition 6.3 and Proposition 6.4.

Proof of Proposition 6.4. We first consider the case when k ≥ l. Then

∥P_{k}D^{−1}P_{l}f∥_{L^{2}(S)} ≤ 2^{−2k}t^{2}∥P_{k}ΔD^{−1}P_{l}f∥_{L^{2}(S)}

We use the representation formula

Δ = ^{⋆}DD ± \underline{K} ± t^{−2}Id

to obtain

∥P_{k}D^{−1}P_{l}f∥_{L^{2}(S)} ≲ 2^{−2k}t^{2}∥P_{k}^{⋆}DP_{l}f∥_{L^{2}(S)} + 2^{−2k}t^{2}∥P_{k}(\underline{K}D^{−1}P_{l}f)∥_{L^{2}(S)}
+ 2^{−2k}∥P_{k}D^{−1}P_{l}f∥_{L^{2}(S)}
≲ 2^{−2k}2^{l}t∥P_{l}f∥_{L^{2}(S)} + 2^{−2k}t^{2}∥P_{k}(\underline{K}D^{−1}P_{l}f)∥_{L^{2}(S)}
+ 2^{−3k}t∥P_{l}f∥_{L^{2}(S)}

According to Lemma 6.4 we have

∥P_{k}(\underline{K}D^{−1}P_{l}f)∥_{L^{2}(S)} ≲ 2^{γk}t^{−γ_{0}}∥P_{l}f∥_{L^{2}(S)}\underline{K}_{γ_{0}}

for some γ > γ_{0} > \frac{1}{2} and the desired estimate follows.

Next, we consider the case 0 < k < l. Using the representation formula Δ = D^{⋆}D ± K, we obtain

∥P_{k}D^{−1}P_{l}f∥_{L_{x}^{2}} ≲ 2^{−2l}t^{2}∥P_{k}^{⋆}DP_{l}f∥_{L_{x}^{2}} + 2^{−2l}t^{2}∥P_{k}D^{−1}(\underline{K}P_{l}f)∥_{L_{x}^{2}} + 2^{−2l}∥P_{k}D^{−1}P_{l}f∥_{L_{x}^{2}}
≲ 2^{−2l}2^{k}t∥P_{l}f∥_{L_{x}^{2}} + 2^{−2l}t^{2}∥P_{k}D^{−1}(\underline{K}P_{l}f)∥_{L_{x}^{2}}.

We claim

∥P_{k}D^{−1}(\underline{K}P_{l}f)∥_{L_{x}^{2}} ≲ 2^{γl}t^{−γ_{0}} \underline{K}_{γ_{0}}∥P_{l}f∥_{L_{x}^{2}}.  (6.20)

(6.20) gives the desired result immediately.

To prove this claim, we decompose \underline{K} by GLP decomposition ∑_{k∈ℤ} P_{s} = Id. For simplicity, we can ignore the low frequency term P_{s<0}\underline{K}.

Case 1 When s > l > k > 0, we proceed in the following way.

∥P_{k}D^{−1}(\underline{K}_{s}P_{l}f)∥_{L_{x}^{2}} ≲ 2^{−2s}t^{2}∥P_{k}D^{−1}(Δ\underline{K}_{s}P_{l}f)∥_{L_{x}^{2}}

≲ 2^{−2s}t^{2}∥P_{k}D^{−1} div (∇\underline{K}_{s} · P_{l}f)∥_{L_{x}^{2}} + 2^{−2s}t^{2}∥P_{k}D^{−1}(∇\underline{K}_{s} · ∇P_{l}f)∥_{L_{x}^{2}}  (6.21)

The first term of (6.21) can be estimated by using the elliptic estimate and Bernstein inequality,

2^{−2s}t^{2}∥P_{k}D^{−1} div (∇\underline{K}_{s} · P_{l}f)∥_{L_{x}^{2}} ≲ 2^{−2s}t^{2}∥∇\underline{K}_{s}∥_{L_{x}^{p}}∥P_{l}f∥_{L_{x}^{p∗}}
≲ 2^{−2s+s+s− \frac{2}{p} s}2^{l− \frac{2}{p∗} l}∥P_{l}f∥_{L_{x}^{2}}∥\underline{K}_{s}∥_{L_{x}^{2}}
≲ 2^{γl}2^{−( \frac{2}{p} −γ)(s−l)}t^{−γ}∥2^{−γs}t^{γ}\underline{K}_{s}∥_{L_{x}^{2}}∥P_{l}f∥_{L_{x}^{2}}  (6.22)

where \frac{1}{p} + \frac{1}{p∗} = \frac{1}{2} and p > 2 is sufficiently close to 2.

The second term of (6.21) can be estimated by using the estimate in Lemma 6.1,

∥P_{k}D^{−1}F ∥_{L_{x}^{2}} ≲ 2^{−(2− \frac{2}{p})k}∥s^{2− \frac{2}{p}} F ∥_{L_{x}^{p}} with p close to 1.

With its aid, we derive

2^{−2s}t^{2}∥P_{k}D^{−1}(∇\underline{K}_{s} · ∇P_{l}f)∥_{L_{x}^{2}} ≲ 2^{−2s−2k+ \frac{2}{p} k}t^{4− \frac{2}{p}} ∥(∇\underline{K}_{s} · ∇P_{l}f)∥_{L_{x}^{p}}
≲ 2^{−2s−2k+ \frac{2}{p} k}t^{4− \frac{2}{p}} ∥∇\underline{K}_{s}∥_{L_{x}^{q}}∥∇P_{l}f∥_{L_{x}^{2}}.

where \frac{1}{q} + \frac{1}{2} = \frac{1}{p}. By finite band property, we get that it is bounded by

2^{−2s−2k+ \frac{2}{p} k+2k− \frac{2}{q} k+l}∥\underline{K}_{s}∥_{L_{x}^{2}}∥P_{l}f∥_{L_{x}^{2}} ≲ 2^{l−2s+k+γs}t^{−γ}∥2^{−γs}t^{γ}\underline{K}_{s}∥_{L_{x}^{2}}∥P_{l}f∥_{L_{x}^{2}}  (6.23)

Let γ ≥ γ_{0}, summing (6.22) and (6.23) over s > l > k > 0 gives

∑_{s>l>k>0} ∥P_{k}D^{−1}(\underline{K}_{s}P_{l}f)∥_{L_{x}^{2}} ≲ 2^{γl}t^{−γ}\underline{K}_{γ}∥P_{l}f∥_{L_{x}^{2}}.  (6.24)

Case 2 When l > s > k > 0, we employ Lemma 6.1 to obtain

∥P_{k}D^{−1}(\underline{K}_{s}P_{l}f)∥_{L_{x}^{2}} ≲ 2^{−(2− \frac{2}{p})k}t^{2− \frac{2}{p}} ∥\underline{K}_{s}∥_{L_{x}^{q}}∥P_{l}f∥_{L_{x}^{2}}
≲ 2^{−2k+ \frac{2}{p} k+s− \frac{2}{p∗} s}∥\underline{K}_{s}∥_{L_{x}^{2}}∥P_{l}f∥_{L_{x}^{2}},  (6.25)

where \frac{1}{p∗} + \frac{1}{2} = \frac{1}{p} and p > 1 is sufficiently close to 1. Let γ = 1 + γ_{0} − \frac{2}{p∗}, then summing over (6.25), we get

∑_{l>s>k>0} ∥P_{k}D^{−1}(\underline{K}_{s}P_{l}f)∥_{L_{x}^{2}} ≲ 2^{γl}t^{−γ_{0}} \underline{K}_{γ_{0}}∥P_{l}f∥_{L_{x}^{2}}.  (6.26)

Case 3 When l > k > s > 0, we treat it in the same way as for Case 2.

(6.20) is proved by combining the above three cases.

Proof of Proposition 6.3. By duality we can assume that k ≥ m. Then

∥P_{k}∇P_{m}F ∥_{L_{t}^{2}L_{x}^{2}} ≲ 2^{−2k}∥t^{2}P_{k}Δ∇P_{m}F ∥_{L_{t}^{2}L_{x}^{2}}.

Let us suppose F to be a one form without loss of generality. We denote by \underline{R} the Riemann curvature tensors on 2-surface (S_{s}, γ). clearly

\underline{R}_{abcd} = (γ_{ac}γ_{bd} − γ_{ad}γ_{bc})K, \underline{R}_{ab} = γ_{ab}K,  (6.27)

where K denotes the Gauss curvature. We commute Δ with ∇ according to

∇_{a}(ΔF_{i}) = ∇_{a}(∇_{c}∇_{c}F_{i}) = ∇_{c}∇_{a}∇_{c}F_{i} + [∇_{a}, ∇_{c}]∇_{c}F_{i}
= ∇_{c}∇_{c}∇_{a}F_{i} + ∇_{c}([∇_{a}, ∇_{c}]F_{i}) + \underline{R}_{cdac}∇_{d}F_{i} + \underline{R}_{idac}∇_{c}F_{d}
= Δ(∇_{a}F_{i}) + ∇_{c}(\underline{R}_{idac}F_{d}) − \underline{R}_{da}∇_{d}F_{i} + \underline{R}_{idac}∇_{c}F_{d},

which, combined with (6.27), gives

∥P_{k}∇P_{m}F ∥_{L_{t}^{2}L_{x}^{2}} ≲ 2^{−2k}{∥t^{2}P_{k}∇ΔP_{m}F ∥_{L_{t}^{2}L_{x}^{2}} + ∥P_{k}∇P_{m}F ∥_{L_{t}^{2}L_{x}^{2}}}
+ 2^{−2k}∥t^{2}P_{k}∇(\underline{K}P_{m}F )∥_{L_{t}^{2}L_{x}^{2}} + 2^{−2k}∥t^{2}P_{k}(\underline{K}∇P_{m}F )∥_{L_{t}^{2}L_{x}^{2}}
= A_{1} + A_{2} + A_{3}.

First,

A_{1} = 2^{−2k}{∥t^{2}P_{k}∇ΔP_{m}F ∥_{L_{t}^{2}L_{x}^{2}} + ∥P_{k}∇P_{m}F ∥_{L_{t}^{2}L_{x}^{2}}}
≲ 2^{−2(k−m)}∥P_{k}∇P_{m}F ∥_{L_{t}^{2}L_{x}^{2}} + 2^{−2k+m}∥t^{−1}P_{m}F ∥_{L_{t}^{2}L_{x}^{2}} ≲ 2^{−2(k−m)}2^{m}∥t^{−1}P_{m}F ∥_{L_{t}^{2}L_{x}^{2}}.

Recall Λ = (s^{−2}I − Δ_{γ})^{\frac{1}{2}} . Let γ > \frac{1}{2} be a positive constant to be chosen later. Then

A_{2} ≲ 2^{−2k}∥t^{2}P_{k}∇Λ^{γ} Λ^{−γ}(\underline{K}P_{m}F )∥_{L_{t}^{2}L_{x}^{2}} ≲ 2^{−2k}∥ t^{2}∥P_{k}∇Λ^{γ}∥_{2→2}∥Λ^{−γ}(\underline{K}P_{m}F )∥_{L_{x}^{2}} ∥_{L_{t}^{2}},

(6.28)

where ∥ ∥_{2→2} denotes the operator norm L_{x}^{2} → L_{x}^{2}.

With the help of Lemma 6.3 and Lemma 6.4, we will provide the estimates of A_{2} and A_{3} by proving the following result.

Proposition 6.5. Let F be an m-covariant S-tangent tensor. Then for any γ > γ_{0} > \frac{1}{2}, where γ_{0} is determined by the condition that \underline{K}_{γ_{0}} < ∞, we have the following estimate for j = 2, 3,

∥A_{j}∥_{L_{t}^{2}L_{x}^{2}} ≲ 2^{min(m,k)}2^{−2|m−k|}∥t^{−1}P_{m}F ∥_{L_{t}^{2}L_{x}^{2}}

+ 2^{min(m,k)}2^{−(1−γ) max(m,k)}∥t^{−γ}P_{m}F ∥_{L_{t}^{2}L_{x}^{2}} + 2^{−|m−k|}∥\underline{K}∥_{L^{2}(H)}^{γ}∥P_{m}F ∥_{L \frac{2}{1−γ} ; t L_{x}^{2}}

Proof of Proposition 6.5. Choosing a constant γ > \frac{1}{2} of the form required in Proposition 4.5 we obtain that

∥P_{k}∇(\underline{K}P_{m}F )∥_{L_{x}^{2}} ≲ 2^{m}(2^{(1+γ)k}s^{−2−γ} + 2^{k}s^{−2}∥\underline{K}∥_{L_{x}^{2}}^{γ})∥P_{m}F ∥_{L_{x}^{2}} \underline{K}_{γ_{0}}.

Thus, in view of (6.28)

A_{2} = 2^{−2k}∥t^{2}P_{k}∇(\underline{K}P_{m}F )∥_{L_{t}^{2}L_{x}^{2}}

≲ 2^{−(1−γ)k}2^{m}∥t^{−γ}P_{m}F ∥_{L_{t}^{2}L_{x}^{2}} \underline{K}_{γ_{0}} + 2^{m−k}∥\underline{K}∥_{L_{t}^{2}L_{x}^{2}}^{γ}∥P_{m}F ∥_{L \frac{2}{1−γ} ; t L_{x}^{2}} \underline{K}_{γ_{0}}.

It only remains to estimate A_{3}. Assuming the following inequality holds true ^{4}

∥P_{k}(K∇P_{m}F)∥_{L_{x}^{2}} ≲ 2^{m}(2^{(1+γ)k}s^{−2−γ} + 2^{k}s^{−2}∥K∥_{L_{x}^{2}}^{γ})∥P_{m}F ∥_{L_{x}^{2}} \underline{K}_{γ_{0}},  (6.29)

4It is easy to observe that since k ≥ m, ∥P_{k}(\underline{K}∇P_{m}F)∥_{L_{x}^{2}} is better than ∥P_{k}∇(\underline{K}P_{m}F )∥_{L_{x}^{2}}.

therefore

A_{3} = 2^{−(1−γ)k}2^{m}∥t^{−γ}P_{m}F ∥_{L_{t}^{2}L_{x}^{2}} \underline{K}_{γ_{0}} + 2^{m−k}∥\underline{K}∥_{L_{t}^{2}L_{x}^{2}}^{γ}∥P_{m}F ∥_{L \frac{2}{1−γ} ; t L_{x}^{2}} \underline{K}_{γ_{0}}.

Now we check (6.29) by using the GLP projections to decompose \underline{K} as follows

∥P_{k}(\underline{K} · ∇P_{m}F)∥_{L_{x}^{2}} ≲ ∥P_{k}(\underline{K}_{<m}∇P_{m}F)∥_{L_{x}^{2}} + ∥P_{k}(\underline{K}_{≥m}∇P_{m}F)∥_{L_{x}^{2}}
= C_{1} + C_{2}.

We estimate C_{1} = ∥P_{k}(\underline{K}_{<m}∇P_{m}F)∥_{L_{x}^{2}} first. Set \frac{2}{p} − 1 = γ, then from Bernstein inequality and the finite band property it follows that for 1 < p < 2

C_{1} = ∑_{m′<m<k} 2^{kγ}s^{−γ}∥\underline{K}_{m′}∇P_{m}F ∥_{L_{x}^{p}}

≲ ∑_{m′<m<k} 2^{kγ}s^{−γ}∥\underline{K}_{m′}∥_{L \frac{2}{γ} ; x} ∥∇P_{m}F ∥_{L_{x}^{2}}

≲ ∑_{m′<m<k} 2^{kγ}2^{(1−γ)m′}s^{−1}∥\underline{K}_{m′}∥_{L_{x}^{2}}∥∇P_{m}F ∥_{L_{x}^{2}}

≲ ∑_{m′<m<k} 2^{m}2^{(1+γ)k}2^{(1−γ+γ_{0})m′−k}s^{−2}∥2^{−γ_{0}m′} \underline{K}_{m′}∥_{L_{x}^{2}}∥P_{m}F ∥_{L_{x}^{2}}

≲ 2^{m}2^{(1+γ)k}s^{−γ_{0}−2}\underline{K}_{γ_{0}}∥P_{m}F ∥_{L_{x}^{2}}.

We then estimate C_{2} as follows,

C_{2} = \{ ∑_{m<k<m′} + ∑_{m≤m′≤k}\}∥P_{k}(\underline{K}_{m′}∇P_{m}F)∥_{L_{x}^{2}} = C_{2}^{(1)} + C_{2}^{(2)}  (6.30)

C_{2}^{(2)} can be estimated in the same manner as for C_{1} since k ≥ m′, while C_{2}^{(1)} should be estimated in a slightly different way.

C_{2}^{(1)} ≲ ∑_{l≤m′;m<k<m′} 2^{−2m′}s^{2}∥P_{k}(Δ\underline{K}_{m′} · P_{l}∇P_{m}F)∥_{L_{x}^{2}}
+ ∑_{m<k<m′<l} 2^{−2l}s^{2}∥P_{k}(\underline{K}_{m′} · ΔP_{l}∇P_{m}F)∥_{L_{x}^{2}}.  (6.31)

We start by estimating the second term of (6.31).

∑_{m<k<m′<l} 2^{−2l}s^{2}∥P_{k}(\underline{K}_{m′} · ΔP_{l}(∇P_{m}F))∥_{L_{x}^{2}}
≲ ∑_{m<k<m′<l} 2^{−2l}s^{2}∥P_{k} div (\underline{K}_{m′} · ∇P_{l}∇P_{m}F)∥_{L_{x}^{2}}  (6.32)
+ ∑_{m<k<m′<l} 2^{−2l}s^{2}∥P_{k}(∇\underline{K}_{m′} · ∇P_{l}∇P_{m}F)∥_{L_{x}^{2}}  (6.33)

Set α > 1 − γ + γ_{0} with 1 < p < 2, α = \frac{2}{p} − 1 and \frac{1}{p∗} = \frac{1}{p} − \frac{1}{2}, (6.33) can be estimated by weak Bernstein inequality and Corollary 6.1 in the following way

∑_{m<k<m′<l} 2^{−2l}s^{2}∥P_{k}(∇\underline{K}_{m′} · ∇P_{l}∇P_{m}F)∥_{L_{x}^{2}}

≲ ∑_{m<k<m′<l} 2^{−2l+kα}s^{2−α}∥∇\underline{K}_{m′}∥_{L_{x}^{p∗}}∥∇P_{l}∇P_{m}F ∥_{L_{x}^{2}}

≲ ∑_{m<k<m′<l} 2^{−2l+kα+(2−α+γ_{0})m′}2^{l(1−γ)}(2^{(1+γ)m}s^{−2} + 2^{m}s^{−2+γ}∥\underline{K}∥_{L_{x}^{2}}^{γ})

· ∥P_{m}F ∥_{L_{x}^{2}}∥2^{−m′γ_{0}} \underline{K}_{m′}∥_{L_{x}^{2}} ≲ (2^{(1+γ)k}s^{−2−γ_{0}} + 2^{k}s^{−2+γ−γ_{0}}∥\underline{K}∥_{L_{x}^{2}}^{γ})2^{m}∥P_{m}F ∥_{L_{x}^{2}} \underline{K}_{γ_{0}}.

Then we estimate (6.32) as follows with α = \frac{2}{p} − 1 and 2 > p > 1

∑_{m<k<m′<l}2^{−2l}s^{2}∥P_{k} div (\underline{K}_{m′} · ∇P_{l}∇P_{m}F)∥_{L_{x}^{2}}

≲ ∑_{m<k<m′<l} 2^{−2l}(2^{k(1+α)}s^{1−α} + 2^{k}s∥\underline{K}∥_{L_{x}^{2}}^{α})∥\underline{K}_{m′} · ∇P_{l}∇P_{m}F ∥_{L_{x}^{p}}

≲ ∑_{m<k<m′<l} 2^{−2l}(2^{k(1+α)}s^{1−α} + 2^{k}s∥\underline{K}∥_{L_{x}^{2}}^{α})∥\underline{K}_{m′}∥_{L_{x}^{p∗}}∥∇P_{l}∇P_{m}F ∥_{L_{x}^{2}}

≲ ∑_{m<k<m′<l} 2^{−2l}(2^{k(1+α)}s^{1−α} + 2^{k}s∥\underline{K}∥_{L_{x}^{2}}^{α})2^{m′(1−α+γ_{0})}s^{−1+α}∥2^{−m′γ_{0}} \underline{K}_{m′}∥_{L_{x}^{2}}

· 2^{(1−γ_{0})l}(2^{m(1+γ_{0})}s^{−2} + 2^{m}s^{−2+γ_{0}}∥\underline{K}∥_{L_{x}^{2}}^{γ_{0}})∥P_{m}F ∥_{L_{x}^{2}} ≲ (2^{k(1+γ)}s^{−2−γ_{0}} + 2^{k}s^{−2+γ−γ_{0}}∥\underline{K}∥_{L_{x}^{2}}^{γ})2^{m}\underline{K}_{γ_{0}}∥P_{m}F ∥_{L_{x}^{2}}

where γ = γ_{0} + α and α can be taken to be a small positive number.

Now we estimate the first term of (6.31),

∑_{l≤m′;m<k<m′}2^{−2m′}s^{2}∥P_{k}(Δ\underline{K}_{m′} · P_{l}∇P_{m}F)∥_{L_{x}^{2}}
≲ ∑_{l<m<k<m′} 2^{−2m′}s^{2}∥P_{k}(Δ\underline{K}_{m′} · P_{l}∇P_{m}F)∥_{L_{x}^{2}}  (6.34)
+ ∑_{m<l≤m′;m<k<m′} 2^{−2m′}s^{2}∥P_{k}(Δ\underline{K}_{m′} · P_{l}∇P_{m}F)∥_{L_{x}^{2}}.  (6.35)

We first estimate (6.35) by using Corollary 6.1 with α = γ > γ_{0} when m < l < k < m′ and with α > 1 − γ + γ_{0} when m < k ≤ l < m′ ,

(6.35) ≤ ∑_{m<l≤m′;m<k<m′} 2^{−2m′}s^{2}\{∥P_{k} div (∇\underline{K}_{m′} · P_{l}∇P_{m}F)∥_{L_{x}^{2}}

+ ∥P_{k}(∇\underline{K}_{m′} · ∇P_{l}∇P_{m}F)∥_{L_{x}^{2}}\}

≲ ∑_{m<l≤m′;m<k<m′} 2^{−2m′}\{(2^{k(1+α)}s^{1−α} + 2^{k}s∥\underline{K}∥_{L_{x}^{2}}^{α})∥∇\underline{K}_{m′} · P_{l}∇P_{m}F ∥_{L_{x}^{p}}

+ 2^{αk}s^{2−α}∥∇\underline{K}_{m′} · ∇P_{l}∇P_{m}F ∥_{L_{x}^{p}}\}

≲ ∑_{m<l≤m′;m<k<m′} 2^{−2m′}\{(2^{k(1+α)}s^{−1} + 2^{k}s^{α−1}∥\underline{K}∥_{L_{x}^{2}}^{α})2^{m′}∥\underline{K}_{m′}∥_{L_{x}^{2}}2^{(1−α)l}∥P_{l}∇P_{m}F ∥_{L_{x}^{2}}

+ 2^{αk}s^{2−α}∥∇\underline{K}_{m′}∥_{L_{x}^{p∗}}∥∇P_{l}∇P_{m}F ∥_{L_{x}^{2}}\}

≲ ∑_{m<l≤m′;m<k<m′} \{2^{−(1−γ_{0})m′}(2^{k(1+α)}s^{−2} + 2^{k}s^{−2+α}∥\underline{K}∥_{L_{x}^{2}}^{α})2^{l(1−α)+m}∥2^{−γ_{0}m′} \underline{K}_{m′}∥_{L_{x}^{2}}∥P_{m}F ∥_{L_{x}^{2}}

+ 2^{αk−(α−γ_{0})m′+(1−γ)l}∥2^{−m′γ_{0}} \underline{K}_{m′}∥_{L_{x}^{2}}(2^{m(1+γ)}s^{−2}

+ 2^{m}s^{−2+γ}∥\underline{K}∥_{L_{x}^{2}}^{γ})∥P_{m}F ∥_{L_{x}^{2}}\}

≲ (2^{k(1+γ)}s^{−2−γ_{0}} + 2^{k}s^{γ−2−γ_{0}}∥\underline{K}∥_{L_{x}^{2}}^{γ})2^{m}\underline{K}_{γ_{0}}∥P_{m}F ∥_{L_{x}^{2}}. At last we estimate (6.34), with α = γ

∑_{l<m<k<m′}2^{−2m′}s^{2}∥P_{k}(Δ\underline{K}_{m′} · P_{l}∇P_{m}F)∥_{L_{x}^{2}}

≲ ∑_{l<m<k<m′} 2^{−2m′}s^{2} \{∥P_{k} div (∇\underline{K}_{m′} · P_{l}∇P_{m}F)∥_{L_{x}^{2}} + ∥P_{k}(∇\underline{K}_{m′} · ∇P_{l}∇P_{m}F)∥_{L_{x}^{2}}\}

≲ ∑_{l<m<k<m′} 2^{−2m′}\{(2^{k(1+α)} + 2^{k}s^{α}∥\underline{K}∥_{L_{x}^{2}}^{α})∥∇\underline{K}_{m′}∥_{L_{x}^{2}}2^{(1−α)l}∥P_{l}∇P_{m}F ∥_{L_{x}^{2}}

+ 2^{αk}s^{2−α}∥∇\underline{K}_{m′} · ∇P_{l}∇P_{m}F ∥_{L_{x}^{p}}\}

≲ ∑_{l<m<k<m′} 2^{−2m′}\{(2^{k(1+α)}s^{−2} + 2^{k}s^{−2+α}∥\underline{K}∥_{L_{x}^{2}}^{α})2^{m′}∥\underline{K}_{m′}∥_{L_{x}^{2}}2^{l(2−α)}∥P_{m}F ∥_{L_{x}^{2}}

+ 2^{kα+m′(2−α)+2l}s^{−2}∥\underline{K}_{m′}∥_{L_{x}^{2}}∥P_{m}F ∥_{L_{x}^{2}}\} ≲ 2^{(1+γ)k}2^{m}s^{−2−γ_{0}} \underline{K}_{γ_{0}}∥P_{m}F ∥_{L_{x}^{2}} + 2^{k+m}s^{−2+α−γ_{0}}∥\underline{K}∥_{L_{x}^{2}}^{γ}∥P_{m}F ∥_{L_{x}^{2}} \underline{K}_{γ_{0}}.

Thus we complete the proof for (6.29).

Error Estimates

7.1 Structure of error terms

In this section, we will study various kinds of error terms which comes from the commutation between ∇_{L} and Hodge operators. Let us recall the renormalized Bianchi equations (3.21), (3.22) and (3.23) as follows

L(ρ̌) + \frac{3}{2}trχ · ρ̌ = div β − ζ · β + \frac{1}{2}χ̂ · (∇ˆ⊗ζ + \frac{1}{2}tr\underline{χ} · χ̂ − ζ ˆ⊗ζ)

L(σ̌) + \frac{3}{2}trχ · σ̌ = − curl β + ζ ∧ β + \frac{1}{2}χ̂ ∧ (∇ˆ⊗ζ − ζ ˆ⊗ζ)

∇_{L} ˇ\underline{β} = −∇ρ + (∇σ)^{⋆} − 2(∇ˆ⊗ζ) · ζ + 3(ζ · ρ − ζ^{⋆}σ) − trχ\underline{β}

+ 2ζ · (−\frac{1}{2}trχ · \underline{χ̂} − \frac{1}{2}tr\underline{χ} · χ̂ + ζ ˆ⊗ζ) − 4χ · \underline{χ̂} · ζ.

Recall that we denote by R the full collection of null curvature components α, β, ρ, σ, \underline{β}, and by Ř either ˇ \underline{β} or (ρ̌, −σ̌). By \underline{A} we denote the collection V,χ̂, ζ, \underline{V}, \underline{χ̂}. We denote by A the collection of V, χ̂, ζ, and by ∇A the collection ∇trχ, ∇χ̂, ∇ζ.

Set R̄ = R_{0} + ∇A + A · \underline{A} + \frac{1}{s} \underline{A}, where R_{0} denotes the collection of the null curvature components β, ρ, σ, \underline{β}. The above null Bianchi equations can be symbolically written as

∇_{L}\underline{β̌} = ^{⋆}D_{1}(ρ, σ) + \frac{1}{s}Ř + A · R̄,  (7.1)
L(ρ̌, −σ̌) = D_{1}β + \frac{1}{s}Ř + A · R̄.  (7.2)

Throughout this chapter, we will use the following conventions:

• Ř denotes either the pair (ρ̌, −σ̌) or ˇ \underline{β}

• D^{−1}Ř denotes either D_{1}^{−1}(ρ̌, −σ̌) or ^{⋆}D_{1}^{−1} ˇ\underline{β}

• D^{−2}Ř denotes either D_{2}^{−1} · D_{1}^{−1}(ρ̌, −σ̌) or D_{1}^{−1} ·^{⋆}D_{1}^{−1} ˇ\underline{β} • D^{−1}∇_{L}Ř denotes either ^{⋆}D_{1}^{−1}∇_{L} ˇ\underline{β} or D_{1}^{−1}L(ρ̌, −σ̌)

• C_{0}(Ř) denotes [∇_{L}, D_{1}^{−1}](ρ̌, −σ̌) or [∇_{L},^{⋆}D_{1}^{−1}]ˇ\underline{β}

• D^{−2} · ∇_{L}Ř denotes D_{2}^{−1} · D_{1}^{−1}∇_{L}(ρ̌, −σ̌) or D_{1}^{−1} ·^{⋆}D_{1}^{−1}∇_{L} ˇ\underline{β}

• D^{−1} · C_{0}(Ř) denotes D_{2}^{−1} · [∇_{L}, D_{1}^{−1}](ρ̌, −σ̌) or D_{1}^{−1} · [∇_{L},^{⋆}D_{1}^{−1}]ˇ\underline{β} According to (3.21) and (3.22), we set

Err = D_{1}^{−1}∇_{L}(ρ̌, −σ̌) − β
= D_{1}^{−1}(\frac{1}{s}Ř) + D_{1}^{−1}(β · A) + D_{1}^{−1}(s^{−1}A · A + A · ∇A + \underline{A} · A · A),  (7.3)

and according to (3.23), we set

Ẽrr = ^{⋆}D_{1}^{−1}∇_{L} ˇ\underline{β} − (ρ, σ)
= ^{⋆}D_{1}^{−1}(s^{−1} ˇ\underline{β} + A · (∇A + A · \underline{A} + \frac{1}{s}\underline{A})) +^{⋆}D_{1}^{−1}(ζ · ρ − ζ^{⋆}σ + V · \underline{β}).  (7.4)

Set F_{1} := (Err, Ẽrr), we can write F_{1} symbolically

F_{1} = D^{−1} (\frac{1}{s}Ř + A · R̄) .

Therefore we infer from (7.1) and (7.2) the symbolic expression

D^{−1}∇_{L}Ř = R_{0} + D^{−1} (\frac{1}{s}Ř + A · R̄) = R_{0} + F_{1}.  (7.5)

We shall consider the following error terms C_{1}(Ř) = ∇ · D_{2}^{−1} · [∇_{L}, D_{1}^{−1}](ρ̌, −σ̌) or ∇ · D_{1}^{−1}[∇_{L},^{⋆}D_{1}^{−1}](ˇ\underline{β}) C_{2}(Ř) = ∇ · [∇_{L}, D_{2}^{−1}] · D_{1}^{−1}(ρ̌, −σ̌) or ∇ · [∇_{L}, D_{1}^{−1}] ·^{⋆}D_{1}^{−1}(ˇ\underline{β})

C_{3}(Ř) = [∇_{L}, ∇] · D_{2}^{−1} · D_{1}^{−1}(ρ̌, −σ̌) or [∇_{L}, ∇] · D_{1}^{−1} ·^{⋆}D_{1}^{−1}(ˇ\underline{β}),  (7.6)

which, for convenience, can be written as follows,

C_{1}(Ř) = ∇ · D^{−1} · [∇_{L}, D^{−1}]Ř,
C_{2}(Ř) = ∇ · D^{−1}[∇_{L}, D] · D^{−2}Ř
C_{3}(Ř) = [∇_{L}, ∇] · D^{−2}Ř.

Symbolically we write the error terms as C(Ř) = (C_{1}(Ř), C_{2}(Ř), C_{3}(Ř)).

We will repeatedly use the commutation formula (3.25) for tensor fields F, which can be written as follows

[∇_{L}, ∇]F = (A + s^{−1}) · ∇F + A · (A + s^{−1}) · F + β · F.  (7.7)

Note the same identity holds for the commutation of ∇_{L} with Hodge operator D, i.e.

[∇_{L}, D]F = (A + s^{−1}) · ∇F + A · (A + s^{−1}) · F + β · F.  (7.8)

We remark that when F are scalar functions, the right hand side of the above two formulas are simply (A + s^{−1}) · ∇F.

We denote by [∇_{L}, ∇]_{g}F and [∇_{L}, D]_{g}F the good part (A+s^{−1})·∇F +A·(A+s^{−1})·F in the above two equations. We also set [∇_{L}, D^{−1}]_{g}F = (A + s^{−1}) · ∇D^{−1}F + (A + s^{−1}) · A · D^{−1}F. In view of the commutation formulae and the definition of C_{2}(Ř) and C_{3}(Ř),

C_{2}(Ř) = ∇ · [∇_{L}, D^{−1}]_{g}D^{−1}Ř + ∇ · D^{−1}(β · D^{−2}Ř)  (7.9)
C_{3}(Ř) = [∇_{L}, ∇]_{g} · D^{−2}Ř + β · D^{−2}Ř.  (7.10)

In this chapter we will establish the P^{0} estimates of the commutators C_{i}(Ř) with i = 0, ⋯ 3. The P^{0} norms of C_{0}(Ř) and C_{1}(Ř) can be directly estimated, however, the terms ∇ · D_{2}^{−1}(β · D^{−2}Ř), β · D^{−2}Ř in C_{2}(Ř) and C_{3}(Ř) can not be bounded in P^{0} norm, to which we refer as “bad” parts. The other terms in C_{2}(Ř) and C_{3}(Ř) will be proved to be bounded in P^{0} norm in Section 7.2.

In fact, in Section 7.2 and 7.3, we prove the following proposition

Proposition 7.1. For the error terms C_{0}(Ř), C_{1}(Ř), C_{2}(Ř) and C_{3}(Ř) there hold

∥C_{0}(Ř)∥_{P^{0}} ≲ Δ_{0}^{2} + R_{0}.  (7.11)
∥C_{1}(Ř)∥_{P^{0}} ≲ Δ_{0}^{2} + R_{0}  (7.12)
C_{2}(Ř) = ∇ · D^{−1}(β · D^{−2}Ř) + err  (7.13)
C_{3}(Ř) = β · D^{−2}(Ř) + err  (7.14)

with

∥err∥_{P^{0}} ≲ Δ_{0}^{2} + R_{0}.

In Section 7.6, we provide the tools to treat the “bad” terms β ·D^{−2}Ř and ∇·D^{−1}(β · D^{−2}Ř). This requires the decomposition lemma as in [12, Section 6.12]. However in [12], the authors relied on the following Hodge-Elliptic P^{0} estimate and product estimate which can not be reproduced

∥∇ · D^{−1}F ∥_{P^{0}} ≲ ∥F ∥_{P^{0}},
∥F · G∥_{P^{0}} ≲ N_{1}(F) · (∥∇G∥_{L_{t}^{2}L_{x}^{2}} + ∥G∥_{L_{t}^{2}L_{x}^{2}}),

where F and G are appropriate tensor fields.

Instead, we need to rely on (6.11) and (5.36), i.e.

∥∇D^{−1}F ∥_{P^{θ}} ≲ ∥F ∥_{P^{θ}} + Δ_{0}∥D^{−1}F ∥_{L_{t}^{a}L_{x}^{2}}^{q}∥F ∥_{L^{2}(H)}^{1−q}
∥F · G∥_{P^{0}} ≲ N_{1}(F)(∥G∥_{L_{t}^{a}L_{x}^{2}} + ∥∇G∥_{L_{t}^{2}L_{x}^{2}})

where 1 − θ > q > γ_{0}, γ_{0} > 1/2 and a > 4. The presence of the terms in L_{t}^{a}L_{x}^{2} norms in the above estimates requires us to modify the proof in [12, Section 6.12] by establishing some L_{t}^{a}L_{x}^{2} estimates for commutators, which will be given in Section 7.3-7.5.

Throughout this chapter, denote by a a number 4 < a < ∞. By \underline{K} we denote K−s^{−2} on each leave S_{s}. From Proposition 4.6 we have ∥\underline{K}∥_{L_{t}^{2}L_{x}^{2}} = ∥K − s^{−2}∥_{L^{2}(H)} ≲ Δ_{0}. According to Proposition 4.5 we have \underline{K}_{γ_{0}} := ∥Λ^{−γ}\underline{K}∥_{L_{t}^{∞}L_{x}^{2}} ≲ Δ_{0}, with 1 > γ_{0} > 1/2.

7.2 Proof of Proposition 7.1

We first record the following simple results obtained in Chapter 4 on the terms A and \underline{A}

∥s^{−1}A∥_{L^{2}(H)}, ∥∇A∥_{L^{2}(H)}, ∥A∥_{L_{t}^{∞}L_{x}^{4}}, ∥A∥_{L_{t}^{6}L_{x}^{6}}, ∥∇trχ∥_{L_{x}^{2}L_{t}^{∞}} ≲ Δ_{0},
∥A · \underline{A}∥_{L^{2}(H)}, ∥A · A∥_{L_{t}^{2+}L_{x}^{2}} ≲ Δ_{0}^{2}, ∥∇_{L}\underline{A}∥_{L_{t}^{2}L_{x}^{2}} ≲ Δ_{0}^{2}  (7.15)

which are based on (BA1), (BA2), (3.39), Hölder inequality and Sobolev inequality. These inequalities will be routinely used. With the help of Proposition 6.1 and Proposition 6.2, we will prove Proposition 7.1 according to the following steps.

Step 1. We first note the following simple estimate for appropriate tensor fields F and G, ^{1}

∥D^{−1}(F · G)∥_{P^{0}} ≲ ∥F ∥_{L_{t}^{∞}L_{x}^{4}}∥t^{\frac{1}{2}} G∥_{L^{2}(H)}.  (7.16)

Indeed, this is obtained by applying (6.3) with θ = 0 and p = \frac{4}{3} to ∥D^{−1}(F · G)∥_{P^{0}} followed by Hölder’s inequality.

Using (7.16), we can easily check the following result.

Proposition 7.2. For F_{1} = (Err, Ẽrr) = D^{−1}(s^{−1}Ř + A · R̄) with Err and Ẽrr given by (7.3) and (7.4)

∥F_{1}∥_{P^{0}} ≲ Δ_{0}^{2} + R_{0}.  (7.17)

1The same estimate holds for ∥D^{−1}(F · G)∥_{L^{2}(H)}. By “appropriate tensor fields”, we mean F and G are tensor fields so that D^{−1}(F · G) on the left hand side of the inequality is well-defined.

Indeed, applying (7.16) to F = A and G = R̄ and using (4.29), we get

∥D^{−1}(A · R̄)∥_{P^{0}} ≲ Δ_{0}^{2} + R_{0}.  (7.18)

∥s^{−1}D^{−1}Ř∥_{P^{0}} can be estimated by using (6.3) with p = 2 followed by the elliptic estimates in Proposition 4.2. We infer ∥t^{−1}D^{−1}Ř∥_{P^{0}} ≲ ∥Ř∥_{L^{2}(H)} ≲ R_{0}.

(7.17) is proved by combining the above two estimates.

Proof of (7.11). By Proposition 4.7 we know

[D^{−1}, ∇_{L}] = D^{−1}[∇_{L}, D]D^{−1}.

Combining this with (7.8), we have

C_{0}(Ř) = D^{−1}((A + \frac{1}{s})(∇ · D^{−1}Ř) + (A + \frac{1}{s}) · A · D^{−1}Ř + β · D^{−1}Ř),  (7.19)

in particular, for D^{−1}Ř being a scalar function,

C_{0}(Ř) = D^{−1}((A + \frac{1}{s})(∇ · D^{−1}Ř)).  (7.20)

By (6.3) followed by Proposition 6.1, we get

∥s^{−1}D^{−1}∇ · D^{−1}Ř∥_{P^{0}} ≲ ∥Ř∥_{L^{2}(H)} ≤ R_{0}.

Applying (7.16) to the term D^{−1}(s^{−1}A · D^{−1}Ř) with F = A and G = s^{−1}D^{−1}Ř, followed by the elliptic estimate ∥t^{−1}D^{−1}Ř∥_{L^{2}(H)} ≲ R_{0} we obtain

∥D^{−1}(s^{−1}A · D^{−1}Ř)∥_{P^{0}} ≲ ∥A∥_{L_{t}^{∞}L_{x}^{4}}∥t^{−1/2}D^{−1}Ř∥_{L^{2}(H)} ≲ Δ_{0}R_{0}.

The remaining terms in (7.19) can be estimated by (7.16) similarly with

(F, G) = (A, ∇ · D^{−1}Ř), (D^{−1}Ř, A · A), (D^{−1}Ř, β).

Consequently we obtain

∥C_{0}(Ř)∥_{P^{0}} ≲ Δ_{0}^{2} + Δ_{0} · N_{1}(D^{−1}Ř).  (7.21)

Clearly we also have

∥C_{0}(Ř)∥_{L^{2}(H)} ≲ Δ_{0}^{2} + Δ_{0}N_{1}(D^{−1}Ř).  (7.22)

Now we assume

N_{1}(D^{−1}Ř) ≲ Δ_{0}^{2} + R_{0},

which will be justified soon. Combining this with (7.21), we can get (7.11).

We may replace Ř by arbitrary S_{s}-tangent tensor fields F in the proof of (7.21), with a slight modification, it is easy to get the following

Lemma 7.1. For appropriate tensor fields F,

∥C_{0}(F)∥_{P^{0}} ≲ N_{1}(D^{−1}F),

and

∥C_{0}(F)∥_{L_{t}^{1}L_{x}^{2}} ≲ ∥F ∥_{L^{2}(H)}.  (7.23)

Now we verify N_{1}(D^{−1}Ř) ≲ Δ_{0}^{2} + R_{0}. To this end, we will establish the following inequalities simultaneously.

Proposition 7.3.

∥D^{−1}∇_{L}Ř∥_{L_{t}^{2}L_{x}^{2}} ≲ R_{0} + Δ_{0}^{2},  (7.24)
∥[∇_{L}, D^{−1}]Ř∥_{L_{t}^{2}L_{x}^{2}} ≲ R_{0} + Δ_{0}^{2},  (7.25)
N_{1}(D^{−1}Ř) ≲ R_{0} + Δ_{0}^{2}.  (7.26)

Combining (7.26) with Lemma 4.4, we obtain

Corollary 7.1.

N_{1}(∇ · D^{−2}Ř) ≲ R_{0} + Δ_{0}^{2}
N_{2}(D^{−2}R) ≲ R_{0} + Δ_{0}^{2}.

Proof of Proposition 7.3. In order to establish (7.24), note we derive from (7.5)

∥D^{−1}∇_{L}Ř∥_{L_{t}^{2}L_{x}^{2}} ≲ ∥R_{0}∥_{L_{t}^{2}L_{x}^{2}} + ∥D^{−1}(t^{−1}Ř)∥_{L_{t}^{2}L_{x}^{2}} + ∥D^{−1}(A · R̄)∥_{L^{2}(H)}.

From (7.18) we obtain

∥D^{−1}(A · R̄)∥_{L^{2}(H)} ≲ Δ_{0}^{2} + R_{0}.  (7.27)

Combining the above two inequalities with the elliptic estimates in Proposition 4.2, we infer

∥D^{−1}∇_{L}Ř∥_{L_{t}^{2}L_{x}^{2}} ≲ ∥R_{0}∥_{L_{t}^{2}L_{x}^{2}} + ∥Ř∥_{L_{t}^{2}L_{x}^{2}} + R_{0} + Δ_{0}^{2} ≲ Δ_{0}^{2} + R_{0}.

In order to establish (7.25) and (7.26), note that N_{1}(D^{−1}Ř) can be estimated, with the help of (7.24) and the elliptic estimates, as follows

N_{1}(D^{−1}Ř) = ∥t^{−1}D^{−1}Ř∥_{L_{t}^{2}L_{x}^{2}} + ∥∇ · D^{−1}Ř∥_{L_{t}^{2}L_{x}^{2}} + ∥∇_{L} · D^{−1}Ř∥_{L_{t}^{2}L_{x}^{2}}
≲ ∥Ř∥_{L_{t}^{2}L_{x}^{2}} + ∥Ř∥_{L_{t}^{2}L_{x}^{2}} + ∥D^{−1}∇_{L}Ř∥_{L_{t}^{2}L_{x}^{2}} + ∥[∇_{L}, D^{−1}]Ř∥_{L_{t}^{2}L_{x}^{2}}
≲ R_{0} + Δ_{0}^{2} + ∥[∇_{L}, D^{−1}]Ř∥_{L_{t}^{2}L_{x}^{2}}.

Combining the above estimate with (7.22), we obtain

∥[∇_{L}, D^{−1}]Ř∥_{L_{t}^{2}L_{x}^{2}} ≲ Δ_{0}^{2} + Δ_{0}(R_{0} + Δ_{0}^{2} + ∥[∇_{L}, D^{−1}]Ř∥_{L_{t}^{2}L_{x}^{2}}).

Since Δ_{0} is small, we therefore obtain (7.25) and (7.26).

Step 2. In the following we will prove (7.13) and (7.14). Set

J_{1} := (A + s^{−1}) · ∇D^{−2}Ř, J_{2} := (A + s^{−1}) · A · D^{−2}Ř,

then it is clear that

J_{1} + J_{2} = [∇_{L}, D]_{g}D^{−2}Ř,
∇ · D^{−1}(J_{1} + J_{2}) = ∇[∇_{L}, D^{−1}]_{g}D^{−1}Ř.  (7.28)

Let us rewrite C_{2}(Ř) and C_{3}(Ř) in view of (7.9) and (7.10)

C_{2}(Ř) = ∇ · D^{−1}(J_{1} + J_{2}) + ∇ · D^{−1}(β · D^{−2}Ř),
C_{3}(Ř) = J_{1} + J_{2} + β · D^{−2}Ř.

By the non-sharp estimates (5.38) and (5.37), we obtain

∥J_{1}∥_{P^{0}} ≲ N_{1}(t^{1/2}∇ · D^{−2}Ř)(∥A∥_{L_{x}^{∞}L_{t}^{2}} + ∥∇A∥_{L_{t}^{2}L_{x}^{2}}) + ∥t^{−1}∇D^{−2}Ř∥_{P^{0}}  (7.29)
∥J_{2}∥_{P^{0}} ≲ N_{2}(D^{−2}Ř)(∥t^{1/2}A · A∥_{P^{0}} + ∥t^{−1/2}A∥_{P^{0}}).  (7.30)

Lemma 6.2 implies ∥t^{−1}∇D^{−2}Ř∥_{P^{0}} ≲ N_{2}(D^{−2}Ř). Thus combining (7.29) and (7.30) with Corollary 7.1, we conclude

∥J_{i}∥_{P^{0}} ≲ Δ_{0}^{2} + R_{0}, i = 1, 2.

In order to get the P^{0} estimates of ∇D^{−1}(J_{1} + J_{2}), we need the first inequality in the following lemma.

Lemma 7.2. For appropriate S_{s}-tangent tensor fields F, the following estimates hold:

∥∇ · [D^{−1}, ∇_{L}]_{g}F ∥_{P^{0}} ≲ N_{1}(F)  (7.31)

and

∥[∇_{L}, ∇ · D^{−1}]_{g}F ∥_{P^{0}} ≲ N_{1}(F).

We postpone the proof of Lemma 7.2 to the end of this section. In view of (7.28), (7.31) immediately gives

∥∇D^{−1}(J_{1} + J_{2})∥_{P^{0}} ≲ N_{1}(D^{−1}Ř) ≲ Δ_{0}^{2} + R_{0}.  (7.32)

Combine (7.29), (7.30) and (7.32), we thus complete the proof of (7.13) and (7.14) in Proposition 7.1.

Corollary 7.2.

∥C(Ř)∥_{L_{t}^{2}L_{x}^{2}} ≲ R_{0} + Δ_{0}^{2}

Proof of Corollary 7.2. We first take L^{2}(H) norms of C_{2}(Ř) and C_{3}(Ř). Using the elliptic estimates in Proposition 4.2 followed by Hölder equality, the L^{2}(H)-norms of the terms β · D^{−2}Ř and ∇D^{−1}(β · D^{−2}Ř) can be bounded by

∥β∥_{L^{2}(H)}∥D^{−2}Ř∥_{L_{t}^{∞}L_{ω}^{∞}} ≲ R_{0}N_{2}(D^{−2}Ř) ≲ Δ_{0}^{2} + R_{0},

where we employed Corollary 7.1 to get the last inequality.

The L^{2}-norms of J_{i} and ∇D^{−1}J_{i} with i = 1, 2 can be estimated by

∥(J_{1} + J_{2}, ∇D^{−1}(J_{1} + J_{2}))∥_{L^{2}(H)} ≲ ∥(J_{1} + J_{2}, ∇D^{−1}(J_{1} + J_{2}))∥_{P^{0}} ≲ Δ_{0}^{2} + R_{0}.

Hence we obtain ∥(C_{2}(Ř), C_{3}(Ř))∥_{L^{2}(H)} ≲ Δ_{0}^{2} + R_{0}.

Combining (7.25) with the estimate

∥C_{1}(Ř)∥_{L^{2}(H)} = ∥∇ · D^{−1}C_{0}(Ř)∥_{L^{2}(H)} ≲ ∥C_{0}(Ř)∥_{L^{2}(H)},

we obtain ∥C_{1}(Ř)∥_{L^{2}(H)} ≲ Δ_{0}^{2} + R_{0}.

In view of L^{2}(H) estimates of C_{2}(Ř), C_{3}(Ř) and C_{1}(Ř), we conclude Corollary 7.2 holds.

At last we check Lemma 7.2.

Proof of Lemma 7.2. We first claim for appropriate S_{s}-tangent tensor fields F, the following inequality holds

∥[∇_{L}, D^{−1}]_{g}F ∥_{L_{t}^{a}L_{x}^{2}} ≲ N_{1}(F), with 4 < a < ∞.  (7.33)

2 To check (7.33), noting that

[∇_{L}, D^{−1}]_{g}F = D^{−1}[D, ∇_{L}]_{g}D^{−1}F
= D^{−1}((A + s^{−1}) · ∇D^{−1}F + A · (A + s^{−1}) · D^{−1}F )

2We will improve the right hand side of (7.33) to be N_{1}(D^{−1}F) in the next section.

by using Proposition 6.1 with p = 4/3, (3.39) and (7.15), we derive that

∥[∇_{L}, D^{−1}]_{g}F ∥_{L_{t}^{a}L_{x}^{2}} ≲ ∥t^{1/2}A · ∇D^{−1}F ∥_{L_{t}^{a}L_{x}^{4/3}} + ∥t^{1/2}A · A · D^{−1}F ∥_{L_{t}^{a}L_{x}^{2}}
+ ∥t^{−1}D^{−1}∇D^{−1}F ∥_{L_{t}^{a}L_{x}^{2}} + ∥t^{−1/2}A · D^{−1}F ∥_{L_{t}^{a}L_{x}^{4/3}}
≲ ∥A∥_{L_{t}^{a}L_{x}^{2}}∥∇ · D^{−1}F ∥_{L_{t}^{∞}L_{x}^{4}} + ∥A · A∥_{L_{t}^{a}L_{x}^{2}}∥D^{−1}F ∥_{L_{t}^{∞}L_{x}^{4}}
+ ∥∇D^{−1}F ∥_{L_{t}^{a}L_{x}^{2}} + ∥A∥_{L_{t}^{a}L_{x}^{2}}∥t^{−1/2}D^{−1}F ∥_{L_{t}^{∞}L_{x}^{4}}
≲ N_{2}(D^{−1}F) + N_{1}(∇D^{−1}F) ≲ N_{1}(F)

where the last inequality was derived from Lemma 4.4.

Noting that by Lemma 5.5, we have

∥[∇_{L}, ∇]_{g}D^{−1}F ∥_{P^{0}} ≲ N_{1}(∇D^{−1}F)(∥∇A∥_{L_{t}^{2}L_{x}^{2}} + ∥A∥_{L_{t}^{a}L_{x}^{2}}) with a > 4
+ N_{2}(D^{−1}F)(∥A · A∥_{P^{0}} + ∥t^{− \frac{1}{2}} A∥_{P^{0}}) + ∥t^{−1}∇D^{−1}F ∥_{P^{0}}
≲ N_{1}(∇D^{−1}F) + N_{2}(D^{−1}F),  (7.34)

where to derive the last inequality we employed Lemma 6.2 to get ∥t^{−1}A∥_{P^{0}} ≲ ∥∇A∥_{L^{2}(H)}+ ∥t^{−1}A∥_{L^{2}(H)} and ∥t^{−1}∇D^{−1}F ∥_{P^{0}} ≲ ∥∇^{2}D^{−1}F ∥_{L^{2}(H)}+∥t^{−1}∇D^{−1}F ∥_{L^{2}(H)}. By Lemma 4.4, we infer from (7.34) that

∥[∇_{L}, ∇]_{g}D^{−1}F ∥_{P^{0}} ≲ N_{1}(F).  (7.35)

The combination of (7.33), (7.35) and (6.11) gives (7.31).

Since

∥[∇_{L}, ∇ · D^{−1}]_{g}F ∥_{P^{0}} ≲ ∥[∇_{L}, ∇]_{g}D^{−1}F ∥_{P^{0}} + ∥∇ · [D^{−1}, ∇_{L}]_{g}F ∥_{P^{0}},

by (7.35) and (7.31) we conclude ∥[∇_{L}, ∇ · D^{−1}]_{g}F ∥_{P^{0}} ≲ N_{1}(F).

3

Step 3. Now it remains to prove (7.12). Observe that the terms

C_{1}(Ř) = ∇ · D_{2}^{−1} · [∇_{L}, D_{1}^{−1}](ρ̌, −σ̌) and ∇ · D_{1}^{−1}[∇_{L},^{⋆}D_{1}^{−1}](ˇ\underline{β})

symbolically can be written in the form C_{1}(Ř) = ∇ · D^{−1}C_{0}(Ř). To prove (7.12), we need to rely on the Hodge-Elliptic estimate (6.11) with F = C_{0}(Ř) and 4 < a < ∞, i.e.

∥∇ · D^{−1}C_{0}(Ř)∥_{P^{0}} ≲ ∥C_{0}(Ř)∥_{P^{0}} + Δ_{0}∥D^{−1}C_{0}(Ř)∥_{L_{t}^{a}L_{x}^{2}}^{q}∥C_{0}(Ř)∥_{L^{2}(H)}^{1−q} with γ_{0} < q < 1 (7.36) where D^{−1}C_{0}(Ř) denotes either D_{2}^{−1} · [∇_{L}, D_{1}^{−1}](ρ̌, −σ̌) or D_{1}^{−1} · [∇_{L},^{⋆}D_{1}^{−1}]ˇ\underline{β}.

3In the proof of Lemma 7.2, we have actually checked a stronger result ∥∇D^{−1}J_{i}∥_{P^{0}} ≲ Δ_{0}^{2} + R_{0}, with i = 1, 2.

It follows from (7.36), (7.11) and (7.22) that

∥C_{1}(Ř)∥_{P^{0}} ≲ Δ_{0}^{2} + R_{0} + Δ_{0}∥D^{−1}C_{0}(Ř)∥_{L_{t}^{a}L_{x}^{2}}^{q}(Δ_{0}^{2} + R_{0})^{1−q} with 4 < a < ∞ (7.37)

where γ_{0} < q < 1. We claim the following estimates hold true

∥D_{2}^{−1}[∇_{L}, D_{1}^{−1}](ρ̌, −σ̌)∥_{L_{t}^{a}L_{x}^{2}} ≲ R_{0} + Δ_{0}^{2},
∥D_{1}^{−1}[∇_{L},^{⋆}D_{1}^{−1}]ˇ\underline{β}∥_{L_{t}^{a}L_{x}^{2}} ≲ R_{0} + Δ_{0}^{2},

or symbolically

∥D^{−1}C_{0}(Ř)∥_{L_{t}^{a}L_{x}^{2}} ≲ Δ_{0}^{2} + R_{0},  (7.38)

where 4 < a < ∞, which will be verified in Section 7.3. Therefore we can conclude (7.12) provided (7.38) holds true. Then we complete the proof of Proposition 7.1.

The next three sections are devoted to estimating the term ∥D^{−1}F ∥_{L_{t}^{a}L_{x}^{2}} in Hodge-Elliptic P^{0} estimates, with F being some tensor fields arising from the commutation.

7.3 L_{t}^{a}L_{x}^{2} Estimates of D^{−1}C_{0}(Ř)

In this section, we complete the proof of P^{0} estimate of C_{1}(Ř) in Proposition 7.1 by establishing the L_{t}^{a}L_{x}^{2} estimates of D^{−1}C_{0}(Ř) .

Proposition 7.4.

∥D^{−1}C_{0}(Ř)∥_{L_{t}^{a}L_{x}^{2}} ≲ Δ_{0}^{2} + R_{0},

where 4 < a < ∞.

In the first subsection, we prove Proposition 7.4 by analyzing various error terms with the help of Lemma 7.3 and Proposition 7.5. In the second subsection, we verify these two results.

Lemma 7.3. For S_{s} tangent tensor fields F

∥t^{−ϵ}Λ^{−ϵ}D_{1}^{−1}F ∥_{L_{x}^{2}} ≲ ∥F ∥_{L_{x}^{1}}  (7.39)
∥t^{−ϵ}Λ^{−ϵ}^{⋆}D_{1}^{−1}F ∥_{L_{t}^{b}L_{x}^{2}} ≲ ∥F ∥_{L_{t}^{b+}L_{x}^{1}}  (7.40)

where b ≥ 1 and 0 < ϵ ≤ 1

Proposition 7.5. For D^{−1} either D_{1}^{−1} or ^{⋆}D_{1}^{−1}, for any S_{s} tangent tensor fields F and H, the following estimate holds

∥D^{−1}(F · ∇H)∥_{L_{t}^{a}L_{x}^{2}} ≲ N_{1}(F)N_{1}(H), with 4 < a < ∞.

Note that for appropriate S_{s} tangent tensor fields F,

∥t^{α−1}Λ^{α}^{⋆}D^{−1}F ∥_{L_{x}^{2}} ≲ ∥F ∥_{L_{x}^{2}} with 1 ≥ α ≥ 0,  (7.41)

which is obtained by using the interpolation property of Λ^{α} and the elliptic estimates of Hodge operators. By dual argument the above estimate implies the following inequality for appropriate S_{s} tangent tensor fields F,

∥t^{α−1}D^{−1}Λ^{α}F ∥_{L_{x}^{2}} ≲ ∥F ∥_{L_{x}^{2}}.  (7.42)

With its aid, we infer from (7.39) and (7.40) the following result

Corollary 7.3. For appropriate S_{s} tangent tensors F, on each leaf S_{s}, there holds

∥D_{2}^{−1}D_{1}^{−1}F ∥_{L_{x}^{2}} ≲ ∥tF ∥_{L_{x}^{1}}.  (7.43)

On H, we have the following estimate for F.

∥D_{1}^{−1}^{⋆}D_{1}^{−1}F ∥_{L_{t}^{b}L_{x}^{2}} ≲ ∥tF ∥_{L_{t}^{b+}L_{x}^{1}}  (7.44)

where 1 ≤ b < ∞.

Remark 7.1. To prove Proposition 7.4, in view of the symbolic formula:

D^{−1}C_{0}(Ř) = D^{−2}(A · ∇D^{−1}Ř) + D^{−2}(β · D^{−1}Ř) + ⋯,

since we only have L_{t}^{2}L_{x}^{2} norms of β, ∇D^{−1}Ř and ∇A, we need to pass the norm L_{t}^{a}L_{x}^{2} to L_{t}^{p}L_{x}^{2} with 1 ≤ p ≤ 2. To implement this, we may use the fact that

∥F ∥_{L_{t}^{a}L_{x}^{2}} ≲ ∥∇_{L}F ∥_{L_{t}^{1}L_{x}^{2}} + l.o.t.  (7.45)

In this way, we have to estimate ∥∇_{L}D^{−2}(A·∇D^{−1}Ř)∥_{L_{t}^{1}L_{x}^{2}} and ∥∇_{L}D^{−2}(β·D^{−1}Ř)∥_{L_{t}^{1}L_{x}^{2}}. Ideally, these terms may be treated by commuting ∇_{L} with D^{−2}. Ignoring commutators, we consider the terms ∥D^{−2}∇_{L}(β · D^{−1}Ř)∥_{L_{t}^{1}L_{x}^{2}} and ∥D^{−2}∇_{L}(A · ∇D^{−1}Ř)∥_{L_{t}^{1}L_{x}^{2}}, which are derived after commutation. By Leibnitz rule, it is necessary to estimate ∥D^{−2}(H · G)∥_{L_{t}^{1}L_{x}^{2}}, where H = (β, ∇_{L}A) and G = (∇_{L}D^{−1}Ř, ∇D^{−1}Ř). It follows from Corollary 7.3 that

∥D^{−2}(H · G)∥_{L_{t}^{1}L_{x}^{2}} ≲ ∥tH · G∥_{L_{t}^{1+}L_{x}^{1}}.

However, for H · G, we can only control the L^{1}(H) norm. Therefore, we can not directly apply (7.45) to ∥D^{−1}C_{0}(Ř)∥L_{t}^{a}L_{x}^{2} from the beginning. This motivates Proposition 7.5.

7.3.1 Error Type Terms

Throughout this subsection, we denote by D one of the operators D_{1}, D_{2} or ^{⋆}D_{1}.

Proposition 7.6. For S_{s} tangent tensors H verifying N_{1}(H) < ∞, set

E_{1}(H) = s^{−1}D^{−1}(A · H), D^{−1}(A · A · H),
E_{2}(H) = D^{−1}(A · ∇H),
E_{3} = D^{−1}(A · \underline{A} · A),
E_{4} = s^{−1}D^{−1}(A · \underline{A}),

the following estimates hold

∥E_{1}(H)∥_{L_{t}^{a}L_{x}^{2}}, ∥E_{2}(H)∥_{L_{t}^{a}L_{x}^{2}} ≲ Δ_{0}N_{1}(H)
∥t^{−ϵ}Λ^{−ϵ}E_{3}∥_{L_{t}^{a}L_{x}^{2}} ≲ Δ_{0}^{2}, ∥E_{4}∥_{L_{t}^{a}L_{x}^{2}} ≲ Δ_{0}^{2}

where 4 < a < ∞, 0 < ϵ ≤ 1.

Proof. In this proof, for simplicity, we write E_{1}(H) and E_{2}(H) to be E_{1} and E_{2} whenever there is no confusion.

For E_{1}, by Proposition 6.1, we can easily get

∥E_{1}∥_{L_{t}^{a}L_{x}^{2}} ≲ ∥t^{1/2}A · A · H∥_{L_{t}^{a}L_{x}^{4/3}} ≲ ∥A∥_{L_{t}^{∞}L_{x}^{4}}^{2}∥H∥_{L_{t}^{∞}L_{x}^{4}} ≲ Δ_{0}^{2}N_{1}(H).

The term s^{−1}D^{−1}(A · H) can be treated similarly.

As to E_{2}, using Proposition 7.5, we infer

∥E_{2}∥_{L_{t}^{a}L_{x}^{2}} ≲ N_{1}(H)N_{1}(A) ≲ Δ_{0}N_{1}(H).

For E_{3}, by (7.39) and (7.40),

∥t^{−ϵ}Λ^{−ϵ}D^{−1}(A · \underline{A} · A)∥_{L_{t}^{a}L_{x}^{2}} ≲ ∥A · \underline{A} · A∥_{L_{t}^{a+}L_{x}^{1}}

Note that we have the simple inequality,

∥F ∥_{L_{t}^{a}L_{x}^{p}} ≲ ∥∇_{L}F ∥_{L_{t}^{1}L_{x}^{p}} + ∥t^{\frac{1}{a} −1}F ∥_{L_{t}^{1}L_{x}^{p}} with p ≥ 1.  (7.46)

Then it suffices to estimate

∥∇_{L}(A · \underline{A} · A)∥_{L_{t}^{1}L_{x}^{1}} + ∥t^{\frac{1}{a+} −1}A · \underline{A} · A∥_{L_{t}^{1}L_{x}^{1}}.

The second term is bounded by

∥A · \underline{A}∥_{L_{t}^{2}L_{x}^{2}}∥t^{−1}A∥_{L_{t}^{2}L_{x}^{2}} ≲ Δ_{0}^{3}.

To control the first term, by Leibnitz rule and Hölder inequality, with the help of (7.15), we derive

∥∇_{L}A · \underline{A} · A∥_{L_{t}^{1}L_{x}^{1}} ≲ ∥∇_{L}A∥_{L_{t}^{2}L_{x}^{2}}∥A · \underline{A}∥_{L_{x}^{2}L_{t}^{2}} ≲ Δ_{0}^{3}
∥∇_{L}\underline{A} · A · A∥_{L_{t}^{1}L_{x}^{1}} ≲ ∥∇_{L}\underline{A}∥_{L_{t}^{2}L_{x}^{2}}∥A · A∥_{L_{t}^{2}L_{x}^{2}} ≲ Δ_{0}^{3}.

Thus

∥t^{−ϵ}Λ^{−ϵ}E_{3}∥_{L_{t}^{a}L_{x}^{2}} ≲ Δ_{0}^{3}.

As to E_{4}, by Proposition 6.1,

∥t^{−1}D^{−1}(A · \underline{A})∥_{L_{t}^{a}L_{x}^{2}} ≲ ∥t^{1− \frac{2}{p}} A · \underline{A}∥_{L_{t}^{a}L_{x}^{p}}

where p > 1 is sufficiently close to 1. We infer by Hölder inequality

∥t^{1− \frac{2}{p}} A · \underline{A}∥_{L_{t}^{a}L_{x}^{p}} ≲ ∥t^{\frac{3}{2} − \frac{2}{p}} A∥_{L_{t}^{a}L_{x}^{p∗}}∥t^{−1/2}\underline{A}∥_{L_{t}^{∞}L_{x}^{2}},

where \frac{1}{p} = \frac{1}{2} + \frac{1}{p∗}. Note, we have ∥t^{\frac{3}{2} − \frac{2}{p}} A∥_{L_{t}^{a}L_{x}^{p∗}} ≲ N_{1}(A), as long as \frac{3}{2} − \frac{2}{p} − \frac{1}{a} ≤ 0. This requirement can be achieved by choosing p to be close to 1. Hence we get ∥E_{4}∥_{L_{t}^{a}L_{x}^{2}} ≲ Δ_{0}^{2}.

In view of the inequality (7.42), we obtain the following estimate for E_{3}.

Corollary 7.4. The following inequality holds for E_{3},

∥t^{−1}D^{−1}E_{3}∥_{L_{t}^{a}L_{x}^{2}} ≲ Δ_{0}^{2}.

By analyzing the expression of β and C_{0}(F), we have the following result.

Corollary 7.5. The following inequalities hold for any S_{s} tangent tensor F,

∥D^{−1}(β · F)∥_{L_{t}^{a}L_{x}^{2}} ≲ N_{1}(F)Δ_{0}  (7.47)
∥C_{0}(F)∥_{L_{t}^{a}L_{x}^{2}} ≲ N_{1}(t^{\frac{1}{a} − \frac{1}{2}} D^{−1}F)  (7.48)

where 4 < a < ∞.

Proof. Using Codazzi equation (3.16), i.e. β = ∇A+A·A+s^{−1}A, we infer D^{−1}(β ·F) = E_{1}(F) + E_{2}(F), whence (7.47) follows.

Note that the expression

C_{0}(F) = D^{−1}(A · ∇D^{−1}F) + D^{−1}(β · D^{−1}F) + D^{−1}(A · A · D^{−1}F)
+ s^{−1}D^{−1}∇D^{−1}F + s^{−1}D^{−1}(A · D^{−1}F)

can be rewritten into the following form

C_{0}(F) = E_{1}(D^{−1}F) + E_{2}(D^{−1}F) + s^{−1}D^{−1}∇D^{−1}F.

For the last term, using elliptic estimates, we infer

∥t^{−1}D^{−1}∇D^{−1}F ∥_{L_{t}^{a}L_{x}^{2}} ≲ ∥t^{−1}D^{−1}F ∥_{L_{t}^{a}L_{x}^{2}} ≲ N_{1}(t^{\frac{1}{a} − \frac{1}{2}} D^{−1}F).

Proof of Proposition 7.4. Combine Proposition 6.1 and (7.48), we derive

∥D^{−1}C_{0}(Ř)∥_{L_{t}^{a}L_{x}^{2}} ≲ ∥tC_{0}(Ř)∥_{L_{t}^{a}L_{x}^{2}} ≲ N_{1}(t^{\frac{1}{a} + \frac{1}{2}} D^{−1}Ř) ≲ Δ_{0}^{2} + R_{0}

as desired.

7.3.2 Proof of Lemma 7.3 and Proposition 7.5

Now we prove Lemma 7.3.

Proof of Lemma 7.3. For any pair of scalar functions F = (f_{1}, f_{2}) in the range of D_{1}, by dual argument, we show for pairs of scalar functions G in the dual space of Λ^{−ϵ}D_{1}^{−1}F

∥t^{−ϵ}^{⋆}D_{1}^{−1}Λ^{−ϵ}G∥_{L_{x}^{∞}} ≲ ∥G∥_{L_{x}^{2}}.  (7.49)

Using GLP decomposition ∑_{k∈ℤ} P_{k}^{2} = Id, (7.49) can be proved by establishing the following inequality for k > 0 ^{4}

∑_{l>0} ∥t^{−ϵ}P_{l}^{⋆}D_{1}^{−1}P_{k}Λ^{−ϵ}G∥_{L_{x}^{∞}} + ∥t^{−ϵ}P_{≤0}^{⋆}D_{1}^{−1}P_{k}Λ^{−ϵ}G∥_{L_{x}^{∞}} ≲ 2^{−ϵk−k}∥P_{k}G∥_{L_{x}^{2}}. (7.50)

Observe that ^{⋆}D_{1}^{−1}G are pairs of scalar functions. This allows us to use sharp Bernstein inequality. Note that for the second term on the left hand side of (7.50), by Bernstein inequality and Proposition 6.1 we have

∥t^{−ϵ}P_{≤0}^{⋆}D_{1}^{−1}P_{k}Λ^{−ϵ}G∥_{L_{x}^{∞}} ≲ ∥t^{−ϵ−1}^{⋆}D_{1}^{−1}P_{k}Λ^{−ϵ}G∥_{L_{x}^{2}} ≲ 2^{−ϵk−k}∥P_{k}G∥_{L_{x}^{2}}.  (7.51)

We apply sharp Bernstein inequality estimate to the first term on the left hand side of (7.50) for k > 0

∥t^{−ϵ}P_{l}^{⋆}D_{1}^{−1}P_{k}Λ^{−ϵ}G∥_{L_{x}^{∞}} ≲ 2^{l}∥t^{−1−ϵ}P_{l}^{⋆}D_{1}^{−1}P_{k}Λ^{−ϵ}G∥_{L_{x}^{2}}.

4We can ignore low frequency terms with k ≤ 0, since they are easier to estimate.

When l ≤ k, by the finite band property for ^{⋆}D_{1}^{−1} in Lemma 6.1,

∥t^{−ϵ}P_{l}^{⋆}D_{1}^{−1}P_{k}Λ^{−ϵ}G∥_{L_{x}^{∞}} ≲ 2^{l−k}t^{−ϵ}∥P_{k}Λ^{−ϵ}G∥_{L_{x}^{2}} ≲ 2^{l−k−ϵk}∥P_{k}G∥_{L_{x}^{2}}.  (7.52)

When l > k, using (6.14),

∥t^{−ϵ}P_{l}^{⋆}D_{1}^{−1}P_{k}Λ^{−ϵ}G∥_{L_{x}^{∞}} ≲ 2^{l−2l+k}t^{−ϵ}∥P_{k}Λ^{−ϵ}G∥_{L_{x}^{2}} ≲ 2^{−l+k−ϵk}∥P_{k}G∥_{L_{x}^{2}}.  (7.53)

Summing over l > k or l ≤ k in the respective case, combined with (7.51), we can obtain (7.50). Thus we finish the proof of (7.39).

We will prove (7.40) by establishing the following inequality for k > 0.

∥t^{−ϵ}Λ^{−ϵ}P_{k}^{⋆}D_{1}^{−1}F ∥_{L_{x}^{2}} ≤ 2^{−kϵ}(1 + ∥t\underline{K}∥_{L_{x}^{2}}^{θ})∥F ∥_{L_{x}^{1}} with 1 ≥ ϵ > 0  (7.54)

where 1 > θ > 0 is close to 0. Summing (7.54) over k > 0 and integrating in t, (7.40) follows.

To check (7.54), by duality, we only need to show for appropriate tensor field G,

∥t^{−ϵ}D_{1}^{−1}P_{k}Λ^{−ϵ}G∥_{L_{x}^{∞}} ≲ 2^{−ϵk}(1 + ∥t\underline{K}∥_{L_{x}^{2}}^{θ})∥G∥_{L_{x}^{2}},  (7.55)

where 0 < θ < 1 is close to 0.

Again, we decompose D_{1}^{−1}P_{k}Λ^{−ϵ}G using ∑_{l} P_{l}^{2} = Id. However, P_{l}D_{1}^{−1}P_{k}Λ^{−ϵ}G are not scalar functions, to which we can not apply sharp Bernstein inequality. Recall that we have the following curvature dependent estimates,

∥P_{l}H∥_{L_{x}^{∞}} ≲ 2^{l}t^{−1}(1 + 2^{−θl}∥t\underline{K}∥_{L_{x}^{2}}^{θ})∥P_{l}H∥_{L_{x}^{2}}, l > 0  (7.56)

and

∥P_{≤0}H∥_{L_{x}^{∞}} ≲ t^{−1}(1 + ∥t\underline{K}∥_{L_{x}^{2}}^{θ})∥P_{l}H∥_{L_{x}^{2}}.  (7.57)

From (7.57), we infer

∥t^{−ϵ}P_{≤0}D_{1}^{−1}P_{k}Λ^{−ϵ}G∥_{L_{x}^{∞}} ≲ 2^{−ϵk−k}(1 + ∥t\underline{K}∥_{L_{x}^{2}}^{θ})∥P_{k}G∥_{L_{x}^{2}}.  (7.58)

When l > 0, it follows from (7.56) that

∥P_{l}D_{1}^{−1}P_{k}Λ^{−ϵ}G∥_{L_{x}^{∞}} ≲ 2^{l}t^{−1}(1 + 2^{−θl}∥t\underline{K}∥_{L_{x}^{2}}^{θ})∥P_{l}D_{1}^{−1}P_{k}Λ^{−ϵ}G∥_{L_{x}^{2}}.

Similar to the derivation of (7.52) and (7.53), we conclude

∥t^{−ϵ}P_{l}D_{1}^{−1}P_{k}Λ^{−ϵ}G∥_{L_{x}^{∞}} ≲ 2^{−ϵk}2^{−|l−k|}(1 + 2^{−θl}∥t\underline{K}∥_{L_{x}^{2}}^{θ})∥P_{k}G∥_{L_{x}^{2}}.

Summing the above inequality over l, k > 0, combined with (7.58), (7.55) follows. We complete the proof of Lemma 7.3 .

Proof of Proposition 7.5. Using GLP decomposition ∑ P_{k}^{2} = Id, ignoring low frequency terms, it suffices to consider I = ∑_{l,m,n>0} I_{lnm} where I_{lnm} := ∥P_{l}D^{−1}(P_{n}F ·∇P_{m}H)∥_{L_{t}^{a}L_{x}^{2}}.

Case 1 l < n: Recall, by Lemma 6.1, the following inequality holds for tensors,

∥P_{k}D^{−1}F ∥_{L_{x}^{2}} ≲ 2^{( \frac{2}{p} −2)k}∥t^{2− \frac{2}{p}} F ∥_{L_{x}^{p}} with 1 < p ≤ 2.  (7.59)

When l < m < n, by Lemma 6.1, we obtain

I_{lnm} ≲ 2^{l( \frac{2}{p} −2)}∥t^{1− \frac{2}{p}} P_{n}F · ∇P_{m}H∥_{L_{t}^{a}L_{x}^{p}}  (7.60)

where 2 > p > 1 is sufficiently close to 1.

The following two inequalities can be easily checked by using Sobolev inequality, Lemma 5.3 and Lemma 5.4.

∥t^{−( \frac{2}{p∗} + \frac{1}{a} − \frac{1}{2} )}P_{n}F ∥_{L_{t}^{a}L_{x}^{p∗}} ≲ 2^{n( \frac{1}{2} − \frac{1}{a} − \frac{2}{p∗})}N_{1}[F_{n}]  (7.61)
∥t^{1/2}∇P_{m}H∥_{L_{t}^{∞}L_{x}^{2}} ≲ 2^{\frac{m}{2}} N_{1}[H_{m}],  (7.62)

where \frac{1}{p∗} + \frac{1}{2} = \frac{1}{p}. Using both of them, also using Hölder inequality, (7.60) can be estimated as follows,

I_{lnm} ≲ 2^{(−2+ \frac{2}{p})l+ \frac{m}{2} + \frac{n}{2} − \frac{n}{a} − \frac{2n}{p∗}} N_{1}[F_{n}]N_{1}[H_{m}].

We can obtain

∑_{0<l<m<n} I_{lnm} ≲ N_{1}(F)N_{1}(H),

as long as 1 − \frac{1}{a} − \frac{2}{p∗} < 0. When p is sufficiently close to 1, this requirement can be achieved.

Next, we consider the case when l < n < m. Note that

P_{n}F · ∇P_{m}H = ∇(P_{n}F · P_{m}H) − ∇P_{n}F · P_{m}H,

thus we need to consider both terms

I_{l<n<m}^{2} = ∥P_{l}D^{−1}(∇P_{n}F · P_{m}H)∥_{L_{t}^{a}L_{x}^{2}}
I_{l<n<m}^{3} = ∥P_{l}D^{−1}∇(P_{n}F · P_{m}H)∥_{L_{t}^{a}L_{x}^{2}}.

Observe that ∑_{l<n<m} I_{l<n<m}^{2} ≲ N_{1}(F)N_{1}(H) can be obtained in the same way as for ∑_{l<m<n} I_{l<m<n}^{1}. Thus it suffices to consider I_{l<n<m}^{3}.

I_{l<n<m}^{3} ≲ ∥D^{−1}∇(P_{n}F · P_{m}H)∥_{L_{t}^{a}L_{x}^{2}}
≲ ∥t^{−( \frac{2}{p∗} + \frac{1}{a} − \frac{1}{2} )}P_{n}F ∥_{L_{t}^{a}L_{x}^{p∗}}∥t^{( \frac{2}{p∗} + \frac{1}{a} − \frac{1}{2} )}P_{m}H∥_{L_{t}^{∞}L_{x}^{p}}
≲ 2^{− \frac{n}{a}} 2^{( \frac{1}{2} − \frac{2}{p∗})(n−m)}N_{1}[F_{n}]N_{1}[H_{m}],

where \frac{1}{p∗} + \frac{1}{p} = \frac{1}{2} and p is sufficiently close to 2. Summing over l < n < m implies

∑_{l<n<m} I_{l<n<m}^{3} ≲ N_{1}(F)N_{1}(H).

When m < l < n, with \frac{1}{p∗} + \frac{1}{2} = \frac{1}{p} and 1 < p < 2, we derive

I_{lnm} ≲ 2^{−(2− \frac{2}{p})l}∥t^{2− \frac{2}{p}} P_{n}F · ∇P_{m}H∥_{L_{t}^{a}L_{x}^{p}}
≲ 2^{−(2− \frac{2}{p})l}∥t^{1/2}∇P_{m}H∥_{L_{t}^{∞}L_{x}^{2}}∥t^{\frac{3}{2} − \frac{2}{p}} P_{n}F ∥_{L_{t}^{a}L_{x}^{p∗}}
≲ 2^{−(2− \frac{2}{p})l+ \frac{m}{2} + \frac{n}{2} − \frac{n}{a} − \frac{2}{p∗} n}N_{1}[H_{m}]N_{1}[F_{n}],

which implies

∑_{m<l<n} I_{lnm} ≲ N_{1}(H)N_{1}(F).

Case 2 l > n: First we consider the case when m > l > n. Following exactly the same way as for the case l < n < m, we obtain by elliptic estimates, (5.11) and (5.16)

I_{n<l<m}^{2} ≲ 2^{(−2+ \frac{2}{p})l+ \frac{n}{2} + \frac{m}{2} − \frac{m}{a} − \frac{2m}{p∗}} N_{1}[F_{m}]N_{1}[H_{n}]
I_{n<l<m}^{3} = ∥P_{l}D^{−1}∇(P_{m}F · P_{n}H)∥_{L_{t}^{a}L_{x}^{2}} ≲ ∥P_{m}F · P_{n}H∥_{L_{t}^{a}L_{x}^{2}}
≲ ∥t^{− \frac{1}{a′} − \frac{1}{2}} P_{m}F ∥_{L_{t}^{a′}L_{x}^{2}}∥t^{\frac{1}{a′} + \frac{1}{2}} P_{n}H∥_{L_{t}^{a′′}L_{x}^{∞}}
≲ 2^{− \frac{m}{a′} − \frac{m}{2} + \frac{n}{2} − \frac{n}{a′′}} N_{1}[F_{m}]N_{1}[G_{n}],

where \frac{1}{a′} + \frac{1}{a′′} = \frac{1}{a}.

Then, we obtain ∑_{n<l<m} I_{lnm} ≲ N_{1}(F)N_{1}(H).

Next, when l > m, we derive by Lemma 6.1, (7.62) and (5.16)

I_{lnm} ≲ 2^{−l}∥tP_{n}F · ∇P_{m}H∥_{L_{t}^{a}L_{x}^{2}}
≲ 2^{−l}∥t^{1/2}P_{n}F ∥_{L_{t}^{a}L_{x}^{∞}}∥t^{1/2}∇P_{m}H∥_{L_{t}^{∞}L_{x}^{2}}
≲ 2^{−l+ \frac{m}{2} +n( \frac{1}{2} − \frac{1}{a})}N_{1}[F_{n}]N_{1}[H_{m}],

which yields ∑_{l>m,l>n} I_{lnm} ≲ N_{1}(F)N_{1}(H). Thus we conclude I ≲ N_{1}(F)N_{1}(H).

7.4 L_{t}^{a}L_{x}^{2} Estimates for D^{−1}(Err · F)

In this section, we provide L_{t}^{a}L_{x}^{2} estimates for D^{−1}(C_{0}(Ř)·F) and D^{−1}(Err·F), which are important for the Hodge-Elliptic P^{0} estimates involved in the decomposition procedure in Section 7.6.

The main result of this section is the following two estimates.

Proposition 7.7. Denote by D either D_{1} or D_{2}, for appropriate S_{s} tangent tensor fields F, the following estimates hold

∥D^{−1}(Err · F)∥_{L_{t}^{a}L_{x}^{2}} ≲ Δ_{0}N_{2}(F)  (7.63)
∥D^{−1}([∇_{L}, D_{1}^{−1}](ρ̌, −σ̌) · F)∥_{L_{t}^{a}L_{x}^{2}} ≲ Δ_{0}N_{1}(F).  (7.64)

where Err is defined in (7.3) and 4 < a < ∞.

We will show Proposition 7.7 in the following two steps.

Step 1. Note that (7.3) can be rewritten by Proposition 7.6 as

Err = D_{1}^{−1}(\frac{1}{s}Ř) + E_{1}(A) + E_{2}(A) + E_{3},  (7.65)

we first establish the following lemma which will be used to treat the E_{3} type terms.

Lemma 7.4. Let D denote one of the operators D_{1}, D_{2} or ^{⋆}D_{1} for appropriate S_{s} tangent tensor fields W and F,

∥D^{−1}(W · F)∥_{L_{t}^{a}L_{x}^{2}} ≲ ∥t^{1−ϵ}Λ^{−ϵ}W ∥_{L_{t}^{a}L_{x}^{2}}N_{2}(F).  (7.66)

where 0 < ϵ < 1.

Proof of Lemma 7.4. We will show (7.66) by using geometric Littlewood Paley decomposition P_{k} with ∑_{k∈ℤ} P_{k}^{2} = Id. By W_{n} we denote P_{n}(W), note D^{−1}(W · F) = ∑_{k,n∈ℤ} P_{k}D^{−1}(W_{n} · F). After dropping the low frequency terms, we consider the following terms

I = ∑_{0<k<n} ∥P_{k}D^{−1}(W_{n} · F)∥_{L_{x}^{2}}, II = ∑_{0<n≤k} ∥P_{k}D^{−1}(W_{n} · F)∥_{L_{x}^{2}}.

Combining (7.59) and the elliptic estimates in Proposition 4.2, I can be estimated by using the finite band property

∥P_{k}D^{−1}(W_{n} · F)∥_{L_{x}^{2}} ≲ 2^{−2n}∥t^{2}P_{k}D^{−1}(ΔW_{n} · F)∥_{L_{x}^{2}}
≲ 2^{−2n}{∥t^{2}P_{k}D^{−1} div (∇W_{n} · F)∥_{L_{x}^{2}} + ∥t^{2}P_{k}D^{−1}(∇W_{n} · ∇F)∥_{L_{x}^{2}}}
≲ 2^{−2n}\{∥t^{2}∇W_{n} · F ∥_{L_{x}^{2}} + 2^{( \frac{2}{p} −2)k}∥t^{4− \frac{2}{p}} ∇W_{n} · ∇F ∥_{L_{x}^{p}}\}.

Then by the finite band property and (3.39) we obtain with \frac{1}{p∗} + \frac{1}{2} = \frac{1}{p} and 1 < p ≤ 4/3

∥P_{k}D^{−1}(W_{n} · F)∥_{L_{x}^{2}} ≲ 2^{−2n+n+ϵn}∥2^{−ϵn}tW_{n}∥_{L_{x}^{2}}∥F ∥_{L_{x}^{∞}}
+ 2^{−2n+( \frac{2}{p} −2)k+n+ϵn}∥2^{−ϵn}tW_{n}∥_{L_{x}^{2}}∥t^{2− \frac{2}{p}} ∇F ∥_{L_{x}^{p∗}}.

Summing over 0 < k < n, we conclude for 0 < ϵ < 1,

I ≲ ∥t^{1−ϵ}Λ^{−ϵ}W ∥_{L_{x}^{2}}(∥F ∥_{L_{x}^{∞}} + ∥t^{2− \frac{2}{p}} ∇F ∥_{L_{x}^{p∗}}).

Taking L_{t}^{1} norm of I, since ∥t^{2− \frac{2}{p}} ∇F ∥_{L_{t}^{∞}L_{x}^{p∗}} ≲ N_{2}(t^{1/2}F), also using (3.39), we infer

∥I∥_{L_{t}^{a}} ≲ N_{2}(t^{1/2}F)∥t^{1−ϵ}Λ^{−ϵ}W ∥_{L_{t}^{a}L_{x}^{2}}.  (7.67)

As to II, by the finite band property and (3.39) we infer for k ≥ n > 0

∥P_{k}D^{−1}(W_{n} · F)∥_{L_{x}^{2}} ≲ 2^{−k}∥tW_{n} · F ∥_{L_{x}^{2}} ≲ 2^{−k}∥tW_{n}∥_{L_{x}^{2}}∥F ∥_{L_{x}^{∞}}
≲ 2^{−k+ϵn}∥2^{−nϵ}tW_{n}∥_{L_{x}^{2}}∥F ∥_{L_{x}^{∞}}.

Summing over k > n > 0, then taking L_{t}^{1} norm, by (3.39) we conclude for 0 < ϵ < 1,

∥II∥_{L_{t}^{a}} ≲ ∥t^{1−ϵ}Λ^{−ϵ}W ∥_{L_{t}^{a}L_{x}^{2}}N_{2}(t^{1/2}F).

Combine the estimates of I and II, we get (7.66) as desired. Remark 7.2. We actually have proved a stronger result

∥D^{−1}(W · F)∥_{L_{t}^{a}L_{x}^{2}} ≲ ∥t^{1−ϵ}Λ^{−ϵ}W ∥_{L_{t}^{a}L_{x}^{2}}N_{2}(t^{1/2}F)  (7.68)

where 0 < ϵ < 1.

Step 2. Proof of Proposition 7.7. (7.64) can be obtained by using Proposition 6.1 and (7.48) as follows,

∥D^{−1}(C_{0}(Ř) · F)∥_{L_{t}^{a}L_{x}^{2}} ≲ ∥t^{\frac{1}{2}} C_{0}(Ř)∥_{L_{t}^{a}L_{x}^{2}}∥F ∥_{L_{t}^{∞}L_{x}^{4}} ≲ N_{1}(F)Δ_{0}.

To prove (7.63), in view of (7.65), we need to show

∥D^{−1}((D_{1}^{−1}(\frac{1}{s}Ř) + E_{1}(A) + E_{2}(A) + E_{3}) · F )∥_{L_{t}^{a}L_{x}^{2}} ≲ N_{2}(F)Δ_{0}.  (7.69)

Similar to (7.64), we can easily check

∥D^{−1}(E_{1}(A) · F)∥_{L_{t}^{a}L_{x}^{2}} + ∥D^{−1}(E_{2}(A) · F)∥_{L_{t}^{a}L_{x}^{2}} ≲ Δ_{0}N_{1}(F).

Using Proposition 6.1, we infer

∥t^{−1}D^{−1}(D_{1}^{−1}Ř · F )∥_{L_{t}^{a}L_{x}^{2}} ≲ ∥t^{−1/2}D_{1}^{−1}Ř · F ∥_{L_{t}^{a}L_{x}^{4/3}} ≲ ∥D_{1}^{−1}Ř∥_{L_{t}^{∞}L_{x}^{4}}∥t^{−1/2}F ∥_{L_{t}^{a}L_{x}^{2}}
≲ N_{1}(D_{1}^{−1}Ř)N_{1}(F).

Thus, it remains to estimate the term ∥D^{−1}(E_{3} · F)∥_{L_{t}^{a}L_{x}^{2}}. By using Lemma 7.4 and Proposition 7.6, we derive

∥D^{−1}(E_{3} · F)∥_{L_{t}^{a}L_{x}^{2}} ≲ ∥t^{1−ϵ}Λ^{−ϵ}E_{3}∥_{L_{t}^{a}L_{x}^{2}}N_{2}(F) ≲ Δ_{0}^{2}N_{2}(F).

Thus Proposition 7.7 is proved.

7.5 L_{t}^{a}L_{x}^{2} Estimates for ∇_{L}D^{−2}Ř

In this section, we prove ∥∇_{L}D^{−2}Ř∥_{L_{t}^{a}L_{x}^{2}} ≲ Δ_{0}^{2} + R_{0} with 4 < a < ∞, which, together with N_{2}(D^{−2}Ř) ≲ Δ_{0}^{2} + R_{0} in Corollary 7.1 and the decomposition theorem in Section 7.6, gives the ultimate result of this chapter, i.e, there exists two tensor fields P and E with appropriate estimates such that C(Ř) = ∇_{L}P + E.

Proposition 7.8. The following estimate holds

∥∇_{L}D^{−2}Ř∥_{L_{t}^{a}L_{x}^{2}} ≲ R_{0} + Δ_{0}^{2}.

where 4 < a < ∞.

To prove Proposition 7.8, we will rely on the following estimates of D^{−1}F_{1}, which will be justified at the end of this section. ^{5}

Proposition 7.9. For F_{1} = (Err, Ẽrr) with Err and Ẽrr given by (7.3) and (7.4), the following estimate holds

∥D^{−1}F_{1}∥_{L_{t}^{a}L_{x}^{2}} ≲ Δ_{0}^{2} + R_{0} with 4 < a < ∞.  (7.70)

Using Hodge-Elliptic P^{σ} estimate (6.11), combining Proposition 7.9 with (7.17), we can obtain

Corollary 7.6.

∥∇ · D^{−1}F_{1}∥_{P^{0}} ≲ Δ_{0}^{2} + R_{0},  (7.71)

where F_{1} = (Err, Ẽrr).

The following proof is based on Proposition 7.9.

Proof of Proposition 7.8. In view of the formula

∇_{L}D^{−2}Ř = [∇_{L}, D^{−1}]D^{−1}Ř + D^{−1}[∇_{L}, D^{−1}]Ř + D^{−2}∇_{L}Ř,

we only need to show

∥[∇_{L}, D^{−1}]D^{−1}Ř∥_{L_{t}^{a}L_{x}^{2}} ≲ Δ_{0}^{2} + R_{0}  (7.72)
∥D^{−1}[∇_{L}, D^{−1}]Ř∥_{L_{t}^{a}L_{x}^{2}} ≲ Δ_{0}^{2} + R_{0}  (7.73)
∥D^{−2}∇_{L}Ř∥_{L_{t}^{a}L_{x}^{2}} ≲ Δ_{0}^{2} + R_{0}.  (7.74)

5By D^{−1}F_{1}, we denote either D_{2}^{−1}Err or ^{⋆}D_{1}^{−1}Ẽrr.

By using the fact that N_{1}(t^{\frac{1}{a} − \frac{1}{2}} D^{−2}Ř) ≲ N_{2}(D^{−2}Ř) ≲ Δ_{0}^{2} + R_{0}, (7.72) follows from (7.48) with F = D^{−1}Ř. (7.73) is identical to Proposition 7.4. Thus, it only remains to prove (7.74).

First we verify (7.74) in the case when D^{−2}∇_{L}Ř = D_{2}^{−1}D_{1}^{−1}∇_{L}(ρ̌, σ̌).

It is clear that

∥∇_{L}D^{−1}β∥_{L_{t}^{1}L_{x}^{2}} ≲ ∥[∇_{L}, D^{−1}]β∥_{L_{t}^{1}L_{x}^{2}} + ∥D^{−1}∇_{L}β∥_{L_{t}^{1}L_{x}^{2}}.

Applying (7.23) to the commutator, and applying (3.20) and Proposition 6.1 to the other term, we obtain ∥∇_{L}D^{−1}β∥_{L_{t}^{1}L_{x}^{2}} ≲ Δ_{0}^{2} + R_{0}. With its aid, by (7.70) and (7.46), we obtain

∥D^{−1}(Err + β)∥_{L_{t}^{a}L_{x}^{2}} ≲ Δ_{0}^{2} + R_{0}.

Therefore, in view of the expression Err = D_{1}^{−1}∇_{L}(ρ̌, −σ̌) − β, (7.74) is proved in this case.

Next, for Ř = ˇ \underline{β}, we estimate the L_{t}^{a}L_{x}^{2} norm of the term D^{−2}∇_{L} ˇ\underline{β}. Using

⋆D_{1}^{−1}∇_{L} ˇ\underline{β} = (ρ, σ) + Ẽrr,

we obtain

∥D^{−2}∇_{L} ˇ\underline{β}∥_{L_{t}^{a}L_{x}^{2}} ≲ ∥D_{1}^{−1}((ρ̌, σ̌) + Ẽrr)∥_{L_{t}^{a}L_{x}^{2}} + ∥tE_{4}∥_{L_{t}^{a}L_{x}^{2}}.

By Proposition 7.6, the second term is bounded by Δ_{0}^{2}. The first term is bounded by

∥D^{−1}Ř∥_{L_{t}^{a}L_{x}^{2}} + ∥D^{−1}Ẽrr∥_{L_{t}^{a}L_{x}^{2}}.

It is bounded by Δ_{0}^{2} + R_{0} by using the inequality ∥D^{−1}Ř∥_{L_{t}^{a}L_{x}^{2}} ≲ N_{1}(D^{−1}Ř), (7.26) and (7.70). Thus we conclude ∥D^{−2}∇_{L} ˇ\underline{β}∥_{L_{t}^{a}L_{x}^{2}} ≲ Δ_{0}^{2} + R_{0}.

We will rely on the following two results to prove Proposition 7.9.

Lemma 7.5. Let D^{−1} denote one of the operators D_{1}^{−1}, D_{2}^{−1} or ^{⋆}D_{1}^{−1}. For any S_{s} tangent tensor H, the following inequality holds

∥D^{−1}(\underline{K} · H)∥_{L_{t}^{a}L_{x}^{2}} ≲ \underline{K}_{γ_{0}}N_{1}(H), with 4 < a < ∞,  (7.75)

where γ_{0} > 1/2.

Proof of Lemma 7.5. Set Ω_{n,l} := P_{l}D^{−1}(\underline{K} · P_{n}H), it suffices to prove

∑_{l,n>0} ∥Ω_{n,l}∥_{L_{t}^{a}L_{x}^{2}} ≲ \underline{K}_{γ_{0}}N_{1}(H).

We first consider the case n > l > 0, when we set Ω_{n>l} = Ω_{n,l}. By (6.20), we have

∥Ω_{n>l}∥_{L_{x}^{2}} ≲ t^{−γ}2^{γn}∥P_{n}H∥_{L_{x}^{2}} \underline{K}_{γ_{0}} with γ > γ_{0}.

Due to (5.11), i.e. ∥t^{− \frac{1}{2} − \frac{1}{a}} P_{n}H∥_{L_{t}^{a}L_{x}^{2}} ≲ 2^{− \frac{n}{2} − \frac{n}{a}} N_{1}[H_{n}], we infer that

∥Ω_{n>l}∥_{L_{t}^{a}L_{x}^{2}} ≲ 2^{n(γ− \frac{1}{2} − \frac{1}{a})}N_{1}[H_{n}]\underline{K}_{γ_{0}}

where γ > γ_{0}.

Since γ < \frac{1}{2} + \frac{1}{a} can be achieved when γ is sufficiently close to \frac{1}{2}, we can obtain

∑_{n>l} ∥Ω_{n>l}∥_{L_{t}^{a}L_{x}^{2}} ≲ N_{1}(H)\underline{K}_{γ_{0}}.

Next, we consider the case 0 < n < l, when we set Ω_{n<l} = Ω_{n,l}. Combine (6.17), i.e.

∥Λ^{−γ}(\underline{K} · P_{n}H)∥_{L_{x}^{2}} ≲ \underline{K}_{γ_{0}}2^{n}t^{−1}∥P_{n}H∥_{L^{2}(S)}.

with the fact that for tensors F,

∥P_{l}D^{−1}Λ^{γ}F ∥_{L_{x}^{2}} ≲ 2^{(−1+γ)l}t^{1−γ}∥F ∥_{L_{x}^{2}},

we infer

∥Ω_{n<l}∥_{L_{x}^{2}} ≲ 2^{(−1+γ)l+n}t^{−γ}∥P_{n}H∥_{L_{x}^{2}} \underline{K}_{γ_{0}}.

Since γ < \frac{1}{2} + \frac{1}{a}, following the same treatment as for ∥Ω_{n>l}∥_{L_{t}^{a}L_{x}^{2}}, we derive

∑_{n<l} ∥Ω_{n<l}∥_{L_{t}^{a}L_{x}^{2}} ≲ ∑_{n<l} 2^{(−1+γ)l+ \frac{n}{2} − \frac{n}{a}} N_{1}[H_{n}]\underline{K}_{γ_{0}} ≲ N_{1}(H)\underline{K}_{γ_{0}}.

Therefore (7.75) is proved.

Lemma 7.6. For tensor fields G, the following estimate holds true

∥[∇_{L}, D_{1}^{−1}^{⋆}D_{1}^{−1}]G∥_{L_{t}^{1}L_{x}^{2}} ≲ ∥^{⋆}D_{1}^{−1}G∥_{L_{t}^{2}L_{x}^{p}} with p > 2.  (7.76)

Proof of Lemma 7.6. In this proof, we denote by D^{−2} the operator D_{1}^{−1}^{⋆}D_{1}^{−1}. In view of

[∇_{L}, D^{−2}]G = C_{0}(^{⋆}D_{1}^{−1}G) + D_{1}^{−1}C_{0}(G),

it suffices to estimate the L_{t}^{1}L_{x}^{2} norms of ∥C_{0}(^{⋆}D_{1}^{−1}G)∥_{L_{t}^{1}L_{x}^{2}} and ∥D_{1}^{−1}C_{0}(G)∥_{L_{t}^{1}L_{x}^{2}}. Recall (7.23), for appropriate tensor fields G,

∥C_{0}(^{⋆}D_{1}^{−1}G)∥_{L_{t}^{1}L_{x}^{2}} ≲ ∥^{⋆}D_{1}^{−1}G∥_{L_{t}^{2}L_{x}^{2}}.  (7.77)

Then we can obtain (7.76) by combining (7.77) with the following estimate which will be verified soon

∥D_{1}^{−1}C_{0}(G)∥_{L_{t}^{1}L_{x}^{2}} ≲ ∥^{⋆}D_{1}^{−1}G∥_{L_{t}^{2}L_{x}^{p}},  (7.78)

where p > 2. Now we check (7.78). By (3.26), we have

D_{1}^{−1}C_{0}(G) = D^{−2}((A + s^{−1})∇^{⋆}D_{1}^{−1}G).

Noting that by Proposition 6.1 and Proposition 4.2,

∥D^{−2}(\frac{1}{t} div^{⋆}D_{1}^{−1}G)∥_{L_{t}^{1}L_{x}^{2}} ≲ ∥^{⋆}D_{1}^{−1}G∥_{L_{t}^{1}L_{x}^{2}},

it remains to estimate ∥D^{−2}(A · ∇^{⋆}D_{1}^{−1}G)∥_{L_{t}^{1}L_{x}^{2}}. Clearly, we have the following identity.

D^{−2}(A · ∇^{⋆}D_{1}^{−1}G) = D^{−2}( div (A ·^{⋆}D_{1}^{−1}G) − ∇A ·^{⋆}D_{1}^{−1}G),  (7.79)

The first term of (7.79) can be estimated by using Proposition 6.1 and Proposition 4.2 as follows

∥D^{−2}∇(A ·^{⋆}D_{1}^{−1}G)∥_{L_{t}^{1}L_{x}^{2}} ≲ ∥tA ·^{⋆}D_{1}^{−1}G∥_{L_{t}^{1}L_{x}^{2}} ≲ ∥A∥_{L_{t}^{2}L_{x}^{p∗}}∥t^{⋆}D_{1}^{−1}G∥_{L_{t}^{2}L_{x}^{p}}
≲ Δ_{0}∥^{⋆}D_{1}^{−1}G∥_{L_{t}^{2}L_{x}^{p}},

where \frac{1}{p} + \frac{1}{p∗} = \frac{1}{2}, and p is close to 2. To derive the last inequality , we employed Sobolev inequality, ∥A∥_{L_{t}^{2}L_{x}^{p∗}} ≲ ∥∇A∥_{L_{t}^{2}L_{x}^{2}} + ∥t^{−1}A∥_{L_{t}^{2}L_{x}^{2}} ≲ Δ_{0}.

Using Proposition 6.1, we infer

∥D_{1}^{−1}^{⋆}D_{1}^{−1}( div A ·^{⋆}D_{1}^{−1}G)∥_{L_{t}^{1}L_{x}^{2}} ≲ ∥t^{3− \frac{2}{p}} ^{⋆}D_{1}^{−1}G · div A∥_{L_{t}^{1}L_{x}^{p}} ≲ Δ_{0}∥^{⋆}D_{1}^{−1}G∥_{L_{t}^{2}L_{x}^{p∗}},

where \frac{1}{p∗} + \frac{1}{2} = \frac{1}{p} and p is close to 1. Thus (7.78) is proved. This completes the proof of (7.76).

Now we are ready to prove Proposition 7.9.

Proof of Proposition 7.9. ∥D_{2}^{−1}Err∥_{L_{t}^{a}L_{x}^{2}} ≲ Δ_{0}^{2} + R_{0} can be obtained by letting F = 1 in (7.63). Thus we only need to consider D_{1}^{−1}Ẽrr.

In view of the formula (7.4), by Proposition 7.6, we have

∥D^{−2}(A · (∇A + \frac{1}{t}\underline{A} + A · \underline{A}))∥_{L_{t}^{a}L_{x}^{2}} ≲ Δ_{0}^{2}, with 4 < a < ∞.  (7.80)

Note that (7.80) allows us to renormalize the curvature terms ρ, σ and \underline{β}. Therefore, it remains to estimate the following three terms:

U = ∥D^{−2}(ζ · ρ̌)∥_{L_{t}^{a}L_{x}^{2}}, V = ∥D^{−2}(ζ^{⋆}σ̌)∥_{L_{t}^{a}L_{x}^{2}}, W = ∥D^{−2}(V ˇ \underline{β})∥_{L_{t}^{a}L_{x}^{2}}.

By (3.17), clearly σ̌ = curlζ. Thus we obtain

V = ∥D^{−1}E_{2}(ζ)∥_{L_{t}^{a}L_{x}^{2}} ≲ N_{1}(ζ)^{2}.

Now we consider the term U. By (3.19), i.e. \underline{K} = A · \underline{A} + s^{−1}\underline{A} + ρ̌,

U ≲ ∥D^{−2}(\underline{K} · ζ)∥_{L_{t}^{a}L_{x}^{2}} + ∥D^{−2}(A · (A · \underline{A} + t^{−1}\underline{A}))∥_{L_{t}^{a}L_{x}^{2}}.  (7.81)

The second term on the right hand side of (7.81) can be bounded by Δ_{0}^{2} by using (7.80). Due to (7.75), the first term can be bounded by \underline{K}_{γ_{0}}N_{1}(ζ), whence U ≲ Δ_{0}^{2} follows. To estimate W, using (7.46), it suffices to show

∥∇_{L}D^{−2}(V ˇ \underline{β})∥_{L_{t}^{1}L_{x}^{2}} ≲ Δ_{0}^{2} + R_{0}.  (7.82)

Note that

∥∇_{L}D^{−2}(V ˇ \underline{β})∥_{L_{t}^{1}L_{x}^{2}} ≤ ∥D^{−2}∇_{L}(V ˇ \underline{β})∥_{L_{t}^{1}L_{x}^{2}} + ∥[∇_{L}, D^{−2}](V ˇ \underline{β})∥_{L_{t}^{1}L_{x}^{2}} = W_{1} + W_{2}. (7.83)

First, by (7.76) and (7.46), we can estimate W_{2} as follows

W_{2} ≲ ∥^{⋆}D_{1}^{−1}(V ˇ \underline{β})∥_{L_{t}^{2}L_{x}^{2}+}
≲ ∥V ˇ \underline{β}∥_{L_{t}^{2}L_{x}^{2}} + ∥t^{0−}D_{1}^{−1}(V ˇ \underline{β})∥_{L_{t}^{2}L_{x}^{2}}
≲ Δ_{0}R_{0},

where we employed Proposition 6.1 to obtain the last inequality.

It remains to prove

W_{1} = ∥D^{−2}∇_{L}(V ˇ \underline{β})∥_{L_{t}^{1}L_{x}^{2}} ≲ Δ_{0}^{2} + R_{0}.

Set

W_{1}^{(1)} = ∥D^{−2}(∇_{L}V · ˇ \underline{β})∥_{L_{t}^{1}L_{x}^{2}}, W_{1}^{(2)} = ∥D^{−2}(V · ∇_{L} ˇ\underline{β})∥_{L_{t}^{1}L_{x}^{2}},

we have W_{1} ≤ W_{1}^{(1)} + W_{1}^{(2)}.

By (7.44), we infer

W_{1}^{(1)} ≲ ∥t∇_{L}V · ˇ \underline{β}∥_{L_{t}^{1+}L_{x}^{2}} ≲ ∥ˇ\underline{β}∥_{L_{t}^{2}L_{x}^{2}}∥t∇_{L}V ∥_{L_{t}^{2+}L_{x}^{2}}.

According to (3.10), t∇_{L}V ≈ A + tA · A, by (7.15), ∥t∇_{L}V ∥_{L_{t}^{2+}L_{x}^{2}} ≲ Δ_{0}. Thus W_{1}^{(1)} ≲ Δ_{0}R_{0}.

Finally, we will use the symbolic null Bianchi identity (7.1) to estimate W_{1}^{(2)}. Using the formula

∇_{L}\underline{β̌} = ^{⋆}D_{1}(ρ, σ) + \frac{1}{s}Ř + A · R̄,

we infer

W_{1}^{(2)} ≲ ∥D_{1}^{−1}^{⋆}D_{1}^{−1}(V ·^{⋆}D_{1}(ρ, σ))∥_{L_{t}^{1}L_{x}^{2}} + ∥t^{−1}D_{1}^{−1}Ř∥_{L_{t}^{1}L_{x}^{2}}
+ ∥D_{1}^{−1}^{⋆}D_{1}^{−1}(V · (A · R̄))∥_{L_{t}^{1}L_{x}^{2}}.  (7.84)

By using Proposition 6.1, the second term on the right hand side is bounded by R_{0}. The last term, in view of (7.44), can be bounded by

∥tV · A · R̄∥_{L_{t}^{1+}L_{x}^{1}} ≲ ∥tV · A∥_{L_{t}^{2+}L_{x}^{2}}∥R̄∥_{L_{t}^{2}L_{x}^{2}} ≲ Δ_{0}^{2} + R_{0}.

We will estimate the first term on the right hand side of (7.84) with the help of the formula

⋆D_{1}(V (ρ, σ)) = V^{⋆}D_{1}(ρ, σ) − ρ∇V + (σ∇V)^{⋆}.  (7.85)

Using (7.44), we obtain

∥D_{1}^{−1}^{⋆}D_{1}^{−1}(−ρ∇V + (σ∇V)^{⋆})∥_{L_{t}^{1}L_{x}^{2}} ≲ ∥tρ∇V ∥_{L_{t}^{1+}L_{x}^{1}} + ∥t(σ∇V)^{⋆}∥_{L_{t}^{1+}L_{x}^{1}}
≲ R_{0}∥∇V ∥_{L_{t}^{∞}L_{x}^{2}} ≲ R_{0}Δ_{0}.

Note the following inequality holds for V (ρ, σ), which can be easily derived by Proposition 6.1,

∥D_{1}^{−1}^{⋆}D_{1}^{−1}^{⋆}D_{1}(V (ρ, σ))∥_{L_{t}^{1}L_{x}^{2}} = ∥D_{1}^{−1}^{⋆}D_{1}^{−1}^{⋆}D_{1}(V (ρ, σ))∥_{L_{t}^{1}L_{x}^{2}}
≲ ∥tV (ρ, σ)∥_{L_{t}^{1}L_{x}^{2}} ≲ Δ_{0}R_{0}.

Hence, for the first term on the right hand side of (7.84), we have the estimate

∥D_{1}^{−1}^{⋆}D_{1}^{−1}(V^{⋆}D_{1}(ρ, σ))∥_{L_{t}^{1}L_{x}^{2}} ≲ Δ_{0}R_{0}.

Therefore, W_{1}^{(2)} ≲ Δ_{0}^{2} + R_{0}. We complete the proof of Proposition 7.9.

7.6 Decomposition and correction estimates for C_{2}(Ř) and C_{3}(Ř).

In this section we show for commutators C(Ř), there exist appropriate tensor fields P and E so that C(Ř) = ∇_{L}P + E with N_{1}(F) ≲ Δ_{0}^{2} + R_{0}, and ∥E∥_{P^{0}} ≲ Δ_{0}^{2} + R_{0}. To this end, according to Proposition 7.1, it remains to consider the “bad” terms β · D^{−2}Ř and ∇ · D^{−1}(β · D^{−2}Ř), for which we establish the following result.

Theorem 7.1. Consider the expression β ·F with F an S-tangent tensor field of appropriate order on H verifying N_{2}(F) < ∞ and ∥∇_{L}F ∥_{L_{t}^{a}L_{x}^{2}} < ∞ with 4 < a < ∞. Then, the following estimates hold true, (i) There exists a 1-form E_{0} such that

β = ∇_{L} · D^{−1}Ř + E_{0} with ∥E_{0}∥_{P^{0}} ≲ Δ_{0}^{2} + R_{0}  (7.86)

(ii) There exists a decomposition β · F = ∇_{L}P + E, where P and E are tensor fields of the same type as β · F with the estimates ^{6}

N_{1}(P) ≲ Δ_{0}N_{2}(F), ∥E∥_{P^{0}} ≲ Δ_{0} · (N_{2}(F) + ∥∇_{L}F ∥_{L_{t}^{a}L_{x}^{2}}). (7.87)

(iii) There exist tensors P̄ and Ē verifying (7.87) so that

∇ · D^{−1}(β · F) = ∇_{L}P̄ + Ē.  (7.88)

(iv) There exists a decomposition D^{−1}(β · F) = ∇_{L}P̃ + Ẽ, where P̃ and Ẽ are tensor fields verifying the estimates

N_{2}(P̃) ≲ Δ_{0}N_{2}(F), ∥∇Ẽ∥_{P^{0}} ≲ Δ_{0} · (N_{2}(F) + ∥∇_{L}F ∥_{L_{t}^{a}L_{x}^{2}}). (7.89)

Throughout this section, by C_{0}(Ř) we denote the commutator [∇_{L}, D_{1}^{−1}](ρ̌, −σ̌).

Proof of Theorem 7.1. In view of (7.3), we have

β = D_{1}^{−1}∇_{L}Ř + Err with Err = D_{1}^{−1}(A · R̄ + s^{−1}Ř).

Set E_{0} = Err + C_{0}(Ř), combine (7.17) with (7.11), we get ∥E_{0}∥_{P^{0}} ≲ Δ_{0}^{2} + R_{0}. This proves (i).

Now we prove (ii). Note that

β · F = (∇_{L}D_{1}^{−1}Ř + Err + C_{0}(Ř)) · F
= ∇_{L}(D_{1}^{−1}Ř · F) − D_{1}^{−1}Ř · ∇_{L}F + Err · F + C_{0}(Ř) · F.

Set

E_{1}^{B} := D_{1}^{−1}Ř · ∇_{L}F and E_{1}^{G} := (Err + C_{0}(Ř)) · F.

By (5.37), (7.11) and (7.17) we obtain

∥Err · F ∥_{P^{0}} ≲ N_{2}(F)∥Err∥_{P^{0}} ≲ Δ_{0}N_{2}(F),
∥C_{0}(Ř) · F ∥_{P^{0}} ≲ N_{2}(F)∥C_{0}(Ř)∥_{P^{0}} ≲ Δ_{0}N_{2}(F).

6In particular, for F being a 1-form, the following estimate holds

∥E∥_{P^{0}} ≲ Δ_{0}N_{2}(F).

Therefore ∥E_{1}^{G}∥_{P^{0}} ≲ Δ_{0}N_{2}(F). By (5.36) we get the estimate for ∥E_{1}^{B}∥_{P^{0}} as follows,^{7}

∥E_{1}^{B}∥_{P^{0}} = ∥D_{1}^{−1}Ř · ∇_{L}F ∥_{P^{0}} ≲ N_{1}(D^{−1}Ř)(∥∇_{L}F ∥_{L_{t}^{a}L_{x}^{2}} + ∥∇∇_{L}F ∥_{L_{t}^{2}L_{x}^{2}})
≲ (R_{0} + Δ_{0}^{2})(N_{2}(F) + ∥∇_{L}F ∥_{L_{t}^{a}L_{x}^{2}}).  (7.92)

Set

P_{1} = D_{1}^{−1}Ř · F E_{1} = E_{1}^{B} + E_{1}^{G} = D_{1}^{−1}Ř · ∇_{L}F + Err · F + C_{0}(Ř) · F, (7.93)

from the above estimates we have

∥E_{1}∥_{P^{0}} ≲ Δ_{0}(N_{2}(F) + ∥∇_{L}F ∥_{L_{t}^{a}L_{x}^{2}}) with Δ_{0} < 1/2.

In order to estimate N_{1}(P_{1}), let us estimate ∥E_{1}∥_{L^{2}(H)} first. By using Hölder’s inequality and Sobolev inequalities, we can obtain

∥E_{1}^{B}∥_{L^{2}(H)} = ∥D^{−1}Ř · ∇_{L}F ∥_{L^{2}(H)} ≲ ∥D^{−1}Ř∥_{L_{t}^{∞}L_{x}^{4}}∥∇_{L}F ∥_{L_{t}^{2}L_{x}^{4}}
≲ N_{1}(D^{−1}Ř)(∥∇∇_{L}F ∥_{L^{2}(H)} + ∥t^{−1}∇_{L}F ∥_{L^{2}(H)}),

and by using ∥E_{1}^{G}∥_{L^{2}(H)} ≲ ∥E_{1}^{G}∥_{P^{0}} we can obtain ∥E_{1}^{G}∥_{L^{2}(H)} ≲ Δ_{0}N_{2}(F). Therefore

∥E_{1}∥_{L^{2}(H)} ≲ Δ_{0}N_{2}(F).  (7.94)

Now we show

N_{1}(P_{1}) ≲ N_{2}(F)(Δ_{0} + R_{0}).  (7.95)

With the help of ∇_{L}P_{1} = β · F − E_{1} and (7.94) we can estimate ∥∇_{L}P_{1}∥_{L_{t}^{2}L_{x}^{2}} as follows

∥∇_{L}P_{1}∥_{L_{t}^{2}L_{x}^{2}} ≲ ∥β · F ∥_{L_{t}^{2}L_{x}^{2}} + ∥E_{1}∥_{L_{t}^{2}L_{x}^{2}} ≲ (R_{0} + Δ_{0})N_{2}(F).

In view of the expression of P_{1} in (7.93) we obtain

∥∇P_{1}∥_{L_{t}^{2}L_{x}^{2}} = ∥∇D^{−1}Ř · F + D^{−1}Ř · ∇F ∥_{L_{t}^{2}L_{x}^{2}}
≲ ∥∇D^{−1}Ř · F ∥_{L_{t}^{2}L_{x}^{2}} + ∥D^{−1}Ř · ∇F ∥_{L_{t}^{2}L_{x}^{2}}
≲ ∥∇ · D^{−1}Ř∥_{L_{t}^{2}L_{x}^{2}}∥F ∥_{L_{t}^{∞}L_{ω}^{∞}} + ∥∇F ∥_{L_{t}^{4}L_{x}^{4}}∥D^{−1}Ř∥_{L_{t}^{4}L_{x}^{4}}
≲ (Δ_{0}^{2} + R_{0})N_{2}(F),

7If F is a 1-form, we obtain by using (5.39)

∥E_{1}^{B}∥_{P^{0}} = ∥D_{1}^{−1}Ř · ∇_{L}F ∥_{P^{0}} ≲ N_{1}(D_{1}^{−1}Ř)∥∇∇_{L}F ∥_{L_{t}^{2}L_{x}^{2}}  (7.90)

≲ (Δ_{0}^{2} + R_{0})N_{2}(F).  (7.91)

where the last inequality follows from Proposition 4.2 and (3.39) and the following

∥t^{−1/4}D^{−1}Ř∥_{L_{t}^{4}L_{x}^{4}} ≲ N_{1}(D^{−1}Ř)^{3/4}∥t^{−1}D^{−1}Ř∥_{L_{t}^{2}L_{x}^{2}}^{1/4} ≲ Δ_{0}^{2} + R_{0},

and

∥t^{−1/4}∇F ∥_{L_{t}^{4}L_{x}^{4}}^{4} ≲ ∥t^{−1}∇F ∥_{L_{t}^{2}L_{x}^{2}} · ∥∇F ∥_{L_{t}^{6}L_{x}^{6}}^{3} ≲ N_{1}(∇F)^{3} · ∥t^{−1}∇F ∥_{L_{t}^{2}L_{x}^{2}},

which immediately implies

∥t^{−1/4}∇F ∥_{L_{t}^{4}L_{x}^{4}} ≲ N_{2}(F).

Therefore (ii) is proved.

In the following, we will prove (iii) and (iv) in three steps.

Step 1. We will sketch the derivation of the decomposition of ∇·D^{−1}(β·F) and figure out the estimates we need to establish. For S-tangent tensor fields F, set P_{0} = DF, in view of the decomposition (7.87), there exists P_{1} and E_{1} which are defined in (7.93) so that

β · F = ∇_{L}P_{1} + E_{1},

with

N_{1}(P_{1}) ≲ (Δ_{0} + R_{0})N_{2}(F) and ∥E_{1}∥_{P^{0}} ≲ Δ_{0} · (N_{2}(F) + ∥∇_{L}F ∥_{L_{t}^{a}L_{x}^{2}}),

and ∥E_{1}∥_{P^{0}} ≲ Δ_{0}N_{2}(F) for 1-forms F.

We will construct by recurrence two sequences P_{2}, P_{3}, ⋯ P_{k}, and E_{2}, E_{3}, ⋯ E_{k} so that

β · D^{−1}P_{i−1} = ∇_{L}P_{i} + E_{i}, i = 2, ⋯ k

with

N_{1}(P_{i}) ≤ cΔ_{0}N_{2}(D^{−1}P_{i−1}) ≤ (cΔ_{0})^{i}N_{2}(F),
∥E_{i}∥_{P^{0}} ≤ (cΔ_{0})^{i}(N_{2}(F) + ∥∇_{L}F ∥_{L_{t}^{a}L_{x}^{2}}),

and ∥E_{i}∥_{P^{0}} ≲ (cΔ_{0})^{i}N_{2}(F) for 1-forms F, where c is some universal constant. Assuming P_{i}, E_{i} have been constructed for i = 2, 3, ⋯, k, we will construct P_{k+1} and E_{k+1} by applying Theorem 7.1 (ii) to decompose β·D^{−1}P_{k} in the form β·D^{−1}P_{k} = ∇_{L}P_{k+1}+E_{k+1}.

Moreover we will assume the following result which will be proved in Step 2.

Proposition 7.10. For {P_{k}}_{k=1}^{∞} and {E_{k}}_{k=1}^{∞}, the inequalities hold

∥∇_{L}D^{−1}P_{k}∥_{L_{t}^{a}L_{x}^{2}} ≲ Δ_{0}(N_{2}(D^{−1}P_{k−1}) + ∥∇_{L}D^{−1}P_{k−1}∥_{L_{t}^{a}L_{x}^{2}}),  (7.96)
∥∇ · D^{−1}E_{k}∥_{P^{0}} ≲ ∥E_{k}∥_{P^{0}} + Δ_{0}(N_{2}(D^{−1}P_{k−1}) + ∥∇_{L}D^{−1}P_{k−1}∥_{L_{t}^{a}L_{x}^{2}}).  (7.97)

Also using Lemma 4.4, we can derive the following estimates for P_{k+1} and E_{k+1},^{8}

N_{1}(P_{k+1}) ≲ Δ_{0} · N_{2}(D^{−1}P_{k}) ≤ cΔ_{0} · N_{1}(P_{k}) ≤ (cΔ_{0})^{k}N_{2}(F),
∥E_{k+1}∥_{P^{0}} ≤ cΔ_{0} · (N_{2}(D^{−1}P_{k}) + ∥∇_{L}D^{−1}P_{k}∥_{L_{t}^{a}L_{x}^{2}})
≤ (cΔ_{0})^{k}(N_{2}(F) + ∥∇_{L}F ∥_{L_{t}^{a}L_{x}^{2}}),

Now we consider

∇ · D^{−1}(β · D^{−1}P_{k}) = ∇ · D^{−1}(∇_{L}P_{k+1} + E_{k+1})
= ∇_{L}(∇ · D^{−1}P_{k+1}) + [∇ · D^{−1}, ∇_{L}]P_{k+1} + ∇ · D^{−1}E_{k+1}.

Note that

[∇_{L}, ∇ · D^{−1}]P = [∇_{L}, ∇]D^{−1}P + ∇ · [∇_{L}, D^{−1}]P

= (A + \frac{1}{s}) · ∇D^{−1}P + (A + \frac{1}{s}) · A · D^{−1}P + β · D^{−1}P

+ ∇D^{−1}((A + \frac{1}{s}) · ∇ · D^{−1}P + (A + \frac{1}{s}) · A · D^{−1}P + β · D^{−1}P )

= [∇_{L}, ∇ · D^{−1}]_{g}P + β · D^{−1}P + ∇ · D^{−1}(β · D^{−1}P),  (7.98)

where [∇_{L}, ∇ · D^{−1}]_{g} represents the good part of the commutator [∇_{L}, ∇ · D^{−1}]P, for which, by Lemma 7.2 we have

∥[∇_{L}, ∇ · D^{−1}]_{g}P ∥_{P^{0}} ≲ N_{1}(P).  (7.99)

Then we have

∇ · D^{−1}(β · D^{−1}P_{k}) = ∇_{L}(∇ · D^{−1}P_{k+1}) + β · D^{−1}P_{k+1} + ∇ · D^{−1}(β · D^{−1}P_{k+1})
+ [∇_{L}, ∇ · D^{−1}]_{g}P_{k+1} + ∇ · D^{−1}E_{k+1}.

We will repeatedly use this formula in the following iteration.

With the help of (7.99) and (7.96) we decompose I := ∇ · D^{−1}(β · F) as

I = ∇ · D^{−1}(∇_{L}P_{1} + E_{1})

= ∇_{L}(∇ · D^{−1}P_{1}) + β · D^{−1}P_{1} + ∇ · D^{−1}(β · D^{−1}P_{1}) + [∇ · D^{−1}, ∇_{L}]_{g}P_{1} + ∇ · D^{−1}E_{1} = ∇_{L}(∇ · D^{−1}P_{1}) + ∇_{L}P_{2} + E_{2} + ∇ · D^{−1}(∇_{L}P_{2} + E_{2})

+ [∇ · D^{−1}, ∇_{L}]_{g}P_{1} + ∇ · D^{−1}E_{1} = ∇_{L}P̄_{2} + ∇ · D^{−1}(∇_{L}P_{2}) + Ē_{2}, where

8In particular for F being 1-forms

∥E_{k+1}∥_{P^{0}} ≤ (cΔ_{0})^{k}N_{2}(F).

where c is some positive universal constant.

P̄_{2} = ∇ · D^{−1}P_{1} + P_{2},
Ē_{2} = [∇ · D^{−1}, ∇_{L}]_{g}P_{1} + E_{2} + ∇ · D^{−1}E_{2} + ∇ · D^{−1}E_{1}.

We continue to decompose ∇ · D^{−1}(∇_{L}P_{2}) as above and derive

I = ∇_{L}P̄_{k} + ∇ · D^{−1}(∇_{L}P_{k}) + Ē_{k}
P̄_{k} = ∇ · D^{−1}(P_{1} + P_{2} + ... + P_{k−1}) + P_{2} + ... + P_{k}
Ē_{k} = [∇ · D^{−1}, ∇_{L}]_{g}(P_{1} + P_{2} + ... + P_{k−1}) + ∇ · D^{−1}(E_{1} + E_{2} + ... + E_{k−1})
+ E_{2} + ... + E_{k},

with

N_{1}(P̄_{k} − P̄_{j}) ≤ N_{2}(F) ∑_{k≥m≥j+1}(cΔ_{0})^{m} ≲ Δ_{0}N_{2}(F),

and

∥Ē_{k} − Ē_{j}∥_{P^{0}} ≲ ∥E_{j+1}∥_{P^{0}} + ⋯ + ∥E_{k}∥_{P^{0}} + ∥∇ · D^{−1}E_{j}∥_{P^{0}} + ⋯ + ∥∇ · D^{−1}E_{k−1}∥_{P^{0}}

+ N_{1}(P_{j}) + ⋯ + N_{1}(P_{k−1})

≤ (N_{2}(F) + ∥∇_{L}F ∥_{L_{t}^{a}L_{x}^{2}}) ∑_{m≥j−1}(cΔ_{0})^{m}

≲ Δ_{0}(N_{2}(F) + ∥∇_{L}F ∥_{L_{t}^{a}L_{x}^{2}}),

The second inequality is derived by using (7.99), (7.96) and (7.97).

Therefore we can conclude {P̄_{k}} forms a Cauchy sequence relative to the norm N_{1}(·), while {Ē_{k}} forms a Cauchy sequence relative to the norm ∥ · ∥_{P^{0}}. Denote by P̄ and Ē their corresponding limits, we have

N_{1}(P̄) ≲ Δ_{0}N_{2}(F) and ∥Ē∥_{P^{0}} ≲ Δ_{0}(N_{2}(F) + ∥∇_{L}F ∥_{L_{t}^{a}L_{x}^{2}}).

We also observe that for sufficiently small Δ_{0},

∥I − ∇_{L}P̄_{k} − Ē_{k}∥_{L_{t}^{2}L_{x}^{2}} = ∥∇ · D^{−1}(∇_{L}P_{k})∥_{L_{t}^{2}L_{x}^{2}} ≤ N_{1}(P_{k}) ≤ (cΔ_{0})^{k−1}N_{2}(F).

When k → +∞,we get

∥I − ∇_{L}P̄ − Ē∥_{L_{t}^{2}L_{x}^{2}} = 0,

whence I = ∇_{L}P̄ + Ē. This completes the proof of Theorem 7.1 (iii).

Step 2. We will show Proposition 7.10 with the help of Proposition 7.7.

Proof of (7.97). Let us start with estimating ∥∇D^{−1}E_{k}∥_{P^{0}} by using (6.11), i.e.

∥∇D^{−1}E_{k}∥_{P^{0}} ≲ ∥E_{k}∥_{P^{0}} + Δ_{0}∥D^{−1}E_{k}∥_{L_{t}^{a}L_{x}^{2}}^{q}∥E_{k}∥_{L^{2}(H)}^{1−q} with 4 < a < ∞, 1 > q > γ_{0}.

Thus in order to prove (7.97), it suffices to show

∥D^{−1}E_{k}∥_{L_{t}^{a}L_{x}^{2}} ≲ Δ_{0}(N_{2}(D^{−1}P_{k−1}) + ∥∇_{L}D^{−1}P_{k−1}∥_{L_{t}^{a}L_{x}^{2}}) with 4 < a < ∞. (7.100)

We will first show it holds for k = 1, i.e.

∥D^{−1}E_{1}∥_{L_{t}^{a}L_{x}^{2}} ≲ Δ_{0}(N_{2}(F) + ∥∇_{L}F ∥_{L_{t}^{a}L_{x}^{2}}).  (7.101)

To this end, in view of E_{1} = E_{1}^{G} + E_{1}^{B}, we start with estimating ∥D^{−1}E_{1}^{B}∥_{L_{t}^{a}L_{x}^{2}} by using Proposition 6.1 followed by Hölder inequality

∥D^{−1}E_{1}^{B}∥_{L_{t}^{a}L_{x}^{2}} ≲ ∥t^{\frac{1}{2}} E_{1}^{B}∥_{L_{t}^{a}L_{x}^{4/3}} ≲ ∥D_{1}^{−1}Ř∥_{L_{t}^{∞}L_{x}^{4}}∥∇_{L}F ∥_{L_{t}^{a}L_{x}^{2}}
≲ (Δ_{0}^{2} + R_{0})∥∇_{L}F ∥_{L_{t}^{a}L_{x}^{2}}.

By Proposition 7.7,

∥D^{−1}E_{1}^{G}∥_{L_{t}^{a}L_{x}^{2}} ≲ Δ_{0}N_{2}(F).

Thus (7.101) is proved.

(7.100) can be proved similarly by taking F to be D^{−1}P_{k−1}. This completes the proof of (7.97).

Remark 7.3. Note that we only need to use N_{2}(F) to control ∥D^{−1}E_{1}^{G}∥_{L_{t}^{a}L_{x}^{2}}. However, to bound ∥D^{−1}E_{1}^{B}∥_{L_{t}^{a}L_{x}^{2}}, an estimate of ∥∇_{L}F ∥_{L_{t}^{a}L_{x}^{2}} is unavoidable.

Proof of (7.96). We prove (7.96) for the case k = 1 first. Note that

∥∇_{L}D^{−1}P_{1}∥_{L_{t}^{a}L_{x}^{2}} ≲ ∥[∇_{L}, D^{−1}]P_{1}∥_{L_{t}^{a}L_{x}^{2}} + ∥D^{−1}∇_{L}P_{1}∥_{L_{t}^{a}L_{x}^{2}}.  (7.102)

The first term of the right hand side of (7.102) can be estimated by (7.48)

∥C_{0}(P_{1})∥_{L_{t}^{a}L_{x}^{2}} ≲ N_{1}(t^{\frac{1}{a} − \frac{1}{2}} D^{−1}P_{1}) ≲ N_{2}(D^{−1}P_{1}) ≲ N_{1}(P_{1}).

By using the relation ∇_{L}P_{1} = β ·F −E_{1}, the second term of (7.102) can be estimated as

∥D^{−1}∇_{L}P_{1}∥_{L_{t}^{a}L_{x}^{2}} ≲ ∥D^{−1}(β · F − E_{1})∥_{L_{t}^{a}L_{x}^{2}}
≲ ∥D^{−1}(β · F)∥_{L_{t}^{a}L_{x}^{2}} + ∥D^{−1}E_{1}∥_{L_{t}^{a}L_{x}^{2}}.

Then according to (7.101) and (7.47) we conclude

∥D^{−1}∇_{L}P_{1}∥_{L_{t}^{a}L_{x}^{2}} ≲ (R_{0} + Δ_{0})(N_{2}(F) + ∥∇_{L}F ∥_{L_{t}^{a}L_{x}^{2}}).

Therefore (7.96) is proved for the case k = 1. Actually (7.48) implies the more general estimates for {P_{k}}_{k=0}^{∞} as follows

∥[∇_{L}, D^{−1}]P_{k}∥_{L_{t}^{a}L_{x}^{2}} ≲ N_{1}(t^{\frac{1}{a} − \frac{1}{2}} D^{−1}P_{k}) ≲ N_{2}(D^{−1}P_{k}) ≲ N_{1}(P_{k}).  (7.103)

For k > 1, combine (7.103) with the formula β · D^{−1}P_{k−1} = ∇_{L}P_{k} + E_{k}, (7.100) and

N_{2}(D^{−1}P_{k}) ≤ CN_{1}(P_{k}) ≤ (cΔ_{0})N_{2}(D^{−1}P_{k−1}),

where C, c are some universal positive constants, we can get (7.96) by replacing F by D^{−1}P_{k−1} and repeating the above procedure.

Step 3. It remains to prove part (iv) of Theorem 7.1. From (7.87), we can write β · H = ∇_{L}P + E with N_{1}(P) ≲ Δ_{0}N_{2}(F) and ∥E∥_{P^{0}} ≲ Δ_{0} · (N_{2}(F) + ∥∇_{L}F ∥_{L_{t}^{a}L_{x}^{2}}). As before we can construct two Cauchy sequences {P_{k}} and {E_{k}} such that β · (D^{−1}P_{k}) = ∇_{L}P_{k+1} + E_{k+1} with

N_{1}(P_{k+1}) ≤ (cΔ_{0})^{k}N_{2}(F) and ∥E_{k+1}∥_{P^{0}} ≤ (cΔ_{0})^{k} · (N_{2}(F) + ∥∇_{L}F ∥_{L_{t}^{a}L_{x}^{2}}),

where c is a universal constant. By induction, we can continue to decompose I_{k+1} := D^{−1}(β · D^{−1}P_{k}) as follows

I_{k+1} = D^{−1}(∇_{L}P_{k+1} + E_{k+1}) = ∇_{L}(D^{−1}P_{k+1}) + [D^{−1}, ∇_{L}]P_{k+1} + D^{−1}E_{k+1}
= ∇_{L}(D^{−1}P_{k+1}) + D^{−1}E_{k+1} + [D^{−1}, ∇_{L}]_{g}P_{k+1} + D^{−1}(β · D^{−1}P_{k+1}).

Since (7.31) implies

∥∇[D^{−1}, ∇_{L}]_{g}P_{k+1}∥_{P^{0}} ≲ N_{1}(P_{k+1}),

by setting

Ẽ_{k+1} = [D^{−1}, ∇_{L}]_{g}P_{k+1} + D^{−1}E_{k+1} and P̃_{k+1} = D^{−1}P_{k+1}

we can rewrite I_{k+1} as

I_{k+1} = ∇_{L}P̃_{k+1} + Ẽ_{k+1} + D^{−1}(β · D^{−1}P_{k+1}),

where

∥∇D^{−1}Ẽ_{k+1}∥_{P^{0}} ≤ (cΔ_{0})^{k} · (N_{2}(F) + ∥∇_{L}F ∥_{L_{t}^{a}L_{x}^{2}}),

and by Lemma 4.4

N_{2}(P̃_{k+1}) ≲ N_{1}(P_{k+1}) ≤ (cΔ_{0})^{k}N_{2}(F).

Following the same steps as before we can easily get (iv). Thus we complete the proof of Theorem 7.1.

In view of Corollary 7.1, Proposition 7.8, we can apply Theorem 7.1 to renormalize the “bad” terms such as β · F and ∇ · D^{−1}(β · F) with F = D^{−2}Ř. Also using Proposition 7.1, we deduce that all the commutators which we have denoted by C(Ř) can be decomposed into the sum between an error term and another which is a perfect ∇_{L} derivative. More precisely,

Proposition 7.11. All the commutators C(Ř) can be expressed as follows

C(Ř) = ∇_{L}P + E,

where P and E are tensors verifying N_{1}(P), ∥E∥_{P^{0}} ≲ Δ_{0}^{2} + R_{0}.

Main Estimate

In this chapter, we will prove the main result of this thesis, Theorem 4.1. We will write the equations satisfied by our main quantities in a schematic form to which we can apply Theorems 5.1–5.3. We will also rely on (BA1), (BA2) and the following auxiliary bootstrap assumption

∥A · \underline{A}∥_{P^{0}} ≲ Δ_{0}^{2}.  (BA4)

This additional assumption will be used only to derive the desired estimates for ζ, μ, tr\underline{χ}, \underline{χ̂}, and not for χ̂, trχ, ∇trχ. The assumption (BA4) will be justified at the end of this chapter.

8.1 Estimates for trχ, χ̂

Step 1. Recall (3.10), for V := trχ − \frac{2}{s} we have the transport equation

∇_{L}V = −\frac{2}{s}V − \frac{1}{2}V^{2} − |χ̂|^{2}  (8.1)

with the initial condition strχ − 2 = O(s^{2}) as s → 0 given by Proposition 3.1. Integrate along any null geodesic, i.e. integrate in the affine parameter 0 < s ≤ 1, we obtain

∥trχ − \frac{2}{t}∥_{L_{t}^{∞}L_{ω}^{∞}} ≲ ∥χ̂∥_{L_{ω}^{∞}L_{t}^{2}}^{2} + ∥V ∥_{L_{ω}^{∞}L_{t}^{2}}^{2}.

By (BA1) we have

∥trχ − \frac{2}{t}∥_{L_{t}^{∞}L_{ω}^{∞}} ≲ Δ_{0}^{2}.  (8.2)

We can easily get from (8.1), with the help of (BA1), (3.39) and Hölder inequality, that

∥∇_{L}V ∥_{L_{x}^{2}L_{t}^{∞}} ≲ ∥t^{−1}V ∥_{L_{x}^{2}L_{t}^{∞}} + ∥V^{2}∥_{L_{x}^{2}L_{t}^{∞}} + ∥|χ̂|^{2}∥_{L_{x}^{2}L_{t}^{∞}} ≲ Δ_{0}^{2} + Δ_{0}

Step 2. Estimates for ∇trχ.

We recall the related transport equation (3.12)

∇_{L}(∇trχ) = −\frac{3}{2}trχ · ∇trχ − χ̂ · ∇trχ − 2χ̂ · ∇χ̂

and the Codazzi equation (3.16)

div χ̂ = \frac{1}{2}∇trχ − \frac{1}{2}trχ · ζ − χ̂ · ζ − β

From the renormalized null Bianchi equations (3.21) and (3.22), which schematically can be written into the form

D_{1}β = L(ρ̌, −σ̌) + \frac{1}{s}Ř + A · R̄

we have β = D_{1}^{−1}L(ρ̌, −σ̌) + F_{1}, with F_{1} = Err. We derive from the Codazzi equation (3.16) that

χ̂ = −D_{2}^{−1}D_{1}^{−1}∇_{L}(ρ̌, −σ̌) + D_{2}^{−1}F_{1} + D_{2}^{−1}(∇trχ + A · A + s^{−1}ζ).

Taking covariant derivative, we obtain by setting D^{−2} = D_{2}^{−1}D_{1}^{−1} and denoting by D^{−1} the operator D_{2}^{−1},

∇χ̂ = −∇ · D^{−2}∇_{L}(Ř) + ∇D^{−1}(F_{1} + F_{2}) + ∇ · D^{−1}M,  (8.3)

where F_{2} = A · A + s^{−1}ζ and M = ∇trχ.

Denote by F the term ∇ · D^{−1}(F_{1} + F_{2}), we claim

∥F ∥_{P^{0}} ≲ Δ_{0}^{2}.  (8.4)

Indeed, according to (7.71), ∥∇D^{−1}F_{1}∥_{P^{0}} ≲ Δ_{0}^{2}. We only need to show for F_{2} = A · A + s^{−1}ζ

∥∇ · D^{−1}F_{2}∥_{P^{0}} ≲ Δ_{0}^{2}.  (8.5)

By non-sharp product estimate,

∥A · A∥_{P^{0}} ≲ (∥∇A∥_{L^{2}(H)} + ∥A∥_{L_{x}^{∞}L_{t}^{2}})N_{1}(A) ≲ Δ_{0}^{2}.

It remains to estimate ∥s^{−1}ζ∥_{P^{0}}. It follows from Lemma 6.2 that

∥s^{−1}ζ∥_{P^{0}} ≲ ∥∇ζ∥_{L_{t}^{2}L_{x}^{2}} + ∥s^{−1}ζ∥_{L_{t}^{2}L_{x}^{2}}.

In view of the Hodge system

div ζ = −μ − ρ̌ + |ζ|^{2}
curl ζ = σ̌,

by Proposition 4.2 (i),

∥∇ζ∥_{L^{2}(H)} + ∥t^{−1}ζ∥_{L^{2}(H)} ≲ ∥μ∥_{L^{2}(H)} + ∥R∥_{L^{2}(H)} + ∥A · A∥_{L^{2}(H)}.

Thus, provided ∥μ∥_{L^{2}(H)} ≲ Δ_{0}^{2} + R_{0}, we can conclude

∥s^{−1}ζ∥_{P^{0}(H)} ≲ ∥∇ζ∥_{L_{t}^{2}L_{x}^{2}} + ∥s^{−1}ζ∥_{L_{t}^{2}L_{x}^{2}} ≲ Δ_{0}^{2} + R_{0}.  (8.6)

For μ, we have the transport equation (3.24), which symbolically can be written as follows

\frac{d}{ds}μ + \frac{3}{s}μ =χ̂ · ∇ˆ⊗ζ + \frac{1}{s}Ř + A · R̄.

Integrating the above equation in s, we derive

∥μ∥_{L^{2}(H)} ≲ ∥t^{−2} ∫_{0}^{t} s^{3}(A · ∇A + s^{−1}Ř + A · R̄)ds∥_{L_{t}^{2}L_{ω}^{2}}.  (8.7)

By the inequality for scalar functions f,

∥\frac{1}{t} ∫_{0}^{t} fds∥_{L_{t}^{2}} ≲ ∥f∥_{L_{t}^{2}},  (8.8)

we deduce from (8.7) and lim_{s→0} sμ = 0 in Proposition 3.2 that

∥μ∥_{L^{2}(H)} ≲ ∥t^{2}(A · ∇A + t^{−1}Ř + A · R̄)∥_{L_{t}^{2}L_{ω}^{2}}
≲ ∥A · ∇A∥_{L^{2}(H)} + ∥Ř∥_{L^{2}(H)} + ∥A · R̄∥_{L^{2}(H)}
≲ ∥A∥_{L_{x}^{∞}L_{t}^{2}}(∥∇A∥_{L^{2}(H)} + ∥R̄∥_{L^{2}(H)}) + R_{0} ≲ Δ_{0}^{2} + R_{0}.

Therefore (8.4), (8.5) and (8.6) are proved.

Now, we consider the first term of (8.3). By using the notations of commutators (7.6), we can write it (8.3) as

∇ · D^{−2}∇_{L}(Ř) = ∇_{L}(∇ · D^{−2}Ř) + C(Ř).

According to Proposition 7.11, there exist tensors P′ and E′ so that

C(Ř) = ∇_{L}P′ + E′

with N_{1}(P′), ∥E′∥_{P^{0}} ≲ Δ_{0}^{2}. Set P = ∇ · D^{−2}Ř + P′, due to Corollary 7.1 which says N_{1}(∇·D^{−2}Ř) ≲ R_{0} +Δ_{0}^{2}, we then infer N_{1}(P) ≲ Δ_{0}^{2} +R_{0}. Therefore ∇χ̂ can be written schematically as

∇χ̂ = ∇_{L}P + ∇ · D^{−1}M + E  (8.9)

where E = F + E′, with N_{1}(P), ∥E∥_{P^{0}} ≲ Δ_{0}^{2} + R_{0}.

From the transport equation (3.12) and lim_{s→0} s∇trχ = 0 in Proposition 3.2,

∥M∥_{L_{x}^{2}L_{t}^{∞}} ≲ ∥t^{−3} ∫_{0}^{t} s^{3}A · ∇Ads∥_{L_{x}^{2}L_{t}^{∞}}
≲ ∥∇A · A∥_{L_{x}^{2}L_{t}^{1}} ≲ Δ_{0}^{2}.

Below we prove ∥∇ · D^{−1}M∥_{P^{0}} ≲ Δ_{0}^{2} + R_{0}.

By Proposition 6.1, we can derive

∥D^{−1}M∥_{L_{t}^{a}L_{x}^{2}} ≲ ∥tM∥_{L_{t}^{a}L_{x}^{2}} ≲ ∥M∥_{L_{x}^{2}L_{t}^{∞}} ≲ Δ_{0}^{2}.

Then, we infer from (6.11) that

∥∇ · D^{−1}M∥_{P^{0}} ≲ ∥M∥_{P^{0}} + Δ_{0}^{2}.

Next, we estimate ∥M∥_{P^{0}}. This can be achieved by using the symbolic transport equation,

∇_{L}M + \frac{3}{s}M = A · (∇_{L}P + ∇ · D^{−1}M + E) − \frac{3}{2}V · M

which is derived from (3.12). Utilizing Proposition 5.4, (BA1) and lim_{s→0} s∇trχ = 0 in Proposition 3.2, we infer

∥M∥_{B^{0}} ≲ ∥t^{−3} ∫_{0}^{t} s^{3}A(∇_{L}P + ∇ · D^{−1}M + E)ds∥_{B^{0}} + ∥t^{−3} ∫_{0}^{t} s^{3}V · Mds∥_{B^{0}}
≲ (N_{1}(P) + ∥E∥_{P^{0}} + ∥M∥_{P^{0}} + Δ_{0}^{2})(N_{1}(A) + ∥A∥_{L_{ω}^{∞}L_{t}^{2}}).

Since ∥M∥_{P^{0}} ≲ ∥M∥_{B^{0}}, we obtain

∥M∥_{B^{0}} ≲ Δ_{0}(∥M∥_{B^{0}} + Δ_{0}^{2} + R_{0}).  (8.10)

Hence we get

∥M∥_{B^{0}}, ∥M∥_{P^{0}}, ∥∇ · D^{−1}M∥_{P^{0}} ≲ Δ_{0}^{2} + R_{0}.

Then set Ẽ = E + ∇ · D^{−1}M where E is defined in (8.9), we have the following decomposition for ∇χ̂,

∇χ̂ = ∇_{L}P + Ẽ,  (8.11)

with N_{1}(P) ≲ Δ_{0}^{2} + R_{0} and ∥Ẽ∥_{P^{0}} ≲ Δ_{0}^{2} + R_{0}.

Step 3. Estimates for N_{1}(χ̂) and ∥χ̂∥_{L_{ω}^{∞}L_{t}^{2}}.

We will first estimate N_{1}(χ̂) = ∥t^{−1}χ̂∥_{L_{t}^{2}L_{x}^{2}} + ∥∇χ̂∥_{L_{t}^{2}L_{x}^{2}} + ∥∇_{L}χ̂∥_{L_{t}^{2}L_{x}^{2}}. In order to estimate ∥t^{−1}χ̂∥_{L_{t}^{2}L_{x}^{2}} + ∥∇χ̂∥_{L_{t}^{2}L_{x}^{2}}, we utilize the elliptic estimates in Proposition 4.2 and the Codazzi equation (3.16) to get

∥t^{−1}χ̂∥_{L_{t}^{2}L_{x}^{2}} + ∥∇χ̂∥_{L_{t}^{2}L_{x}^{2}}
≲ ∥β∥_{L_{t}^{2}L_{x}^{2}} + ∥∇trχ∥_{L_{t}^{2}L_{x}^{2}} + ∥A · A∥_{L_{t}^{2}L_{x}^{2}}
≲ R_{0} + Δ_{0}^{2}

By the transport equation (3.11), it is straightforward to derive ∥∇_{L}χ̂∥_{L_{t}^{2}L_{x}^{2}} by taking the L^{2}(H) norm, i.e.

∥∇_{L}χ̂∥_{L_{t}^{2}L_{x}^{2}} ≲ ∥trχ · χ̂∥_{L_{t}^{2}L_{x}^{2}} + ∥α∥_{L_{t}^{2}L_{x}^{2}}
≲ ∥V · χ̂∥_{L_{t}^{2}L_{x}^{2}} + ∥t^{−1}χ̂∥_{L_{t}^{2}L_{x}^{2}} + R_{0}
≲ Δ_{0}^{2} + R_{0}.

Hence we conclude that

N_{1}(χ̂) ≲ R_{0} + Δ_{0}^{2}.

To estimate the trace norm ∥χ̂∥_{L_{ω}^{∞}L_{t}^{2}}, we use Theorem 5.4 and (8.11) to get

∥χ̂∥_{L_{ω}^{∞}L_{t}^{2}} ≲ N_{1}(χ̂) + N_{1}(P) + ∥E∥_{P^{0}} ≲ Δ_{0}^{2} + R_{0}.

In summary, in this section we have obtained the following results

∥trχ − \frac{2}{t}∥_{L_{t}^{∞}L_{ω}^{∞}} ≲ Δ_{0}^{2} + R_{0} ∥∇_{L}(trχ − \frac{2}{t})∥_{L_{x}^{2}L_{t}^{∞}} ≲ Δ_{0}^{2} + R_{0}
∥∇trχ∥_{L_{x}^{2}L_{t}^{∞}} ≲ Δ_{0}^{2} + R_{0} ∥χ̂∥_{L_{ω}^{∞}L_{t}^{2}} + N_{1}(χ̂) ≲ R_{0} + Δ_{0}^{2}
∥∇trχ∥_{B^{0}} ≲ R_{0} + Δ_{0}^{2}

In particular we can choose R_{0} sufficiently small and check that the bootstrap assumptions (BA1) for trχ and χ̂ are verified with Δ_{0} replaced by Δ_{0}/2. We also deduce all the estimates for trχ and χ̂ in Theorem 4.1.

8.2 Estimates for μ, and ∇ζ

Step 1 Estimates for μ.

Recall the mass aspect function

μ = − div ζ + \frac{1}{2}χ̂ · \underline{χ̂} − ρ + |ζ|^{2}

and the transport equation (3.24),

\frac{d}{ds}μ + \frac{3}{s}μ =χ̂ · ∇ˆ⊗ζ + \frac{1}{s}ρ̌ + A · R̄.

Decompose ∇ζ as we did for ∇χ̂ by using the elliptic system

div ζ = −μ − ρ̌ + |ζ|^{2} curl ζ = σ̌,

which, according to the definition of D_{1}, can be rewritten as follows:

D_{1}ζ = −(ρ̌, −σ̌) − (μ, 0) + A · A.  (8.12)

By using the null Bianchi identity (3.23) and following the same way as in [12, Sec.7.3], we obtain

(−ρ, σ) = J ·^{⋆}D_{1}^{−1} · ∇_{L} ˇ\underline{β} + J · F_{1} + J ·^{⋆}D_{1}^{−1}(\frac{1}{s}R_{0})  (8.13)

where J is the involution (ρ, σ) → (−ρ, σ), R_{0} = \underline{β} and F_{1} = Ẽrr is given by (7.4). Therefore, set F_{2} = A · A, combining (8.13) with (8.12), we get

ζ =D_{1}^{−1} · J ·^{⋆}D_{1}^{−1}(∇_{L} ˇ\underline{β} + \frac{1}{s}R_{0}) + D_{1}^{−1} · J · F_{1} − D_{1}^{−1}(μ, 0) + D_{1}^{−1}F_{2}.

Set D^{−2} = D_{1}^{−1} · J ·^{⋆}D_{1}^{−1} and D^{−1} = D_{1}^{−1}, similar to (8.3), we get

∇ζ = ∇ · D^{−2}(∇_{L} ˇ\underline{β} + \frac{1}{s}R_{0}) + ∇ · D^{−1}(F_{1} + F_{2} + M),

where M = (μ, 0).

We still denote by F the term ∇ · D^{−1}(F_{1} + F_{2}). By (8.4) we have ∥F ∥_{P^{0}} ≲ Δ_{0}^{2} + R_{0}. Next we rewrite ∇ζ as follows

∇ζ = ∇_{L}(∇ · D^{−2} ˇ\underline{β}) + C(Ř) + ∇ · D^{−1}M + F + \frac{1}{s}∇D^{−2}R_{0}.

Since there exists P′ and E′ so that C(Ř) = ∇_{L}P′ +E′ with N_{1}(P′), ∥E′∥_{P^{0}} ≲ Δ_{0}^{2} +R_{0}, set P = −∇ · D^{−2} ˇ\underline{β} + P′ and E = E′ + F + \frac{1}{s}∇D^{−2}R_{0}, we can write ∇ζ in the following form:

∇ζ = ∇_{L}P + ∇ · D^{−1}M + E

where M = (μ, 0). Return to the transport equation of M,

\frac{d}{ds}M + \frac{3}{s}M =χ̂ (∇_{L}P + ∇ · D^{−1}M + E) + \frac{1}{s}ρ̌ + V · ρ̌ + A · (A · \underline{A} + M),

we will decompose ρ̌ by using Bianchi identity again. By the renormalized null Bianchi equation (7.1), we have

(ρ, σ) = ^{⋆}D_{1}^{−1}∇_{L}β̌ + F_{1} = ∇_{L}(^{⋆}D_{1}^{−1} ˇ\underline{β}) + [∇_{L},^{⋆}D_{1}^{−1}]Ř + F_{1}.

Since

∥[∇_{L},^{⋆} D_{1}^{−1}]Ř∥_{P^{0}} ≲ Δ_{0}^{2} + R_{0}

and ∥F_{1}∥_{P^{0}} ≲ Δ_{0}^{2}, we set e′ = [∇_{L},^{⋆}D_{1}^{−1}]Ř + F_{1} and set p′ = ^{⋆}D_{1}^{−1} ˇ\underline{β}, then (ρ, σ) can be rewritten as follows:

(ρ, σ) = ∇_{L}p′ + e′.

Denote χ̂ · \underline{χ̂} + e′ still by e′, by the bootstrap assumption (BA4), we can conclude

(ρ̌, σ̌) = ∇_{L}p′ + e′

with N_{1}(p′) = N_{1}(D^{−1}Ř) ≲ Δ_{0}^{2} + R_{0} and ∥e′∥_{P^{0}} ≲ Δ_{0}^{2} + R_{0}. Therefore the transport equation (3.24) can be rewritten as follows

\frac{d}{ds}M + \frac{3}{s}M = A · ∇_{L}P̃ + A · (∇ · D^{−1}M + Ẽ) + \frac{1}{s}(∇_{L}p′ + e′)

with Ẽ = E + A · \underline{A} + M + e′, P̃ = P + p′ and the estimates

N_{1}(P̃) ≲ R_{0} + Δ_{0}^{2}, ∥Ẽ∥_{P^{0}} ≲ R_{0} + Δ_{0}^{2}.

Set

W = ∇ · D^{−1}M + Ẽ

then

∇_{L}M + \frac{3}{s}M = A · ∇_{L}P̃ + \frac{1}{s}(∇_{L}p′ + e′) + A · W.  (8.14)

We first claim the following estimates hold true

∥\frac{1}{t^{3}} ∫_{0}^{t} s^{2}e′ds∥_{P^{0}}, ∥\frac{1}{t^{3}} ∫_{0}^{t} s^{2}∇_{L}p′ds∥_{P^{0}} ≲ Δ_{0}^{2} + R_{0}.  (8.15)

Indeed, in view of Proposition 4.1, the first term of (8.15) can be estimated as follows

∥\frac{1}{t^{3}} ∫_{0}^{t} s^{2}e′ds∥P̃^{0} = ∑_{k>0} ∥\frac{1}{t^{2}} ∫_{0}^{t} s^{2}E_{k}e′ds∥_{L_{t}^{2}L_{ω}^{2}} + ∥\frac{1}{t^{2}} ∫_{0}^{t} s^{2}e′ds∥_{L_{t}^{2}L_{ω}^{2}}

≲ ∑_{k>0} ∥\frac{1}{t} ∫_{0}^{t} s|E_{k}e′|ds∥_{L_{t}^{2}L_{ω}^{2}} + ∥\frac{1}{t} ∫_{0}^{t} ∥se′∥_{L_{ω}^{2}}∥_{L_{t}^{2}}

≲ ∑_{k>0} ∥sE_{k}e′∥_{L_{t}^{2}L_{ω}^{2}} + ∥e′∥_{L_{t}^{2}L_{x}^{2}} ≲ ∥e′∥_{P^{0}} ≲ Δ_{0}^{2} + R_{0},

where we employed (8.8).

Now we estimate the second term of (8.15). Recall that p′ =^{⋆} D_{1}^{−1} ˇ\underline{β}, then

lim_{s→0} ∥s^{−1}p′∥_{B_{2,1}^{0}} ≲ N_{1}(p′) ≲ Δ_{0}^{2} + R_{0}  (8.16)
∥s^{−1}p′∥_{P^{0}} ≲ N_{1}(p′) ≲ R_{0} + Δ_{0}^{2}.  (8.17)

Applying Proposition 4.1 to the second term of (8.15), with the help of (8.16), (8.17) and (8.8) we derive

∥\frac{1}{t^{2}} ∫_{0}^{t} s^{2}∇_{L}p′ds∥P̃^{0} ≲ ∑_{k>0} ∥E_{k} \frac{1}{t^{2}} ∫_{0}^{t} s^{2}∇_{L}p′ds∥_{L_{t}^{2}L_{ω}^{2}} + ∥\frac{1}{t^{2}} ∫_{0}^{t} s^{2}∇_{L}p′ds∥_{L_{t}^{2}L_{ω}^{2}}

≲ ∑_{k>0}{∥E_{k}(p′ − lim_{s→0} p′)∥_{L_{t}^{2}L_{ω}^{2}} + ∥E_{k}t^{−2} ∫_{o}^{t} sp′ds∥_{L_{t}^{2}L_{ω}^{2}}} + ∥∇_{L}p′∥_{L_{t}^{2}L_{x}^{2}}

≲ ∑_{k>0} ∥E_{k}p′∥_{L_{t}^{2}L_{ω}^{2}} + ∑_{k>0} ∥\frac{1}{t^{2}} ∫_{0}^{t} sE_{k}p′ds∥_{L_{t}^{2}L_{ω}^{2}} + lim_{s→0} ∥s^{−1}p′∥B̃_{2,1}^{0} + ∥∇_{L}p′∥_{L_{t}^{2}L_{x}^{2}}

≲ ∥t^{−1}p′∥_{P^{0}} + ∥∇_{L}p′∥_{L_{t}^{2}L_{x}^{2}} + lim_{s→0} ∥s^{−1}p′∥_{B_{2,1}^{0}} ≤ R_{0} + Δ_{0}^{2}.

Thus by (8.17) and (7.26), we obtain (8.15).

Note following in the same manner as above, using Hölder inequality instead of (8.8), we can get ∥t^{1/2}μ∥_{B^{0}} ≲ Δ_{0}^{2} + R_{0}.

Now we estimate ∥D^{−1}μ∥_{L_{t}^{a}L_{x}^{2}} with 4 < a < ∞. By the definition of μ, Proposition 6.1 and Proposition 7.6

∥D^{−1}μ∥_{L_{t}^{a}L_{x}^{2}} ≲ ∥D^{−1}(ρ̌ + A · \underline{A} + div ζ)∥_{L_{t}^{a}L_{x}^{2}}
≲ N_{1}(D^{−1}Ř) + N_{1}(ζ) + ∥tE_{4}∥_{L_{t}^{a}L_{x}^{2}} ≲ Δ_{0}^{2} + R_{0}.

Thus, in view of (6.11), ∥∇ · D^{−1}M∥_{P^{0}} ≲ ∥M∥_{P^{0}} + Δ_{0}^{2} + R_{0}.

By applying Proposition 5.4 to the symbolic transport equation (8.14), we get by (8.15) that

∥M∥_{P^{0}} ≲ (N_{1}(P̃) + ∥W ∥_{P^{0}})(N_{1}(A) + ∥A∥_{L_{ω}^{∞}L_{t}^{2}}) + R_{0} + Δ_{0}^{2}
≲ (N_{1}(P̃) + ∥M∥_{P^{0}} + Δ_{0}^{2} + ∥Ẽ∥_{P^{0}})Δ_{0} + Δ_{0}^{2} + R_{0}

Note and R_{0} ≤ Δ_{0} < 1/2, we conclude

∥M∥_{P^{0}}, ∥∇ · D^{−1}M∥_{P^{0}}, ∥M∥_{L^{2}(H)} ≲ Δ_{0}^{2} + R_{0},

as desired.

∥t^{1/2}M∥_{L_{x}^{2}L_{t}^{∞}} can be estimated directly from its transport equation. Since in Proposition 3.2, we proved sμ → 0 as s → 0, then

∥t^{1/2}M∥_{L_{x}^{2}L_{t}^{∞}} ≲ R_{0} + ∥χ̂ · ∇ζ∥_{L_{x}^{2}L_{t}^{1}} + ∥A · ∇trχ∥_{L_{x}^{2}L_{t}^{1}} + ∥A · A · \underline{A}∥_{L_{x}^{2}L_{t}^{1}}
≲ R_{0} + Δ_{0}^{2}.

We therefore have verified the bootstrap assumptions for M = (μ, 0). Step 2. Estimates for N_{1}(ζ). From (3.13), we get

∥∇_{L}ζ∥_{L^{2}(H)} ≲ ∥A · A∥_{L^{2}(H)} + ∥β∥_{L^{2}(H)} + ∥s^{−1}ζ∥_{L^{2}(H)} ≲ Δ_{0}^{2} + R_{0}.

Combine this result with (8.6), N_{1}(ζ) ≲ R_{0} + Δ_{0}^{2}. Step 3. Estimates for ∥ζ∥_{L_{ω}^{∞}L_{t}^{2}}. From the calculation in Step 2, we have the following decomposition for ∇ζ,

∇ζ = ∇_{L}P + ∇ · D^{−1}M + E  (8.18)

with N_{1}(P) ≲ R_{0} + Δ_{0}^{2}, and ∥E∥_{P^{0}} ≲ R_{0} + Δ_{0}^{2}. Then by Trace theorem

∥ζ∥_{L_{ω}^{∞}L_{t}^{2}} ≲ N_{1}(ζ) + N_{1}(P) + ∥∇ · D^{−1}M + E∥_{P^{0}}
≲ R_{0} + Δ_{0}^{2}.

We summarize the above results in the following

∥μ∥_{L_{x}^{2}L_{t}^{2}} ≲ Δ_{0}^{2} + R_{0}, ∥μ∥_{P^{0}} ≲ Δ_{0}^{2} + R_{0}
N_{1}(ζ) ≲ R_{0} + Δ_{0}^{2}, ∥ζ∥_{L_{ω}^{∞}L_{t}^{2}} ≲ R_{0} + Δ_{0}^{2}
∥t^{\frac{1}{2}} μ∥_{L_{x}^{2}L_{t}^{∞}} ≲ Δ_{0}^{2} + R_{0}. ∥t^{1/2}μ∥_{B^{0}} ≲ Δ_{0}^{2} + R_{0}  (8.19)

In particular we can choose R_{0} sufficiently small and check that the bootstrap assumptions (BA1) and (BA2) for ζ are verified with Δ_{0} replaced by Δ_{0}/2. We also deduce all the estimates for ζ in the Theorem 4.1.

8.3 Estimates for (tr\underline{χ}, \underline{χ̂})

We consider the equation (3.14) in the symbolic form,

\frac{d}{ds}\underline{V} + \frac{1}{s}\underline{V} = −\frac{1}{2}V · \underline{V} + 2μ + \frac{1}{s}V + 4ρ̌,  (8.20)

according to Proposition 3.1, lim_{s→0} s\underline{V} (s, ω) = 0, we then infer

∥\underline{V} ∥_{L_{ω}^{2}L_{t}^{2}} ≲ ∥\frac{1}{t} ∫_{0}^{t} s\{V · \underline{V} + ρ̌ + μ + \frac{1}{s}V \}ds∥_{L_{t}^{2}L_{ω}^{2}}
≲ ∥tV · \underline{V} ∥_{L_{t}^{2}L_{ω}^{2}} + ∥tρ̌∥_{L_{t}^{2}L_{ω}^{2}} + ∥V ∥_{L_{t}^{2}L_{ω}^{2}} + ∥μ∥_{L_{t}^{2}L_{x}^{2}}.

In the same way as used in deriving ∥μ∥_{P^{0}}, we decompose ρ̌ into the form: ρ̌ = ∇_{L}p′+e′, with p′ = ^{⋆}D_{1}^{−1}\underline{β̌}, and ∥e′∥_{P^{0}} ≲ Δ_{0}^{2}. Using Proposition 4.1, ignoring the low frequency terms, followed by (8.8), we get

∥t^{−1}\underline{V} ∥_{P^{0}} ≲ ∑_{k>0} ∥E_{k} \frac{1}{t} ∫_{0}^{t} V + sμ + sV · \underline{V} + s(∇_{L}p′ + e′)ds∥_{L_{t}^{2}L_{ω}^{2}}
≲ ∥t^{−1}p′∥_{P^{0}} + ∥e′∥_{P^{0}} + ∥t^{−1}V ∥_{P^{0}} + ∥V · \underline{V} ∥_{P^{0}} + ∥μ∥_{P^{0}}.

Note that ∥t^{−1}V ∥_{P^{0}} ≲ ∥∇trχ∥_{L_{t}^{2}L_{x}^{2}} + ∥t^{−1}V ∥_{L^{2}(H)} ≲ Δ_{0}^{2} + R_{0}, by (BA4) we conclude

∥t^{−1}\underline{V} ∥_{P^{0}} ≲ Δ_{0}^{2} + R_{0}.

In the same way, using Hölder inequality instead of (8.8), we can obtain ∥t^{−1/2}\underline{V} ∥_{B^{0}} ≲ Δ_{0}^{2} + R_{0}.

Again, by integrating (8.20) in s, we derive the following,

∥t^{−1/2}\underline{V} ∥_{L_{x}^{2}L_{t}^{∞}} ≲ ∥\frac{1}{t^{1/2}} ∫_{0}^{t} s\{V · \underline{V} + ρ̌ + μ + \frac{1}{s}V \}ds∥_{L_{ω}^{2}L_{t}^{∞}}
≲ ∥tV · \underline{V} ∥_{L_{t}^{2}L_{ω}^{2}} + ∥ρ̌∥_{L_{x}^{2}L_{t}^{2}} + ∥V ∥_{L_{t}^{2}L_{ω}^{2}} + ∥μ∥_{L_{t}^{2}L_{x}^{2}} ≲ Δ_{0}^{2} + R_{0}.

It is also easy to check ∥∇_{L}\underline{V} ∥_{L^{2}(H)} ≲ Δ_{0}^{2} + R_{0} from (8.20).

Now consider the estimates of \underline{χ̂}. By (8.18), the structure equation (3.15) can be rewritten in the following form

∇_{L}\underline{χ̂} + \frac{1}{s}\underline{χ̂} = −(∇_{L}P + E) − \frac{1}{2}(\underline{V} · χ̂ + V · \underline{χ̂} − \frac{2}{s}χ̂) + ζ ˆ⊗ζ.  (8.21)

With the initial data lim_{s→0} |\underline{χ̂} | = 0 given by Proposition 3.1, using Lemma 4.2, we integrate both sides of the transport equation in s and obtain

∥t^{−1/2}\underline{χ̂} ∥B̃^{0} ≲ ∥A · \underline{A}∥P̃^{0} + ∥s^{−1}χ̂∥P̃^{0} + ∥E∥P̃^{0} + ∥t^{−1}P ∥P̃^{0} + ∥t^{−1/2}P ∥B̃^{0}.

By Proposition 4.1, we derive,

∥t^{−1/2}\underline{χ̂} ∥_{B^{0}} ≲ ∥A · \underline{A}∥_{P^{0}} + ∥s^{−1}χ̂∥_{P^{0}} + ∥E∥_{P^{0}} + ∥t^{−1}P ∥_{P^{0}} + ∥t^{−1/2}P ∥_{B^{0}},

where, due to Lemma 6.2, the last two terms are bounded by N_{1}(P). This gives

∥t^{−1/2}\underline{χ̂} ∥_{B^{0}} ≲ Δ_{0}^{2} + R_{0}.

Similarly, we can obtain ∥t^{−1}\underline{χ̂} ∥_{P^{0}} ≲ Δ_{0}^{2} + R_{0} by integrating (8.21) in s and using (8.8).

We will employ (3.15) to estimate L^{2}(H) and L_{x}^{2}L_{t}^{∞} norms of \underline{χ̂}. Using the vector fields verifying the condition (4.14), we still can reduce (3.15) to be a scalar equation. Unlike the above calculations, due to the presence of ∇ζ in (3.15), in view of

∥∇ζ · X ⊗ X∥_{L^{2}(H)} ≲ ∥∇ζ∥_{L^{2}(H)} + ∥ζ∥_{L_{ω}^{∞}L_{t}^{2}},

we derive,

∥t^{−1/2}\underline{χ̂} ∥_{L_{x}^{2}L_{t}^{∞}} ≲ ∥∇ζ∥_{L^{2}(H)} + ∥ζ∥_{L_{ω}^{∞}L_{t}^{2}} + ∥A · \underline{A}∥_{L^{2}(H)} + ∥s^{−1}A∥_{L^{2}(H)} ≲ Δ_{0}^{2} + R_{0}.

In the same way, using (8.8), we can get from (3.15) that

∥s^{−1}\underline{χ̂} ∥_{L^{2}(H)} ≲ ∥∇ζ∥_{L^{2}(H)} + ∥ζ∥_{L_{ω}^{∞}L_{t}^{2}} + ∥A · \underline{A}∥_{L^{2}(H)} + ∥s^{−1}A∥_{L^{2}(H)} ≲ Δ_{0}^{2} + R_{0}. With its aid, using (3.15) again, we obtain ∥∇_{L}\underline{χ̂} ∥_{L^{2}(H)} ≲ Δ_{0}^{2} + R_{0}.

Thus we already have the following results for \underline{χ}, ∥t^{−1}\underline{V} ∥_{P^{0}} + ∥t^{−1}\underline{χ̂} ∥_{P^{0}} ≲ Δ_{0}^{2} + R_{0}

∥t^{−1}\underline{V} ∥_{L^{2}(H)} + ∥t^{−1}\underline{χ̂} ∥_{L^{2}(H)} ≲ Δ_{0}^{2} + R_{0}  (8.22)
∥t^{−1/2}\underline{V} ∥_{L_{x}^{2}L_{t}^{∞}} + ∥t^{−1/2}\underline{χ̂} ∥_{L_{x}^{2}L_{t}^{∞}} ≲ Δ_{0}^{2} + R_{0}
∥∇_{L}\underline{V} ∥_{L^{2}(H)} + ∥∇_{L}\underline{χ̂} ∥_{L^{2}(H)} ≲ Δ_{0}^{2} + R_{0}
∥t^{−1/2}\underline{V} ∥_{B^{0}} + ∥t^{−1/2}\underline{χ̂} ∥_{B^{0}} ≲ Δ_{0}^{2} + R_{0}.  (8.23)

In particular we can choose R_{0} sufficiently small and check that the bootstrap assumptions (BA2) for tr\underline{χ} and \underline{χ̂} are verified with Δ_{0} replaced by Δ_{0}/2. We also deduce all the estimates for \underline{χ} in Theorem 4.1.

It remains to verify the auxiliary assumption (BA4). This part will follow the similar way as in [12, p.524]. For completeness we include the proof in the following. Note for \underline{A} = (\underline{V} , \underline{χ̂}), we have the following structure equation

∇_{L}\underline{A} + \frac{1}{s}\underline{A} = −∇_{L}P + E

N_{1}(P), ∥E∥_{P^{0}} ≲ R_{0} + Δ_{0}^{2}.

To estimate ∥A·\underline{A}∥_{P^{0}} ≲ R_{0}+Δ_{0}^{2}, we use Proposition 5.4 by reducing the above equation for \underline{A} to the following one for W = \underline{A} + P

∇_{L}W + \frac{1}{s}W = \frac{1}{s}P + E.

Since ∥\frac{1}{s}P ∥_{P^{0}} ≲ ∥∇P ∥_{L^{2}(H)} + ∥s^{−1}P ∥_{L^{2}(H)} ≲ Δ_{0}^{2} + R_{0}, according to Theorem 5.3 we infer

∥A · W ∥_{P^{0}} ≲ (N_{1}(A) + ∥A∥_{L_{ω}^{∞}L_{t}^{2}})( lim_{s→0} ∥W ∥_{B_{2,1}^{0}} + ∥E + \frac{1}{s}P ∥_{P^{0}}) ≲ Δ_{0}^{2} + Δ_{0}R_{0}.

The last inequality follows from Lemma 6.2 that ∥W ∥_{B_{2,1}^{0}} ≲ N_{1}(P) + ∥\underline{A}∥_{B_{2,1}^{0}}. When s → 0, we get ∥\underline{A}∥_{B_{2,1}^{0}} → 0 since s|\underline{A}| → 0. Thus

lim_{s→0} ∥W ∥_{B_{2,1}^{0}} ≲ N_{1}(P).

Combine with ∥A · P ∥_{P^{0}} ≲ (∥∇A∥_{L^{2}(H)} + ∥A∥_{L_{t}^{∞}L_{x}^{2}}) · N_{1}(P), we conclude by noting N_{1}(P) ≲ Δ_{0}^{2} + R_{0} that

∥A · \underline{A}∥_{P^{0}} ≲ ∥A · W ∥_{P^{0}} + ∥A · P ∥_{P^{0}} ≲ Δ_{0}^{3} + Δ_{0}R_{0} ≤ \frac{1}{2}Δ_{0}^{2},

provided that R_{0} is sufficiently small relative to Δ_{0}. This justifies (BA4) and shows the result of (8.19) and (8.23) without additional assumption.

Appendix

9.1 Initial Data

In this section, we will use the idea in [20, Chapter 2,3] to prove Proposition 3.1, Proposition 3.2 and Proposition 3.3.

9.1.1 Local expansions of χ, \underline{χ} and ζ

In this subsection, we will prove Proposition 3.1 and Proposition 3.2 by providing the local expansions of χ, \underline{χ} and ζ in the normal convex neighborhood of the vertex p.

Proof of Proposition 3.1. We will prove the estimates by showing the following :

(i) sχ_{ab} = δ_{ab} + \frac{s^{2}}{3} S_{ab} + O(s^{3})

(ii) χ_{ab} + \underline{χ}_{ab} → 0 as s → 0,

(iii) ζ_{a} = −\frac{1}{6}sS_{a} + O(s^{2}), where S_{ab} and S_{a} are some tensor fields, depending on curvature tensors.

In order to obtain (i), we are going to construct a number of bitensors, tensorial functions of two points in spacetimes. The first point is x′, to which we assign indices α′ and β′, etc. The second is x, to which we assign α and β, etc. We assume that x belongs to N (x′), the normal convex neighborhood of x′. The geodesic Γ that links x to x′ is described by relations z^{μ}(λ) in which λ is an affine parameter that ranges from λ_{0} to λ_{1}, with z(λ_{0}) = x′ and z(λ_{1}) = x. To an arbitrary point z on the geodesic we assign indices μ, ν etc. The vector t^{μ} = dz^{μ}/dλ which is tangent to the geodesic, obeys the geodesic equation Dt^{μ}/dλ = 0.

We define Synge’s world function to be a scalar function of the point x′ and x, given by

σ(x, x′) = \frac{1}{2}(λ_{1} − λ_{0}) ∫_{λ_{0}}^{λ_{1}} g_{μν}t^{μ}t^{ν}dλ,  (9.1)

where the integral is evaluated on the geodesic Γ. It is easy to see that σ is invariant under a constant rescaling of the affine parameter λ → λ̄ = C_{1}λ + C_{2}, where C_{1} and C_{2} are constants. By virtue of the geodesic equation, the quantity ϵ = g_{μν}t^{μ}t^{ν} is constant on the geodesic. Therefore the Synge’s world function is numerically equal to \frac{1}{2}ϵ(λ_{1} − λ_{0})^{2}. If the geodesic is a null geodesic, then σ = 0.

σ(x, x′) can be differentiated with respect to either argument. Let σ_{α} = ∂σ/∂x^{α} be its partial derivative with respect to x, and σ_{α′} = ∂σ/∂x^{α′} its partial derivative with respect to x′. σ_{α} behaves as a 1-form with respect to tensorial operations carried out at x, but a scalar with respect to operations carried out at x′. σ_{α′} behaves as a 1-form with respect to tensorial operations carried out at x′ and a scalar with respect to operations carried out at x. Let σ_{αβ} = ∇_{β}σ_{α} be the covariant derivative of σ_{α} with respect to x. Then the tensor is symmetric: σ_{βα} = σ_{αβ}. Let σ_{αβ′} = ∂_{β′}σ_{α} = ∂^{2}σ/∂x^{β′}∂x^{α} be the partial derivative of σ_{α} with respect to x′. We can also define σ_{α′β} = ∂^{2}σ/∂x^{β}∂x^{α′} to be the partial derivative of σ_{α′} with respect to x. We then have σ_{β′α} = σ_{αβ′}. Finally, we let σ_{α′β′} = ∇_{β′}σ_{α′} be the covariant derivative of σ_{α′} with respect to x′. This notation can be extended to any number of derivatives.

Any vector field A^{μ}(z) on Γ whose affine parameter is s, can be decomposed in the orthonormal basis e_{i}^{μ} which is parallely transported along Γ: A^{μ} = A^{i}e_{i}^{μ}, and the vector’s frame components are given by A^{i} = A^{μ}e_{μ}^{i}. If A^{μ} is parallely transported on the geodesic, then the coefficients A^{i} are constants. The vector at x can then be expressed as A^{α} = (A^{α′}e_{α′}^{i})e_{i}^{α}, or

A^{α}(x) = g_{α′}^{α}(x, x′)A^{α′}(x′) g_{α′}^{α}(x, x′) = e_{i}^{α}(x)e_{α′}^{i}(x′).  (9.2)

1 Similarly,

A^{α′}(x′) = g_{α}^{α′}(x′, x)A^{α}(x), g_{α}^{α′}(x′, x) = e_{i}^{α′}(x′)e_{α}^{i}(x)  (9.3)

and we see that g_{α}^{α′} = e_{i}^{α′}e_{α}^{i} performs the inverse operation which takes a vector at x and parallel transport it back to x′. Clearly,

g_{α′}^{α}g_{β}^{α′} = δ_{β}^{α}, g_{α}^{α′}g_{β′}^{α} = δ_{β′}^{α′}, g_{α}^{α′}g_{β}^{β′}g_{α′β′} = g_{αβ}

and these relations formally express the fact that g_{α}^{α′} is the inverse of g_{α′}^{α}.

1The objects g_{α′}^{α} = e_{i}^{α}e_{α′}^{i} is the parallel propagator which takes a vector at x′ and parallel-transports it to x along the unique geodesic that links these points.

Applying the general expansion method to σ_{αβ}, we can derive the following formula (for its proof, see [20, Chapter 2.4]),

σ_{αβ} = g_{α}^{α′}g_{β}^{β′}(g_{α′β′} − \frac{1}{3}R_{α′γ′β′δ′}σ^{γ′}σ^{δ′} + O(s^{3})),  (9.4)

with s the affine parameter of the geodesic Γ. We will introduce the retarded coordinates in the small normal convex neighbourhood of p which is denoted by N (p). We have a timelike curve l_{1} described by relations z^{μ}(τ), its normalized tangent vector u^{μ} = dz^{μ}/dτ, and its acceleration vector a^{μ} = Du^{μ}/dτ. We also have an orthonormal triad e_{i}^{μ} that is transported on the l_{1} according to

\frac{De_{i}^{μ}}{dτ} = a_{i}u^{μ} + w_{i}^{j}e_{j}^{μ}  (9.5)

where a_{i}(τ) = a_{μ}e_{i}^{μ} are the frame component of the acceleration vector and w_{ij}(τ) = −w_{ji}(τ) is the prescribed rotation tensor. (9.5) is compatible with the requirement that the tetrad (u^{μ}, e_{i}^{μ}) be orthonormal everywhere on l_{1}. Finally, we have a dual tetrad (e_{μ}^{0}, e_{μ}^{i}) with e_{μ}^{0} = −u_{μ} and e_{μ}^{i} = δ^{ij}g_{μν}e_{j}^{ν}. The tetrad and its dual give rise to the complete relations,

g^{μν} = −u^{μ}u^{ν} + δ^{ij}e_{i}^{μ}e_{j}^{ν}, g_{μν} = −e_{μ}^{0}e_{ν}^{0} + δ_{ij}e_{μ}^{i}e_{ν}^{j}.

We thus let x be within the normal convex neighborhood of l_{1}. The retarded coordinates are based on a null geodesic going from the world line l_{1} to x. Let l_{2} be the unique future directed null geodesic that goes from the world line to x, and x′ = z(u) be the point at which l_{2} intersects the world line, with u denoting the value of the proper time parameter at the point. Then u is the solution of the eikonal equation,

g^{αβ}∂_{α}u∂_{β}u = 0.

2 From the tetrad at x′ we obtain another tetrad (e_{0}^{α}, e_{i}^{α}) at x by parallel transport on l_{2}. By raising the frame index and lowering the vectorial index we also obtain a dual tetrad at x : e_{α}^{0} = −g_{αβ}e_{0}^{β} and e_{α}^{i} = δ^{ij}g_{αβ}e_{j}^{β}. We define the following coordinates in N (p),

x̂^{0} = u, x̂^{i} = −e_{α′}^{i}(x′)σ^{α′}(x, x′), σ(x, x′) = 0.  (9.6)

2The null vector L can be written as

L = −g^{αβ}∂_{β}u∂_{α}.

Set

s = (δ_{ij}x̂^{i}x̂^{j})^{1/2} = u_{α′}σ^{α′},  (9.7)

then s is a positive function on N (p) by virtue of the fact that l_{2} is a future-directed null geodesic. Then we have the following expression for the null vector L^{α} defined in Chapter 3

L^{α} = \frac{σ^{α}(x, x′)}{s(x)}.  (9.8)

We can compute the gradient of s by finding how s changes under a displacement of x which also induces a displacement of x′,

∂_{β}s = −(σ_{α′}a^{α′} + σ_{α′β′}u^{α′}u^{β′})L_{β} + σ_{α′β}u^{α′}.  (9.9)

Indeed, noting 0 = σ(x + δx, x′ + δx′) = σ_{α}δx^{α} + σ_{α′}δx^{α′}, with the definition of s and the fact that a displacement along the world line is described by

δx^{α′} = u^{α′}δu,  (9.10)

we infer the following relation,

du = −L_{α}dx^{α}.  (9.11)

Then combine

∂_{β}s = D_{β′}(σ_{α′}(x′)u^{α′}(x′))\frac{∂x^{β′}}{∂x^{β}} + D_{β}(σ_{α′}(x)u^{α′}(x′))

with (9.10) and (9.11), we get (9.9).

We compute sL_{α;β} by using (9.8) and the above identity to get the following relation

sL_{α;β} =σ_{αβ} − L_{β}σ_{αγ′}u^{γ′} − ∂_{β}sL_{α}
= σ_{αβ} − L_{α}σ_{βγ′}u^{γ′} − L_{β}σ_{αγ′}u^{γ′} + (σ_{α′}a^{α′} + σ_{α′β′}u^{α′}u^{β′})L_{α}L_{β}.  (9.12)

Note that L(s) = 1 together with (9.9) and ⟨L, L⟩ = 0 yields σ_{α′β}u^{α′}L^{β} = 1, we then infer from (9.12) the following

strχ = σ_{α}^{α} − 2.  (9.13)

Using (9.4), we get

strχ = 2 − \frac{1}{3}R_{α′β′}σ^{α′}σ^{β′} + O(s^{3}).  (9.14)

Note that all the Ricci curvature tensors vanish, we get strχ = 2 + O(s^{3}) as desired.

The quantity strχ is independent of choice of frames and foliations. From the definition of null frames and also noting that D_{L}L = 0, it is easy to observe that strχ coincides with the quantity which we derive by taking the trace of sχ_{ab} in null foliation.

Now we parametrize the congruence of null geodesics emanating from p by

x̂^{0} = u = 0 x̂^{i} = sΩ^{i} i=1,2,3

with δ_{ij}Ω^{i}Ω^{j} = 1. From (9.6) and (9.7), we get

\frac{σ^{α′}}{s} = −(u^{α′} + Ω^{i}e_{i}^{α′}).  (9.15)

Now we take the null frame e_{a}, \underline{L}, L for a = 1, 2. From (9.12) we infer

sχ_{ab} = se_{a}^{α}e_{b}^{β}L_{α;β} = e_{a}^{α}e_{b}^{β}σ_{αβ}  (9.16)

since all other terms vanish because of orthogonality.

From (9.4), we get

sχ_{ab} = δ_{ab} − \frac{1}{3}s^{2}R_{aγ′bδ′}s^{−1}σ^{γ′}s^{−1}σ^{δ′} + O(s^{3}).  (9.17)

Then by (9.15), (9.17) can be rewritten as follows

sχ_{ab} = δ_{ab} − \frac{1}{3}s^{2}(R_{a0b0} + R_{a0bi}Ω^{i} + R_{b0ai}Ω^{i} + R_{aibj}Ω^{i}Ω^{j}) + O(s^{3}),  (9.18)

where

R_{a0b0} = R_{aγ′bδ′}u^{γ′}u^{δ′} = R_{α′γ′β′δ′}e_{a}^{α′}u^{γ′}e_{b}^{β′}u^{δ′}
R_{a0bi} = R_{aγ′bδ′}u^{γ′}e_{i}^{δ′} = R_{α′γ′β′δ′}e_{a}^{α′}u^{γ′}e_{b}^{β′}e_{i}^{δ′}
R_{aibj} = R_{aγ′bδ′}e_{i}^{γ′}e_{j}^{δ′} = R_{α′γ′β′δ′}e_{a}^{α′}e_{i}^{γ′}e_{b}^{β′}e_{j}^{δ′}.

Set S_{ab} = R_{a0b0}+R_{a0bi}Ω^{i}+R_{b0ai}Ω^{i}+R_{aibj}Ω^{i}Ω^{j}, then S_{ab} is a symmetric traceless 2-tensor. (9.18) can be rewritten in the following form

sχ_{ab} = δ_{ab} − \frac{1}{3}s^{2}S_{ab} + O(s^{3})

Note that δ_{ij}Ω^{i}Ω^{j} = 1, due to the assumption that the spacetime is smooth, we obtain that all components of S_{ab} are finite in N (p). Then we conclude that χ̂ → 0 as s → 0.

Now we prove (ii). Let us set the vector field V = g^{αβ}∂_{α}s∂_{β}. We immediately have ⟨V, X⟩ = 0 for any S tangent tensor fields X. Further, in view of g^{αβ}∂_{α}s∂_{β}u = −1, we have ⟨V, L⟩ = 1. Then from ⟨\underline{L}, \underline{L}⟩ = 0 and ⟨L, \underline{L}⟩ = −2 we infer

\underline{L} = −2V + ⟨V, V ⟩L.

Consequently, we get \underline{χ}_{ab} = ⟨D_{a}\underline{L}, e_{b}⟩ as follows,

⟨D_{a}\underline{L}, e_{b}⟩ = −2⟨D_{a}V, e_{b}⟩ + ⟨V, V ⟩⟨D_{a}L, e_{b}⟩ = −2e_{a}^{ν}V_{μ;ν}e_{b}^{μ} + g^{μν}∂_{μ}s∂_{ν}sχ_{ab}.

From (9.9), we get the expression for ⟨V, V ⟩ as follows,

g^{μν}∂_{μ}s∂_{ν}s = −2(σ_{α′}a^{α′} + σ_{α′μ′}u^{α′}u^{μ′}) + σ_{α′μ}u^{α′}σ_{δ′ν}u^{δ′}g^{μν}

Then the local expansion of \underline{χ}_{ab} can be computed in the following way,

\underline{χ}_{ab} = ⟨D_{a}\underline{L}, e_{b}⟩ = −2⟨D_{a}V, e_{b}⟩ − 2(σ_{α′}a^{α′} + σ_{α′μ′}u^{α′}u^{μ′})χ_{ab} + σ_{α′μ}u^{α′}σ_{δ′ν}u^{δ′}g^{μν}χ_{ab}
= 2e_{a}^{ν}e_{b}^{μ}{(σ_{α′}a^{α′} + σ_{α′μ′}u^{α′}u^{μ′})L_{μ;ν} + (σ_{α′}a^{α′} + σ_{α′μ′}u^{α′}u^{μ′})_{,ν}L_{μ} − (σ_{α′μ}u^{α′})_{;ν}}
− 2(σ_{α′}a^{α′} + σ_{α′μ′}u^{α′}u^{μ′})χ_{ab} + σ_{α′μ}u^{α′}σ_{δ′ν}u^{δ′}g^{μν}χ_{ab}.
= −2e_{a}^{ν}e_{b}^{μ}(σ_{α′μ}u^{α′})_{;ν} + σ_{α′μ}u^{α′}σ_{δ′ν}u^{δ′}g^{μν}χ_{ab}
= −2e_{a}^{ν}e_{b}^{μ}(σ_{α′μν}u^{α′}) + σ_{α′μ}u^{α′}σ_{δ′ν}u^{δ′}g^{μν}χ_{ab}.  (9.19)

When x → p, which means s → 0, by the relations ^{3} σ_{α′μν} → 0, and

σ_{α′β} = −g_{β}^{β′}(g_{α′β′} + \frac{1}{6}R_{α′γ′β′δ′}σ^{γ′}σ^{δ′}) + O(s^{3})

we get the first term of (9.19) vanishes and the second term can be expressed in the form of

(g_{α′δ′}u^{α′}u^{δ′} + \frac{1}{3}s^{2}R_{0i0j}Ω_{i}Ω_{j} + O(s^{3}))χ_{ab}.

Also noting g_{α′δ′}u^{α′}u^{δ′} = ∥T ∥_{g} = −1, we conclude \underline{χ}_{ab} + χ_{ab} → 0 as s → 0. Thus we complete the proof of (ii).

Next we prove (iii). Recall that ζ_{a} = \frac{1}{2}⟨D_{a}L, \underline{L}⟩ with \underline{L} = e_{3},

sζ_{a} = \frac{1}{2}e_{a}^{β}e_{3}^{α}sL_{α;β}.  (9.20)

In view of (9.12) and ⟨L, \underline{L}⟩ = −2, we get

sζ_{a} = \frac{1}{2}e_{3}^{α}e_{a}^{β}σ_{αβ} + 2e_{a}^{β}σ_{βγ′}u^{γ′}.  (9.21)

Moreover, from (9.9), noting that e_{a}^{β}L_{β} = 0, we derive that

0 = e_{a}(s) = e_{a}^{β}σ_{α′β}u^{α′}.  (9.22)

Therefore the second term of (9.21) vanishes. We then obtain that

sζ_{a} = \frac{1}{2}e_{3}^{α}e_{a}^{β}σ_{αβ}.  (9.23)

From (9.4), we rewrite (9.23) in the following

sζ_{a} = −\frac{1}{6}s^{2}e_{a}^{β}e_{3}^{α}R_{αγ′βδ′}s^{−1}σ^{γ′}s^{−1}σ^{δ′} + O(s^{3}).  (9.24)

3For their proofs, refer to [20, Chap.2.4].

In the same manner as deriving (9.18), it can be written in the following form,

sζ_{a} = −\frac{1}{6}s^{2}(R_{30a0} + R_{30ai}Ω^{i} + R_{3ia0}Ω^{i} + R_{3iaj}Ω^{i}Ω^{j}) + O(s^{3}).

For simplicity, we set

S_{a} = R_{30a0} + R_{30ai}Ω^{i} + R_{3ia0}Ω^{i} + R_{3iaj}Ω^{i}Ω^{j},

then

ζ_{a} = −\frac{1}{6}sS_{a} + O(s^{2}).  (9.25)

Proof of Proposition 3.2. Now we are going to compute lim_{s→0} s∇_{d}ζ_{a} with d, a = 1, 2. By (9.23), we know

s∇_{d}ζ_{a} = \frac{1}{2}∇_{d}(e_{3}^{α}e_{a}^{β}σ_{αβ}) = \frac{1}{2}e_{d}^{μ}e_{3}^{α}_{;μ}e_{a}^{β}σ_{αβ} + \frac{1}{2}e_{3}^{α}e_{d}^{μ}e_{a}^{β}_{;μ}σ_{αβ} + \frac{1}{2}e_{d}^{μ}e_{3}^{α}e_{a}^{β}σ_{αβ;μ}.

Note that L_{μ}e_{a}^{μ} = 0, the last term in the above is simply \frac{1}{2}e_{3}^{α}e_{a}^{β}e_{d}^{μ}σ_{αβμ}, since all other terms arising in differentiation vanish. When s → 0, \frac{1}{2}e_{3}^{α}e_{a}^{β}e_{a}^{μ}σ_{αβμ} → 0, since we have σ_{αβμ} → 0 (see [20, Chap. 2]). For the first two terms, by using the expression of σ_{αβ} in (9.4) we obtain

\frac{1}{2}e_{d}^{μ}e_{3}^{α}_{;μ}e_{a}^{β}σ_{αβ} + \frac{1}{2}e_{3}^{α}e_{d}^{μ}e_{a}^{β}_{;μ}σ_{αβ}

= \frac{1}{2}⟨∇_{d}e_{3}, e_{a}⟩ + \frac{1}{2}⟨∇_{d}e_{a}, e_{3}⟩  (9.26)
− \frac{1}{6}(g_{α}^{α′}g_{β}^{β′}R_{α′γ′β′δ′}σ^{γ′}σ^{δ′} + O(s^{3}))(e_{d}^{μ}e_{3}^{α}_{;μ}e_{a}^{β} + e_{3}^{α}e_{d}^{μ}e_{a}^{β}_{;μ}).  (9.27)

Since (9.26) vanishes, it remains to estimate (9.27). Noting ∇_{d}e_{3} = \underline{χ}_{db}e_{b}, by using (9.15) and denoting ∇_{d}e_{a} = Γ_{d,a}^{c}e_{c}, we deduce the following,

(g_{α}^{α′}g_{β}^{β′}R_{α′γ′β′δ′}σ^{γ′}σ^{δ′})(e_{d}^{μ}e_{3}^{α}_{;μ}e_{a}^{β} + e_{3}^{α}e_{d}^{μ}e_{a}^{β}_{;μ}) = s^{2}\underline{χ}_{db}S_{ab} + s^{2}Γ_{a,d}^{c}S_{c}.

Thus we get s∇_{d}ζ_{a} → 0, and s div ζ → 0 as desired. According to the definition of μ, we get sμ → 0 as s → 0.

Note that from (9.16), we obtain

s∇_{c}χ_{ab} = e_{c}^{ν}(e_{a}^{α}e_{b}^{β}σ_{αβ})_{;ν}

following exactly the same steps as above, we can get (3.7).

9.1.2 Transformation of angular coordinates

Because the vector Ω^{a} = x̂^{a}/s verifies δ_{ab}Ω^{a}Ω^{b} = 1, it then can be parameterized by two angles θ^{A}. A canonical choice is Ω^{a} = (sin θ cos ϕ, sin θ sin ϕ, cos θ). Then we perform a coordinate transformation from x̂^{a} to (s, θ^{A}), using the relation x̂^{a} = sΩ^{a}(θ^{A}). (The angles θ^{A} are constants on the generators of the null hypersurface u = const., and that s is the affine parameter on these generator.) The differential form of the coordinate transformation is

dx̂^{a} = Ω^{a}ds + sΩ_{A}^{a}dθ^{A}

where the transformation matrix Ω_{A}^{a} := \frac{∂Ω^{a}}{∂θ^{A}} verifies the identity Ω_{a}Ω_{A}^{a} = 0.

Set Ω_{AB} = δ_{ab}Ω_{A}^{a}Ω_{B}^{b}, which acts as a metric in the subspace spanned by the angular coordinates. In the canonical parametrization, Ω_{AB} = diag(1, sin^{2} θ). Then in the angular coordinates, we can check the metric g takes the form of

g_{uu}du^{2} + 2g_{us}duds + 2g_{uA}dudθ^{A} + g_{AB}dθ^{A}dθ^{B},

where g_{AB} = s^{2}[Ω_{AB} − \frac{1}{3}s^{2}S_{ab}Ω_{A}^{a}Ω_{B}^{b} + O(s^{3})]. g_{su} = −1 and g_{ss} = 0 are precise. (Refer to [20, Section 3.37].) We take γ(s) to be the restriction of the spacetime metric g on the slice S_{s}, then γ_{AB}(s) = g_{AB}. At last, we recall that the rescaled metric γ^{∘} is give by γ^{∘}= \frac{1}{s^{2}} γ and the canonical metric on a standard two sphere is given by γ_{AB}^{(0)} = Ω_{AB}. Set a_{s} = ^{\begin{matrix}√ & \overline{|γ_{s}|} \\ √ & \overline{|γ^{(0)}|} \end{matrix}}, we have the following,

γ^{∘}_{AB}= γ_{AB}^{(0)} + O(s^{2}), s^{−2}a_{s} → 1, \frac{r}{s} → 1 as s → 0.

As an easy consequence, ∂_{C} γ^{∘}_{AB}= ∂_{C}γ_{AB}^{(0)} +O(s^{2}) where ∂ denotes the differentiation with respect to the two angular variables on 𝕊^{2}. Although the calculation is carried out with angular coordinates on S_{s}, the results in Proposition 3.3 is independent of choice of coordinate systems on S_{s}.

9.2 Commutator Estimates: Proof of Lemma 5.1

The proof will follow essentially the treatment in [14, Sec.12]. Throughout this section, by \underline{K} we denote K − s^{−2} on each leave S_{s}. The estimates in this section will rely on the bootstrap assumptions (BA1), (BA2), (BA3) and the following inequality

∥\underline{K}∥_{L_{t}^{2}L_{x}^{2}} + ∥β∥_{L_{t}^{2}L_{x}^{2}} ≲ Δ_{0}  (K1)

for the Gauss curvature K of (S_{s}, γ) and the null component of curvature β. This has been verified in Chapter 4 based on (4.8) and the bootstrap assumptions (BA1) and (BA2).

Now we will revisit the properties of the heat flow. Given a tensor F on S = S_{s} we define the corresponding heat flow U(τ)F to be the unique solution of the equation

∂_{τ}U(τ)F − s^{2}Δ_{γ}U(τ)F = 0, U(0)F = F,

In what follows we will write Δ instead of Δ_{γ}, whenever there is no confusion. By applying Proposition 2.1 to g =γ^{∘} we have

Proposition 9.1. The heat flow U(τ)F verifies the following properties:

∥U(τ)F ∥_{L^{p}(S)} ≲ ∥F ∥_{L^{p}(S)}, 1 ≤ p ≤ ∞
∥∇U(τ)F ∥_{L^{2}(S)} ≲ ∥∇F ∥_{L^{2}(S)}  (9.28)
∥∇U(τ)F ∥_{L^{2}(S)} ≲ s^{−1}τ^{− \frac{1}{2}} ∥F ∥_{L^{2}(S)}  (9.29)
∥U(τ)∇F ∥_{L^{2}(S)} ≲ s^{−1}τ^{− \frac{1}{2}} ∥F ∥_{L^{2}(S)}
∥ΔU(τ)F ∥_{L^{2}(S)} ≲ s^{−2}τ^{−1}∥F ∥_{L^{2}(S)}.

Moreover, for 2 ≤ p < ∞ we have

∥U(τ)F ∥_{L^{p}(S)} ≲ s^{−1+ \frac{2}{p}} ⟨τ^{−( \frac{1}{2} − \frac{1}{p})}⟩∥F ∥_{L^{2}(S)}  (9.30)

and for 1 < q ≤ 2 the dual estimate

∥U(τ)F ∥_{L^{2}(S)} ≲ s^{1− \frac{2}{q}} ⟨τ^{( \frac{1}{2} − \frac{1}{q})}⟩∥F ∥_{L^{q}(S)}.  (9.31)

In addition, for given scalar functions f, the following estimates hold true:

∥U(τ)f∥_{L^{∞}(S)} ≲ s^{−1}⟨τ^{−1/2}⟩∥f∥_{L^{2}(S)}  (9.32)
∥U(τ)f∥_{L^{2}(S)} ≲ s^{−1}⟨τ^{−1/2}⟩∥f∥_{L^{1}(S)},  (9.33)

where we use the notation ⟨τ⟩^{p} = 1 + τ^{p} for any p and τ > 0.

We denote arbitrary S-tangent tensorfields by F and scalars by f. First recall the commutation formula (3.26)–(3.28) as follows

[∇_{L}, ∇]f = −\frac{1}{2}trχ · ∇f + A · ∇f  (9.34)
[∇_{L}, Δ]f = −\frac{1}{s}Δf + ∇(A · ∇f) + ∇A · ∇f + (A + s^{−1}) · A · ∇f  (9.35)
[∇_{L}, ∇]F = −\frac{1}{2}trχ · ∇F + A · ∇F + β · F + (A + \frac{1}{s}) · A · F  (9.36)

As a consequence of (9.36) and

[∇_{L}, Δ]F = [∇_{L}, div ]∇F + div [∇_{L}, ∇]F,

we get

[∇_{L}, Δ]F = −trχΔF + A · ∇^{2}F + ∇A · ∇F + (A + \frac{1}{s}) · A · ∇F + β · ∇F
+ ∇(β · F + (A + \frac{1}{s}) · A · F).  (9.37)

Proof of Lemma 5.1. By the definition of the GLP decomposition P_{k}, we obtain

[P_{k}, ∇_{L}]F = ∫_{0}^{∞} m_{k}(τ)[U(τ), ∇_{L}]F dτ.  (9.38)

Set Φ(τ) = [U(τ), ∇_{L}]F, by using the heat equation, we have

\frac{d}{dτ} Φ(τ) − s^{2}ΔΦ(τ) = s^{2}[Δ, ∇_{L}]U(τ)F − 2sΔU(τ)F and Φ(0) = 0.

Then by Duhamel’s principle and the commutator formula (9.37), we infer that

Φ(τ) = ∫_{0}^{τ} U(τ − τ′) {s^{2}[Δ, ∇_{L}]U(τ′)F − 2sΔU(τ′)F } dτ′
= ∫_{0}^{τ} U(τ − τ′)[s^{2}A · ΔU(τ′)F − s^{2}(A · ∇^{2}U(τ′)F + ∇A · ∇U(τ′)F
+ (A + s^{−1}) · A · ∇U(τ′)F + β · ∇U(τ′)F
+ ∇(β · U(τ′)F + (A + s^{−1}) · A · U(τ′)F))].  (9.39)

Observe that the terms (A + \frac{1}{s}) · A and ∇A have same or better estimates than β, therefore we can ignore them and simplify (9.39) as Φ = Φ_{1} + Φ_{2}, where

Φ_{1} := ∫_{0}^{τ} s^{2}U(τ − τ′)∇(A · ∇U(τ′)F + β · U(τ′)F)dτ′,  (9.40)
Φ_{2} := ∫_{0}^{τ} s^{2}U(τ − τ′)(β · ∇U(τ′)F)dτ′.  (9.41)

Now we are in a position to estimate Φ_{1} and Φ_{2}. In what follows we will rely on the following heat flow estimates.

Lemma 9.1. The following estimates hold

(i) ∥U(τ)G∥_{L_{ω}^{∞}} ≲ (1 + ∥t\underline{K}∥L_{x}^{2}^{\frac{1}{p−1}} + ∥t\underline{K}∥L^{2}^{\frac{1}{p}} + τ^{− \frac{1}{2p}} )∥U(τ/2)G∥_{H_{x}^{1}},^{4}

(ii) ∥U(τ)G∥_{L_{ω}^{∞}} ≲ ⟨t^{2}τ⟩^{− \frac{1}{r}} (1 + ∥t\underline{K}∥L_{x}^{2}^{\frac{1}{p−1}} + ∥t\underline{K}∥L_{x}^{2}^{\frac{1}{p}} + τ^{− \frac{1}{2p}} )∥G∥_{L_{x}^{r}} for any 2 ≤ r ≤ ∞,

(iii) ∥∇^{2}U(τ)G∥_{L_{x}^{2}} ≲ ((t^{2}τ)^{− \frac{1}{2}} + t^{\frac{1}{p−1}} ∥\underline{K}∥L_{x}^{2}^{\frac{p}{p−1}} + ∥\underline{K}∥_{L_{x}^{2}} + t^{−1})∥U(τ/2)G∥_{H_{x}^{1}}.

4We define ∥F ∥_{H_{x}^{1}} = ∥∇F ∥_{L_{x}^{2}} + ∥s^{−1}F ∥_{L_{x}^{2}}.

Proof of Lemma 9.1. By Böchner identity (4.45), we infer for a tensor field F the following inequality with γ_{0} > 1/2,

∥∇^{2}F ∥_{L_{x}^{2}} + ∥s^{−1}∇F ∥_{L_{x}^{2}} ≲ ∥ΔF ∥_{L_{x}^{2}} + \underline{K}_{γ_{0}}(∥∇^{2}F ∥_{L_{x}^{2}} + ∥s^{−1}∇F ∥_{L_{x}^{2}})
+ ∥\underline{K}∥_{L_{x}^{2}}∥F ∥_{L_{x}^{∞}} + ∥s^{−1}∇F ∥_{L_{x}^{2}} + ∥s^{−2}F ∥_{L_{x}^{2}}.

By (BA3) which says \underline{K}_{γ_{0}} ≤ Δ_{0} < \frac{1}{2}, we infer

∥∇^{2}F ∥_{L_{x}^{2}} ≲ ∥ΔF ∥_{L_{x}^{2}} + ∥\underline{K}∥_{L_{x}^{2}}∥F ∥_{L_{x}^{∞}} + t^{−1}∥F ∥_{H^{1}}.  (9.42)

Substituting (3.38) or the simplified formulae ∥F ∥_{L_{x}^{∞}} ≲ ∥t∇^{2}F ∥L_{x}^{2}^{\frac{1}{p}}∥F ∥H_{x}^{1}^{\frac{p−1}{p}} + ∥∇F ∥_{L_{x}^{2}}, into (9.42), and using Young’s inequality, we obtain

∥∇^{2}F ∥_{L_{x}^{2}} ≲ ∥ΔF ∥_{L_{x}^{2}} + ∥\underline{K}∥L_{x}^{2}^{\frac{p}{p−1}} t^{\frac{1}{p−1}} ∥F ∥_{H_{x}^{1}} + t^{−1}∥F ∥_{H_{x}^{1}} + ∥\underline{K}∥_{L_{x}^{2}}∥∇F ∥_{L_{x}^{2}}.

Then, using Proposition 9.1, we obtain the desired results.

Below are two basic estimates which will be used frequently.

Proposition 9.2. (i) For tensor fields F there holds for q ≥ 2

∥t^{\frac{1}{2} − \frac{1}{q}} ∇U(τ′)F ∥_{L_{t}^{q}L_{x}^{2}} ≲ ⟨τ′^{\frac{1}{q} − \frac{1}{2}} ⟩N_{1}(F).

(ii) For tensor fields F, there holds for 1 ≤ q < 2,

∥U(τ′/2)F ∥_{H_{x}^{1}} ≲ ⟨τ′^{− \frac{3}{4} + \frac{1}{q}} ⟩∥F ∥_{H_{x}^{1}}^{\frac{2}{q} − \frac{1}{2}} ∥t^{−1}F ∥_{L_{x}^{2}}^{\frac{3}{2} − \frac{2}{q}} .

Proof. For tensor fields F we have

∥F ∥_{L_{t}^{q}L_{x}^{2}} ≲ ∥F ∥L_{t}^{2}L_{x}^{2}^{\frac{2}{q}}∥F ∥_{L_{t}^{∞}L_{x}^{2}}^{1− \frac{2}{q}}.

Using the heat flow estimate (9.29) we can infer when q ≥ 2 that

∥t^{\frac{1}{2} − \frac{1}{q}} ∇U(τ′)F ∥_{L_{t}^{q}L_{x}^{2}} ≲ ∥∇U(τ′)F ∥L_{t}^{2}L_{x}^{2}^{\frac{2}{q}}∥t^{1/2}∇U(τ′)F ∥_{L_{t}^{∞}L_{x}^{2}}^{1− \frac{2}{q}} ≲ ⟨τ′^{\frac{1}{q} − \frac{1}{2}} ⟩N_{1}(F)

which implies (i).

When 1 ≤ q < 2, (ii) can be obtained by using (9.28) and (9.29) and an interpolation argument. Thus we finish the proof of Proposition 9.2.

Now we are able to establish the following estimates for Φ_{1}(τ) and Φ_{2}(τ).

Lemma 9.2. For any 1 ≤ q < 2 close to 2 and τ > 0,

∥t^{\frac{1}{2} − \frac{1}{q}} Φ_{1}(τ)∥_{L_{t}^{q}L_{x}^{2}} ≲ max (τ^{\frac{1}{4} −}, τ^{\frac{1}{2}} ) N_{1}(F),  (9.43)
∥t^{\frac{1}{2} − \frac{1}{q}} Φ_{2}(τ)∥_{L_{t}^{q}L_{x}^{2}} ≲ max (τ^{\frac{1}{q} − \frac{1}{2} −}, τ^{\frac{1}{2} +}) N_{1}(F).  (9.44)

Proof of Lemma 9.2. Set

ϕ_{1}(τ′) = A · ∇U(τ′)F + β · U(τ′)F,

we will estimate ∥t^{\frac{3}{2} − \frac{1}{q}} ϕ_{1}(τ′)∥_{L_{t}^{q}L_{x}^{2}}. By Hölder inequality, we infer, with \frac{1}{q_{3}} = \frac{1}{q} − \frac{1}{2},

∥t^{\frac{3}{2} − \frac{1}{q}} ϕ_{1}(τ′)∥_{L_{t}^{q}L_{x}^{2}} ≲ ∥A∥_{L_{t}^{∞}L_{x}^{4}}∥t^{\frac{3}{2} − \frac{1}{q}} ∇U(τ′)F ∥_{L_{t}^{q}L_{x}^{4}} + ∥β∥_{L_{t}^{2}L_{x}^{2}}∥t^{\frac{3}{2} − \frac{1}{q}} U(τ′)F ∥_{L_{t}^{q_{3}}L_{ω}^{∞}}. (9.45)

Now we will use Proposition 9.2 to derive the estimate for ϕ_{1}. As to the first term in (9.45), noting the following Sobolev inequality

∥∇U(τ′)F ∥_{L_{x}^{4}} ≲ ∥∇^{2}U(τ′)F ∥L_{x}^{2}^{\frac{1}{2}}∥∇U(τ′)F ∥L_{x}^{2}^{\frac{1}{2}} + ∥t^{−1/2}∇U(τ′)F ∥_{L_{x}^{2}},  (9.46)

combined with Lemma 9.1 (iii) and Proposition 9.2 (ii), we infer

∥t^{\frac{3}{2} − \frac{1}{q}} ∇U(τ′)F ∥_{L_{t}^{q}L_{x}^{4}} ≲ τ′^{− \frac{1}{4}} ∥t^{1− \frac{1}{q}} U(τ′/2)F ∥_{L_{t}^{q}H_{x}^{1}}

+ ⟨τ′^{−( \frac{3}{4} − \frac{1}{q})}⟩∥\underline{K}∥L_{t}^{2}L_{x}^{2}^{\frac{1}{2}}∥F ∥_{L_{t}^{2}H_{x}^{1}}^{\frac{2}{q} − \frac{1}{2}}∥F ∥_{L_{t}^{∞}L_{x}^{2}}^{\frac{3}{2} − \frac{2}{q}}

+ ∥t^{1− \frac{1}{q}} U(τ′)F ∥_{L_{t}^{q}H_{x}^{1}}.

By Proposition 9.2 (i) and the Hölder inequality for L_{t}^{q}, for the exponent 1 ≤ q < 2 and sufficiently close to 2 we get ^{5}

∥t^{\frac{3}{2} − \frac{1}{q}} ∇U(τ′)F ∥_{L_{t}^{q}L_{x}^{4}} ≲ ⟨τ′^{− \frac{1}{4} −}⟩N_{1}(F).  (9.47)

As to the second term of (9.45), using Lemma 9.1 (ii) with r = 4, we obtain with \frac{1}{q_{3}} = \frac{1}{q} − \frac{1}{2}

∥t^{\frac{3}{2} − \frac{1}{q}} U(τ)F ∥_{L_{t}^{q_{3}}L_{ω}^{∞}} ≲ ⟨τ′^{− \frac{1}{4} −}⟩∥F ∥_{L_{t}^{∞}L_{x}^{4}}.  (9.48)

Then, combining (9.45), (9.47) and (9.48), with the help of (K1) and (BA1), when q < 2 and is sufficiently close to 2, we get

∥t^{\frac{3}{2} − \frac{1}{q}} ϕ_{1}(τ′)∥_{L_{t}^{q}L_{x}^{2}}dτ′ ≲ ⟨τ′^{− \frac{1}{4} −}⟩N_{1}(F).  (9.49)

Using the definition of Φ_{1}(τ) given in (9.40), we thus obtain

∥t^{\frac{1}{2} − \frac{1}{q}} Φ_{1}(τ)∥_{L_{t}^{q}L_{x}^{2}} ≲ ∫_{0}^{τ}(τ − τ′)^{− \frac{1}{2}} ⟨τ′^{− \frac{1}{4} −}⟩N_{1}(F) ≲ max(τ^{\frac{1}{4} −}, τ^{\frac{1}{2}} )N_{1}(F),

which completes the proof of (9.43).

5When we use the formula in Lemma 9.1 to compute ∥∇^{2}U(τ′)F ∥_{L_{x}^{2}}, we set p = ∞ for simplicity.

Now we estimate Φ_{2}(τ). Set ϕ_{2}(τ′) = β · ∇U(τ′)F and fix r > 1 sufficiently close to 1. Then by (9.31) we have

∥t^{\frac{1}{2} − \frac{1}{q}} Φ_{2}(τ)∥_{L_{t}^{q}L_{x}^{2}} ≲ ∥t^{\frac{1}{2} − \frac{1}{q}} ∫_{0}^{τ} t^{2}U(τ − τ′)ϕ_{2}(τ′)dτ′∥_{L_{t}^{q}L_{x}^{2}}
≲ ∥ ∫_{0}^{τ}⟨(τ − τ′)^{−( \frac{1}{r} − \frac{1}{2} )}⟩t^{\frac{7}{2} − \frac{1}{q} − \frac{2}{r}} ∥ϕ_{2}(τ′)∥_{L_{x}^{r}}dτ′∥_{L_{t}^{q}}.  (9.50)

Thus we need the estimate of ∥t^{\frac{7}{2} − \frac{1}{q} − \frac{2}{r}} ϕ_{2}(τ′)∥_{L_{t}^{q}L_{x}^{r}}. We apply Hölder inequality to get

∥ϕ_{2}(τ′)∥_{L_{x}^{r}} ≲ ∥β∥_{L_{x}^{2}}∥∇U(τ′)F ∥_{L \frac{2r}{2−r} ; x} .  (9.51)

Moreover, by Sobolev inequality

∥∇U(τ′)F ∥_{L \frac{2r}{2−r} ; x} ≲ ∥∇^{2}U(τ′)F ∥_{L_{x}^{2}}^{2− \frac{2}{r}} ∥∇U(τ′)F ∥_{L_{x}^{2}}^{\frac{2}{r} −1} + ∥t^{−(2− \frac{2}{r})}∇U(τ′)F ∥_{L_{x}^{2}}, (9.52)

and Lemma 9.1 (iii) we infer

∥∇U(τ′)F ∥_{L \frac{2r}{2−r} ; x} ≲ ((t^{2}τ′)^{− \frac{1}{2}} + t^{\frac{1}{p−1}} ∥\underline{K}∥L_{x}^{2}^{\frac{p}{p−1}} + ∥\underline{K}∥_{L_{x}^{2}} + t^{−1})^{2− \frac{2}{r}} ∥U(τ′/2)F ∥_{H_{x}^{1}}. (9.53)

By using Hölder’s inequality for L_{t}^{q} , we derive

∥t^{\frac{7}{2} − \frac{1}{q} − \frac{2}{r}} ϕ_{2}(τ′)∥_{L_{t}^{q}L_{x}^{r}} ≲ ⟨(τ′)^{\frac{1}{r} −1}⟩∥t^{\frac{3}{2} − \frac{1}{q}} U(τ′/2)F ∥_{L_{t}^{q_{3}}H_{x}^{1}}
+ ∥\underline{K}∥_{L^{2}(H)}^{(2− \frac{2}{r}) \frac{p}{p−1}} ∥t^{3− \frac{2}{r} − \frac{1}{q_{3}} + \frac{2− \frac{2}{r}}{p−1}} U(τ/2)F ∥_{L_{t}^{a}H_{x}^{1}}

where \frac{1}{q_{3}} = \frac{1}{q} − \frac{1}{2}, \frac{1}{a} = \frac{1}{q_{3}} − \frac{1− \frac{1}{r}}{1− \frac{1}{p}}. According to Proposition 9.2 (i) and (K1), we conclude

∥t^{\frac{7}{2} − \frac{1}{q} − \frac{2}{r}} ϕ_{2}(τ′)∥_{L_{t}^{q}L_{x}^{r}} ≲ ⟨τ′^{\frac{1}{q_{3}} − \frac{1}{2} −}⟩N_{1}(F).  (9.54)

Returning to (9.50), we thus get (9.44). This completes the proof of Lemma 9.2.^{6}

We are going to derive a refined estimate for t^{\frac{1}{2} − \frac{1}{q}} Φ_{2}.

Lemma 9.3.

∥t^{\frac{1}{2} − \frac{1}{q}} Φ_{2}(τ)∥_{L_{t}^{q}L_{x}^{2}} ≲ max (τ^{\frac{1}{4} −}, τ^{\frac{1}{2}} ) N_{1}(F).

Proof. We first assume for given tensor field F there holds

∥t^{\frac{1}{2}} ∇U(τ)F ∥_{L_{t}^{∞}L_{x}^{2}} ≲ ⟨τ^{− \frac{1}{4} −}⟩N_{1}(F),  (9.55)

6We can ignore the term of ∥\underline{K}∥_{L_{x}^{2}} in (9.53.

which will be proved later. By applying (9.55) and (9.52) we obtain

∥t^{\frac{7}{2} − \frac{1}{q} − \frac{2}{r}} ϕ_{2}(τ′)∥_{L_{t}^{q}L_{x}^{r}}

≲ ∥t^{\frac{7}{2} − \frac{2}{r} − \frac{1}{q}} ((t^{2}τ′)^{− \frac{1}{2}} + t^{\frac{1}{p−1}} ∥\underline{K}∥L_{x}^{2}^{\frac{p}{p−1}} + ∥\underline{K}∥_{L_{x}^{2}} + t^{−1})^{2− \frac{2}{r}} ∥∇U(τ′/2)F ∥_{L_{x}^{2}}∥_{L \frac{2q}{2−q} ; t} ∥β∥_{L_{t}^{2}L_{x}^{2}}

≲ τ^{′−1+ \frac{1}{r}} ∥t^{1− \frac{1}{q}} ∥_{L \frac{2q}{2−q} ; t} ∥t^{\frac{1}{2}} ∇U(τ′/2)F ∥_{L_{t}^{∞}L_{x}^{2}}

+ ∥t^{\frac{7}{2} − \frac{2}{r} − \frac{1}{q} + \frac{1}{p−1} (2− \frac{2}{r})}∥\underline{K}∥_{L_{x}^{2}}^{\frac{p}{p−1} (2− \frac{2}{r})}∥∇U(τ′/2)F ∥_{L_{x}^{2}}∥_{L \frac{2q}{2−q} ; t}

+ ∥t^{\frac{3}{2} − \frac{1}{q}} ∥∇U(τ′/2)F ∥_{L_{x}^{2}}∥_{L \frac{2q}{2−q} ; t} + ∥∥\underline{K}∥_{L_{x}^{2}}^{2− \frac{2}{r}} ∥t^{\frac{7}{2} − \frac{2}{r} + \frac{1}{q}} ∇U(τ′/2)F ∥_{L_{x}^{2}}∥_{L \frac{2q}{2−q} ; t} .

The first term can be estimated easily by (9.55). The last two terms can be estimated similarly. The second term can be estimated by Hölder’s inequality since when r, p > 1, we always have \frac{7}{2} − \frac{2}{r} − \frac{1}{q} + \frac{1}{p−1}(2 − \frac{2}{r}) > \frac{p}{p−1}(1 − \frac{1}{r}) + 1 − \frac{1}{q}. Therefore

∥t^{\frac{7}{2} − \frac{2}{r} − \frac{1}{q}} ϕ_{2}(τ′)∥_{L_{t}^{q}L_{x}^{r}} ≲ ⟨τ^{′−1+ \frac{1}{r}} ⟩∥t^{\frac{1}{2}} ∇U(τ′/2)F ∥_{L_{t}^{∞}L_{x}^{2}}.

By using (9.55), we can conclude for r sufficiently close to 1 that

∥t^{\frac{7}{2} − \frac{1}{q} − \frac{2}{r}} ϕ_{2}(τ′)∥_{L_{t}^{q}L_{x}^{r}} ≲ ⟨τ′^{− \frac{1}{4} −}⟩N_{1}(F).  (9.56)

Finally, combining (9.56) with (9.31), we obtain for r > 1 sufficiently close to 1,

∥t^{\frac{1}{2} − \frac{1}{q}} Φ_{2}(τ)∥_{L_{t}^{q}L_{x}^{2}} ≲ ∥∫_{0}^{τ} U(τ − τ′)ϕ_{2}(τ′)dτ′∥_{L_{t}^{q}L_{x}^{2}}
≲ ∫_{0}^{τ}⟨(τ − τ′)^{− \frac{1}{r} + \frac{1}{2}} ⟩∥t^{\frac{7}{2} − \frac{2}{r} − \frac{1}{q}} ϕ_{2}(τ′)∥_{L_{t}^{q}L_{x}^{r}}
≲ ∫_{0}^{τ}⟨(τ − τ′)^{\frac{1}{2} − \frac{1}{r}} ⟩⟨τ′^{− \frac{1}{4} −}⟩N_{1}(F)
≲ max(τ^{\frac{5}{4} − \frac{1}{r} −}, τ^{\frac{3}{2} − \frac{1}{r}})N_{1}(F)
≲ max(τ^{\frac{1}{4} −}, τ^{\frac{1}{2} +})N_{1}(F).

It remains to prove (9.55). We proceed as follows

∥t^{\frac{1}{2}} ∇U(τ′/2)F ∥_{L_{t}^{∞}L_{x}^{2}}^{2}
≲ ∫_{H} 2s∇_{L}∇U(τ′/2)F · ∇U(τ′/2)F + ∇U(τ′/2)F · ∇U(τ′/2)F
≲ ∫_{H} 2s∇U(τ′/2)∇_{L}F · ∇U(τ′/2)F + 2s[U(τ′/2), ∇_{L}]F · ΔU(τ′/2)F
+ 2s[∇_{L}, ∇]U(τ′/2)F · ∇U(τ′/2)F + ∇U(τ′/2)F · ∇U(τ′/2)F
≲ ∥t^{\frac{3}{2}} ΔU(τ′/2)F ∥_{L_{t}^{∞}L_{x}^{2}}∥t^{− \frac{1}{2}} [∇_{L}, U(τ′/2)]F ∥_{L_{t}^{1}L_{x}^{2}}  (9.57)
+ ∥∇U(τ′/2)F ∥_{L_{t}^{2}L_{x}^{2}}∥t∇U(τ′/2)∇_{L}F ∥_{L_{t}^{2}L_{x}^{2}} + ∥∇U(τ′/2)F ∥_{L_{t}^{2}L_{x}^{2}}^{2} (9.58)
+ ∥2(t[∇_{L}, ∇] + ∇)U(τ′/2)F · ∇U(τ′/2)F ∥_{L_{t}^{1}L_{x}^{1}}.  (9.59)

It is easy to see that (9.58) is bounded by ⟨τ′^{− \frac{1}{2}} ⟩N_{1}(F)^{2}. To estimate (9.57), observing

∥t^{\frac{3}{2}} ΔU(τ′/2)F ∥_{L_{t}^{∞}L_{x}^{2}} ≲ τ′^{−1}∥t^{−1/2}F ∥_{L_{t}^{∞}L_{x}^{2}} ≲ τ′^{−1}N_{1}(F)

and the following estimate

∥t^{− \frac{1}{2}} [∇_{L}, U(τ′/2)]F ∥_{L_{t}^{1}L_{x}^{2}} ≲ max(τ′^{\frac{1}{2} −}, τ′^{\frac{1}{2} +})N_{1}(F),  (9.60)

which will be verified soon, we can see (9.57) is bounded by max(τ′^{− \frac{1}{2} +}, τ′^{− \frac{1}{2} −})N_{1}(F)^{2}.

The term (9.59) can be estimated by using the commutation formula (9.36) and the definition of ϕ_{1}(τ). Since

(s[∇_{L}, ∇] + ∇)U(τ′/2)F ≈ sϕ_{1}(τ′/2),

we conclude that (9.59) is bounded by

∥tϕ_{1}(τ′/2)∇U(τ′/2)F ∥_{L_{t}^{1}L_{x}^{1}} ≲ ∥t^{\frac{1}{2}} ϕ_{1}(τ′/2)∥_{L_{t}^{1}L_{x}^{2}}∥t^{\frac{1}{2}} ∇U(τ′/2)F ∥_{L_{t}^{∞}L_{x}^{2}}
≲ ⟨τ^{− \frac{1}{2} −}⟩N_{1}(F)^{2}

where the last inequality follows from the following estimate, for tensor fields F,

∥t^{\frac{1}{2}} ϕ_{1}(τ′/2)∥_{L_{t}^{1}L_{x}^{2}} ≲ ⟨τ′^{0−}⟩N_{1}(F),  (9.61)

which will be justified soon. Combine the above estimates for (9.57), (9.58) and (9.59), we get(9.55).

It remains to verify (9.60) and (9.61). Let us start with (9.61). Fix a sufficiently large exponent r < ∞. Then

∥t^{\frac{2}{r}} ϕ_{1}(τ)∥_{L_{x}^{2}} ≲ ∥t^{\frac{2}{r}} A · ∇U(τ)F ∥_{L_{x}^{2}} + ∥t^{\frac{2}{r}} β · U(τ)F ∥_{L_{x}^{2}}

≲ t^{\frac{2}{r}} ∥A∥_{L_{x}^{r}}∥∇U(τ)F ∥_{L \frac{2r}{r−2} ; x} + t^{\frac{2}{r}} ∥β∥_{L_{x}^{2}}∥U(τ)F ∥_{L_{x}^{∞}}

≲ t^{\frac{2}{r}} ∥A∥_{L_{x}^{r}}∥∇^{2}U(τ)F ∥L_{x}^{2}^{\frac{2}{r}}∥∇U(τ)F ∥_{L_{x}^{2}}^{1− \frac{2}{r}} + t^{\frac{2}{r}} ∥β∥_{L_{x}^{2}}∥U(τ)F ∥_{L_{x}^{∞}}

≲ ∥A∥_{L_{x}^{r}}((t^{2}τ)^{− \frac{1}{r}} + t^{\frac{2}{r(p−1)}} ∥\underline{K}∥L_{x}^{2}^{\frac{2p}{r(p−1)}} + ∥\underline{K}∥L_{x}^{2}^{\frac{2}{r}} + t^{− \frac{2}{r}})t^{\frac{2}{r}} ∥U(τ/2)F ∥_{H_{x}^{1}}

+ ∥β∥_{L_{x}^{2}}⟨τ^{− \frac{1}{r}} ⟩(1 + ∥t\underline{K}∥L_{x}^{2}^{\frac{1}{p−1}} + ∥t\underline{K}∥L_{x}^{2}^{\frac{1}{p}} + τ^{− \frac{1}{2p}})∥F ∥_{L_{x}^{r}}.

For simplicity, we set p = ∞,

∥t^{\frac{2}{r}} ϕ_{1}(τ)∥_{L_{x}^{2}} ≲ (τ^{− \frac{1}{r}} + ∥\underline{K}∥L_{x}^{2}^{\frac{2}{r}}) ∥A∥_{L_{x}^{r}}∥F ∥_{H_{x}^{1}} + ⟨τ^{− \frac{1}{r}} ⟩∥β∥_{L_{x}^{2}}∥F ∥_{L_{x}^{r}}.

Taking L_{t}^{1} norm, we obtain

∥t^{\frac{1}{2}} ϕ_{1}(τ)∥_{L_{t}^{1}L_{x}^{2}} ≲ τ^{− \frac{1}{r}} ∥t^{\frac{1}{2} − \frac{2}{r}} A∥_{L_{t}^{2}L_{x}^{r}}∥F ∥_{L_{t}^{2}H_{x}^{1}} + ⟨τ^{− \frac{1}{r}} ⟩∥β∥_{L_{t}^{2}L_{x}^{2}}∥t^{\frac{1}{2} − \frac{2}{r}} F ∥_{L_{t}^{2}L_{x}^{r}}

+ ∥t^{\frac{1}{2} − \frac{2}{r}} A∥_{L \frac{2r}{r−2} ; t L_{x}^{r}}∥\underline{K}∥L_{t}^{2}L_{x}^{2}^{\frac{2}{r}}∥F ∥_{L_{t}^{2}H_{x}^{1}}.

By Sobolev inequality and (3.39), both ∥t^{\frac{1}{2} − \frac{2}{r}} A∥_{L_{t}^{2}L_{x}^{r}} and ∥t^{\frac{1}{2} − \frac{2}{r}} A∥_{L \frac{2r}{r−2} ; t L_{x}^{r}} can be bounded

by N_{1}(A). Consequently, it follows from (BA1) and (K1) that ∥t^{\frac{1}{2}} ϕ_{1}(τ′/2)∥_{L_{t}^{1}L_{x}^{2}} ≲ ⟨τ′^{0−}⟩ · N_{1}(F). Thus we complete the proof of (9.61), which then implies

∥t^{− \frac{1}{2}} Φ_{1}∥_{L_{t}^{1}L_{x}^{2}} ≲ max(τ^{\frac{1}{2} −}, τ^{\frac{1}{2}} )N_{1}(F).  (9.62)

Combine this with (9.44) which says ∥t^{− \frac{1}{2}} Φ_{2}(τ)∥_{L_{t}^{1}L_{x}^{2}} ≲ max(τ^{\frac{1}{2} −}, τ^{\frac{1}{2} +})N_{1}(F), we obtain

∥t^{− \frac{1}{2}} [∇_{L}, U(τ′/2)]F ∥_{L_{t}^{1}L_{x}^{2}} ≲ max(τ′^{\frac{1}{2} −}, τ′^{\frac{1}{2} +})N_{1}(F).

This completes the proof of (9.60). Lemma 9.3 is therefore proved.

Now we are ready to complete the proof of Lemma 5.1. It follows from Lemma 9.2 that

∥t^{\frac{1}{q} − \frac{1}{2}} [P_{k}, ∇_{L}]F ∥_{L_{t}^{q}L_{x}^{2}} ≲ N_{1}(F) (∫_{0}^{1} m_{k}(τ)τ^{\frac{1}{4} −} + ∫_{1}^{∞} m_{k}(τ)τ^{\frac{1}{2} +})
≲ N_{1}(F) (2^{\frac{−k}{2} +} ∫_{0}^{2^{2k}} m(τ)τ^{\frac{1}{4} −} + 2^{−k−} ∫_{2^{2k}}^{∞} m(τ)τ^{\frac{1}{2} +})
≲ 2^{− \frac{k}{2} +}N_{1}(F)

and

∥t^{− \frac{1}{2}} [P_{k}, ∇_{L}]F ∥_{L_{t}^{1}L_{x}^{2}} ≲ N_{1}(F) · ∫_{0}^{∞} m_{k}(τ) max(τ^{\frac{1}{2} −}, τ^{\frac{1}{2} +})
≲ N_{1}(F) · (∫_{0}^{1} m_{k}(τ)τ^{\frac{1}{2} −} + ∫_{1}^{∞} m_{k}(τ) · τ^{\frac{1}{2} +})
≲ N_{1}(F) · (2^{−k+} ∫_{0}^{2^{2k}} m(τ)τ^{\frac{1}{2} −} + 2^{−k−} ∫_{2^{2k}}^{∞} m(τ)τ^{\frac{1}{2} +})
≲ 2^{−k+} · N_{1}(F).

Next we will prove

∥t^{\frac{3}{2} − \frac{1}{q}} ∇[P_{k}, ∇_{L}]F ∥_{L_{t}^{q}L_{x}^{2}} ≲ 2^{k− \frac{k}{2} +}N_{1}(F).  (9.63)

To obtain the estimate for the term ∇[P_{k}, ∇_{L}]F, we need to control ∇Φ_{1}(τ) and ∇Φ_{2}(τ). Observe that the tensor fields Φ_{j}(τ), j = 1, 2 verify the repective heat equations

∂_{τ}Φ_{1} − s^{2}ΔΦ_{1} = s^{2}∇ϕ_{1} Φ_{1}|_{τ=0} = 0,
∂_{τ}Φ_{2} − s^{2}ΔΦ_{2} = s^{2}ϕ_{2} Φ_{2}|_{τ=0} = 0.

By multiplying the equations for Φ_{j} by Φ_{j} and integrating by parts, we can obtain

∥t^{\frac{3}{2} − \frac{1}{q}} ∇Φ_{1}∥_{L_{τ}^{2}L_{x}^{2}} ≲ ∥t^{\frac{3}{2} − \frac{1}{q}} ϕ_{1}∥_{L_{τ}^{2}L_{x}^{2}},  (9.64)
∥t^{\frac{3}{2} − \frac{1}{q}} ∇Φ_{2}∥_{L_{τ}^{2}L_{x}^{2}} ≲ ∥t^{\frac{7}{2} − \frac{2}{r} − \frac{1}{q}} ϕ_{2}∥_{L \frac{2r}{3r−2} ; τ L_{x}^{r}}, 1 ≤ r < 2,  (9.65)

where ∥ · ∥_{L_{τ}^{2}} refers to the L^{2} norm on the interval [0, τ]. Since we have (9.56) for ϕ_{2} and (9.49) for ϕ_{1}, we then obtain

∥t^{\frac{3}{2} − \frac{1}{q}} ϕ_{1}∥_{L_{t}^{q}L_{τ}^{2}L_{x}^{2}} ≲ max(τ^{\frac{1}{2}} , τ^{\frac{1}{4} −})N_{1}(F)  (9.66)

∥t^{\frac{7}{2} − \frac{2}{r} − \frac{1}{q}} ϕ_{2}∥_{L_{t}^{q}L \frac{2r}{3r−2} ; τ L_{x}^{r}} ≲ max(τ^{\frac{1}{4} −}, τ^{\frac{1}{2}} )N_{1}(F).

By a standard argument, we can convert the above into the following weighted estimates

∥t^{\frac{3}{2} − \frac{1}{q}} min(τ^{− \frac{1}{4} +}, τ^{− \frac{1}{2} +})∇Φ_{1}∥_{L_{t}^{q}L_{τ}^{2}L_{x}^{2}} ≲ N_{1}(F)
∥t^{\frac{3}{2} − \frac{1}{q}} min(τ^{− \frac{1}{4} +}, τ^{− \frac{1}{2} +})∇Φ_{2}∥_{L_{t}^{q}L_{τ}^{2}L_{x}^{2}} ≲ N_{1}(F).

Therefore

∥t^{\frac{3}{2} − \frac{1}{q}} ∇[P_{k}, ∇_{L}]F ∥_{L_{t}^{q}L_{x}^{2}} ≲ ∥ ∫_{0}^{∞} m_{k}(τ)t^{\frac{3}{2} − \frac{1}{q}} ∇Φ_{j}(τ)∥_{L_{t}^{q}L_{x}^{2}}
≲ ∥ max(τ^{\frac{1}{4} −}, τ^{\frac{1}{2} −})m_{k}(τ)∥_{L_{τ}^{2}}∥t^{\frac{3}{2} − \frac{1}{q}} min(τ^{− \frac{1}{4} +}, τ^{− \frac{1}{2} +})∇Φ_{j}∥_{L_{t}^{q}L_{τ}^{2}L_{x}^{2}}
≲ 2^{k}2^{− \frac{k}{2} +}N_{1}(F).

As to the case when q = 1, taking L_{τ}^{2} norm of (9.61) and Lτ^{\frac{2r}{3r−2}} norm of (9.54), we obtain

∥t^{\frac{1}{2}} ϕ_{1}∥_{L_{t}^{1}L_{τ}^{2}L_{x}^{2}} ≲ max(τ^{\frac{1}{2}} , τ^{\frac{1}{2} −}) · N_{1}[F]
∥t^{\frac{5}{2} − \frac{2}{r}} ϕ_{2}∥_{L_{t}^{1}L \frac{2r}{3r−2} ; τ L_{x}^{r}} ≲ τ^{\frac{3r−2}{2r}} · N_{1}[F].

Hence

∥t^{\frac{1}{2}} ∇Φ_{1}∥_{L_{t}^{1}L_{τ}^{2}L_{x}^{2}} ≲ max(τ^{\frac{1}{2}} , τ^{\frac{1}{2} −}) · N_{1}(F)
∥t^{\frac{1}{2}} ∇Φ_{2}∥_{L_{t}^{1}L_{τ}^{2}L_{x}^{2}} ≲ max(τ^{\frac{1}{2}} , τ^{\frac{1}{2} −}) · N_{1}(F).

Then in the similar manner as before we conclude

∥t^{\frac{1}{2}} ∇[P_{k}, ∇_{L}]F ∥_{L_{t}^{1}L_{x}^{2}} ≲ 2^{k}2^{−k+}N_{1}[F].

The proof of Lemma 5.1 is thus complete.

Bibliography

[1] Choquet-Bruhat, Y., Theoreme d’Existence pour certains systemes d’equations aux derivees partielles nonlineaires, Acta Math., 88(1952), 141-225.

[2] Christodoulou, D. and Klainerman, S. The global nonlinear stability of the Minkowski space, Princeton Mathematical Series 41, 1993.

[3] Friedrich, H., and Stewart, J. M., Characteristic initial data and wave front singularities in general relativity, Proc. R.Soc. Lond., Ser. A. Math. Phys. Eng. Sci. 385(1983), 341-371.

[4] Hawking, S.W. and Ellis, G. F. R., The Large Scale Structure of Space-Time, Cambridge Monographs on Mathematical Physics, 1973.

[5] Hideo, K and Masao, Y., Semilinear Heat equations and the Navier-Stokes equation with distributions in new function spaces as initial data, Commun in PDE, 19(1994), 959-1014.

[6] Hughes, T. J. R., Kato, T. and Marsden, J., Well posed quasilinear second order hyperbolic systems, Arch. Ration. Maech. Anal. 63(1976), 273-294.

[7] Klainerman, S. and Francesco N., The evolution Problem in General Relativity, Birkhäuser, 2003.

[8] Klainerman, S., Machedon, M. Space-time estimates for null forms and the local existence theorem, CPAM 46, 1221 C1268 (1993).

[9] Klainerman, S. and Rodnianski, I., Unpublished notes, 2003.

[10] Klainerman, S. and Rodnianski, I., Rough Solutions to the Einstein Vacuum Equations, Annals of Math., 161 (2005), 1143-1193.

[11] Klainerman, S. and Rodnianski, I., The Causal Structure of Microlocalized Rough Einstein Metrics, Annals of Math., 161 (2005), 1195-1243.

[12] Klainerman, S. and Rodnianski, I., Causal geometry of Einstein-vacuum spacetimes with finite curvature flux, Invent. Math., 159 (2005), no. 3, 437–529.

[13] Klainerman, S. and Rodnianski, I., Bilinear estimates on curved space-times, J. Hyperb. Diff. Eqs. 2 2005), 279-291.

[14] Klainerman, S. and Rodnianski, I., Sharp Trace theorems for null hypersurfaces on Einstein metrics with finite curvature flux, Geom. Funct. Anal., to appear.

[15] Klainerman, S. and Rodnianski, I., A geometric Littlewood-Paley theory, Geom. Funct. Anal., to appear.

[16] Klainerman, S. and Rodnianski, I., On the radius of injectivity of null hypersurfaces

[17] Klainerman, S. and Rodnianski, I. A Kirchoff-Sobolev parametrix for the wave equation and applications

[18] Klainerman, S. and Rodnianski, I. A large data break-down criterion in General relativity

[19] Lindblad H. and Rodnianski, I., Global Existence for the Einstein Vacuum Equations in Wave Coordinates, Comm. Math. Phys. 256 (2005), 43-110.

[20] Poisson, E., The motion of Point Particles in Curved Spacetimes, www.livingreviews.org/lrr-2004-6

[21] Seeger, A. and Sogge, C. D., On the boundedness of function of (pseudo-) differential operators on compact manifolds, Duke Math J., 59:3(1989), 1297–1364.

[22] Stein, E. M., Topics in harmonic analysis related to the Littlewood-Paley theory, Annals of Mathematics Studies, No. 63, Princeton University Press, 1970.

[23] Stein, E. M., Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, With the assistance of Timothy S. Murphy, Princeton Mathematical Series 43, Monographs in Harmonic Analysis III, Princeton University Press, Princeton, NJ, 1993.

[24] Tao, T., Harmonic analysis in the phase plane, Lecture notes 254A, http:// www.math.ucla.edu/∼tao.

[25] Triebel, H., Theory of function spaces, Monographs in Mathematics 78, Birkhäuser Verlag, Basel, 1983.

[26] Triebel, H., Interpolation theory, function spaces, differential operators, Second edition, Johann Ambrosius Barth, Heidelberg, 1995.

[27] Wald R.M. General relativity, University of Chicago Press, 1984.