Mémoire présenté à l’Université Paris-Nordpour l’obtention de l’habilitation à diriger des recherchesSpécialité : Mathématiques

Qualitative properties of some nonlinear partial
differential equations

parJérémie Szeftel

Soutenu publiquement le 8 octobre 2012, après avis deJean-Yves Chemin
Professeur à l’Université Pierre et Marie Curie
MihalisDafermos Professeur à University of Cambridge
GillesLebeauProfesseur à l’Université de Nice Sophia-Antipolis
devant le jury composé deJean-Yves
MihalisJean-Marc
Laurence
GillesPhilippe
Fabrice

CheminDafermosDelortHalpern
LebeauLeFlochPlanchon

Professeur à l’Université Pierre et Marie Curie
Professeur à University of Cambridge
Professeur à l’Université Paris-NordProfesseur à l’Université Paris-NordProfesseur à l’Université de Nice Sophia-Antipolis
Directeur de recherche à l’Université Pierre et Marie Curie
Professeur à l’Université de Nice Sophia-Antipolis

Remerciements
Je voudrais tout d’abord remercier Jean-Yves Chemin, Mihalis Dafermos et Gilles Lebeau
qui m’ont fait l’honneur de rapporter sur mon habilitation. Vu la longueur du mémoire et lesdélais très courts imposés, je suis bien conscient que la tâche était ardue! Je suis également
reconnaissant à Philippe LeFloch et Fabrice Planchon pour le plaisir qu’ils me font en prenantpart à ce jury.
Je remercie particulièrement Laurence Halpern que j’ai eu beaucoup de plaisir à avoir nonseulement comme directrice de thèse, mais également comme collaboratrice depuis. Je lui envie
son enthousiasme à toute épreuve et son infatigable énergie.
Je remercie chaleureusement Jean-Marc Delort pour sa grande disponiblilité et la qualité deses conseils avisés. J’ai fortement apprécié de collaborer avec lui, et mon parcours scientifiquelui doit beaucoup.
Je tiens à exprimer ma profonde gratitude à Sergiu Klainerman et Igor Rodnianski pourm’avoir introduit à la relativité générale, en particulier la conjecture de courbure L2 , et pour lesnombreuses discussions scientifiques (et politiques) passionnantes que nous avons eues depuis.
Je tiens aussi à remercier du fond du coeur Pierre Raphaël pour m’avoir formé aux problèmes
d’explosion en temps fini pour l’équation de Schrödinger non linéaire, ainsi que Frank Merle dequi j’ai également beaucoup appris. Je suis très admiratif de la fulgurance et de la profondeurde ces quatre là, et j’espère avoir le privilège de collaborer avec eux pendant encore longtemps!
Je remercie sincèrement Zaina Elmir et Bénédicte Auffray pour leur efficacité sans failleet leur aide précieuse. Je tiens également à remercier les directeurs du DMA et de l’équiped’analyse, Olivier Debarre et David Lannes, qui me permettent de travailler dans de très bonnes
conditions. Je remercie aussi tous mes collègues du DMA qui contribuent à la bonne ambiancequi y règne, et plus particulièrement Thomas Alazard et Gilles Stoltz pour m’avoir aidé à naviguer à travers les nombreuses démarches administratives que comporte une HDR.
Tous les jours de ma vie ne suffiront pas à remercier Emilie pour tout ce qu’elle m’a apporté
et continue de m’apporter, à commencer par mes deux petits coquins adorés!

List of the works
Works presented in this memoir
The bounded L2 curvature conjecture in general relativity.
• Parametrix for wave equations on a rough background I: regularity of the phase atinitial time. Preprint, 145 pp.
• Parametrix for wave equations on a rough background II: control of the parametrix at
initial time. Preprint, 84 pp.
• Parametrix for wave equations on a rough background III: space-time regularity of thephase. Preprint, 276 pp.
• Parametrix for wave equations on a rough background IV: Control of the error term.
Preprint, 284 pp.
• The bounded L2 curvature conjecture (with S. Klainerman and I. Rodnianski). Preprint,
79 pp.
Formation of singularities for the nonlinear Schrödinger equation.
• Standing ring blow up solutions to the N-dimensional quintic nonlinear Schrödinger
equation (with P. Raphaël), Comm. Math. Phys., 290 (3), 973-996, 2009.
• Stable self similar blow up dynamics for slightly L2 supercritical NLS equations (withF. Merle and P. Raphaël). Geom. Funct. Anal., 20 (4), 1028-1071, 2010.
• Existence and uniqueness of minimal blow up solutions to an inhomogeneous masscritical NLS (with P. Raphaël). J. Amer. Math. Soc., 24, 471-546, 2011.
• On the instability of the pseudoconformal blow up for the L2 critical NLS (with F.
Merle and P. Raphaël). Accepted for publication in Amer. J. Math, 41 pp.
Domain decomposition methods.
• Nonlinear Nonoverlapping Schwarz Waveform Relaxation for Semilinear Wave Propagation (with L. Halpern), Math. Comp., 78, 865-889, 2009.
• Schwarz waveform relaxation algorithms for semilinear reaction-diffusion equations(with F. Caetano, M. Gander and L. Halpern). Netw. Heterog. Media, 5 (3), 487505, 2010.
• Optimized and Quasi-Optimal Schwarz Waveform Relaxation for the One Dimensional
Schrödinger equation (with L. Halpern). Math. Models Methods Appl. Sci, 20 (12),2167-2199, 2010.
• Optimized Schwarz waveform relaxation and discontinuous Galerkin time stepping forheterogeneous problems (with C. Japhet and L. Halpern). Submitted, 33 pp.
iii

iv

LIST OF THE WORKS

Long-time existence for the Klein-Gordon equation on compact manifolds.
• Long time existence for small data nonlinear Klein-Gordon equations on tori andspheres (with J.M. Delort), Int. Math. Res. Not. 37, 1897-1966, 2004.
• Almost orthogonality properties of products of eigenfunctions and applications to longtime existence for semi-linear Klein-Gordon equations on Zoll Manifolds (with J.M.
Delort), Amer. J. Math., 128, 1187-1218, 2006.
• Bounded almost global solutions for non hamiltonian semi-linear Klein-Gordon equations with radial data on compact revolution hypersurfaces (with J. M. Delort), Ann.
Inst. Fourier (Grenoble), 56, 1419-1456, 2006.
• Almost global existence for Hamiltonian semi-linear Klein-Gordon equations with small
Cauchy data on Zoll manifolds (with D. Bambusi, J. M. Delort and B. Grébert), Comm.
Pure. Appl. Math., 60, 1665-1690, 2007.
Other works (mostly PhD)
Absorbing boundary conditions.
• Absorbing boundary conditions for reaction diffusion equation, IMA J. Appl. Math.,
68 (2), 167-184, 2003.
• Design of absorbing boundary conditions for Schrödinger equations in Rd , SIAM J.
Numer. Anal., 42 (4), 1527-1551, 2004.
• A nonlinear approach to absorbing boundary conditions for the semilinear wave equation, Math. Comp., 75, 565-594, 2006.
• Absorbing boundary conditions for nonlinear scalar partial differential equations, Comput. Methods Appl. Mech. Engrg., 195, 3760-3775, 2006.
• Absorbing boundary conditions for nonlinear Schrödinger equations, Numerische Mathematik, 104, 103-127, 2006.
• Towards accurate artificial boundary conditions for nonlinear PDEs through examples(with X. Antoine and C. Besse), Cubo, 11 (4), 29-48, 2009.
Propagation of singularities and smoothing effect.
• Réflection des singularités pour l’équation de Schrödinger, Comm. Partial Differential
Equations, 29 (5-6), 707-761, 2004.
• Propagation et réflexion des singularités pour l’équation de Schrödinger non linéaire,
Ann. Inst. Fourier (Grenoble), 55 (2), 573-671, 2005.
• Microlocal smoothing effect for the nonlinear Schrödinger equation, SIAM J. Math.
Anal., 37 (2), 549-597, 2005.

Summary
This memoir contains an overview of my research since the completion of my PhD in 2004.
I present this overview in inverse chronological order. The topics covered are respectively:• The bounded L2 curvature conjecture in general relativity• Formation of singularities for the nonlinear Schrödinger equation• Domain decomposition methods• Long-time existence for the Klein-Gordon equation on compact manifolds
I choose here to focus on the most recent work which is the one on the bounded L2 curvature
conjecture in general relativity. This is covered in details in part 1 of this memoir. In part 2,
the other three topics are then covered with much less details.

vContents
List of the works
Works presented in this memoir
Other works (mostly PhD)

iii
iii
iv

Summaryv

Part 1.

The Bounded L2 Curvature Conjecture

1

Chapter 1. Introduction
1.1. General Introduction
1.2. Statement of the main results

338

Chapter 2. Einstein vacuum equations as a Yang-Mills gauge theory
2.1. Yang-Mills formalism
2.2. Strategy of the proof of the main Theorem
2.3. Simple consequences of the bootstrap assumptions
2.4. Estimates for B
2.5. Energy estimate for the wave equation on a curved background2.6. Improvement of the bootstrap assumptions (part 1)
2.7. Parametrix for the wave equation
2.8. Improvement of the bootstrap assumptions (part 2)
2.9. Improvement of the bootstrap assumptions (part 3)

19
19
22
27
29
31
33
34
37
41

Chapter 3. Control of the error term
3.1. Geometric set-up and main results
3.2. Control of the error term
3.3. Almost orthogonality in frequency
3.4. Control of the diagonal term
3.5. Almost orthogonality in angle

45
45
49
52
57
62

Chapter 4. Control of the space-time foliation
4.1. Geometric set-up and main results
4.2. Regularity of the foliation with respect to (t, x)4.3. An estimate for LLtrχ
4.4. Regularity of the foliation with respect to ω
4.5. Additional decompositions

69
69
76
83
84
86

Chapter 5.

89Construction and control of the parametrix at initial time
vii

viii

CONTENTS

5.1.
5.2.
5.3.

Geometric set-up and main results
Control of Fourier integral operators
Control of the parametrix at initial time

89
94
101

Chapter 6. Control of the foliation at initial time
6.1. Geometric set-up and main results
6.2. Construction of the foliation and regularity with respect to x
6.3. Estimates for ∇N a and ∇2N a
6.4. Regularity of the foliation with respect to ω
6.5. A global coordinate system on Pu and Σ
6.6. Additional estimates

105
105
109
111
112
113
114

Part 2. Other works
Formation of singularities for the nonlinear Schrödinger equation
Domain decomposition methods
Long-time existence for the Klein-Gordon equation on compact manifolds

115
116
122
124

Bibliography

129

Part 1The Bounded L2 Curvature Conjecture

CHAPTER 1

Introduction
1.1. General Introduction
We present a summary of our proof of the bounded L2 -curvature conjecture in General
Relativity. According to the conjecture the time of existence of a classical solution to the
Einstein-vacuum equations depends only on the L2 -norm of the curvature and a lower boundon the volume radius of the corresponding initial data set. At a deep level the L2 curvature
conjecture concerns the relationship between the curvature tensor and the causal geometry ofan Einstein vacuum space-time. Thus, though the result is not optimal with respect to thestandard scaling of the Einstein equations, it is nevertheless critical to the presence of anotherscaling tied to its causal properties. More precisely, L2 curvature bounds are strictly necessary
to obtain lower bounds on the radius of injectivity of causal boundaries. We note also that,while the first nontrivial improvements for well posedness for quasilinear hyperbolic systems, inspacetime dimensions greater than 1 + 1 (based on Strichartz estimates) were obtained in [2],[3], [79], [78], [35] and optimized in [36], [66], the result we present here is the first in which
the full structure of the quasilinear hyperbolic system, not just its principal part, plays a crucialrole.
As a helpful analogy consider the case of the Cheeger-Gromov non-collapse theory in Riemannian geometry where, based on Lp bounds on the curvature tensor, lower bounds on the
diameter and upper bounds for the volume of a compact manifold, one can control the geometryof the manifold. Note however that among Lp bounds only the exponent p = 2 is relevant
to hyperbolic equations. Note also that, unlike the case of Riemannian geometry, the proper
setting for the conjecture is in the context of the initial value formulation of General Relativity.
1.1.1. Initial value problem. We consider the Einstein vacuum equations (EVE),
Rαβ = 0(1.1)

where Rαβ denotes the Ricci curvature tensor of a four dimensional Lorentzian space time(M, g). An initial data set for (1.1) consists of a three dimensional 3-surface Σ0 together witha Riemannian metric g and a symmetric 2-tensor k verifying the constraint equations, j∇ kij − ∇i trk = 0,
(1.2)
R − |k|2 + (trk)2 = 0,
where the covariant derivative ∇ is defined with respect to the metric g, R is the scalar curvatureof g, and trk is the trace of k with respect to the metric g. In this work we restrict ourselvesto asymptotically flat initial data sets with one end. For a given initial data set the Cauchy
problem consists in finding a metric g satisfying (1.1) and an embedding of Σ0 in M such thatthe metric induced by g on Σ0 coincides with g and the 2-tensor k is the second fundamental
form of the hypersurface Σ0 ⊂ M. The first local existence and uniqueness result for (EVE)34

1. INTRODUCTION

was established by Y.C. Bruhat, see [11], with the help of wave coordinates which allowed her
to cast the Einstein vacuum equations in the form of a system of nonlinear wave equations towhich one can apply1 the standard theory theory of nonlinear hyperbolic systems. The optimal,classical2 result states the following,
Theorem 1.1 (Classical local existence [21] [29]). Let (Σ0 , g, k) be an initial data set for
the Einstein vacuum equations (1.1). Assume that Σ0 can be covered by a locally finite systems (Σ ) ×of coordinate charts, related to each other by C 1 diffeomorphisms, such that (g, k) ∈ Hloc
0
s−1
Hloc
(Σ0 ) with s > 25 . Then there exists a unique3 (up to an isometry) globally hyperbolic
development (M, g), verifying (1.1), for which Σ0 is a Cauchy hypersurface4.
1.1.2. Bounded L2 curvature conjecture. The classical exponents s > 5/2 are clearly
not optimal. By straightforward scaling considerations one might expect to make sense of theinitial value problem for s ≥ sc = 3/2, with sc the natural scaling exponent for L2 based
Sobolev norms. Note that for s = sc = 3/2 a local in time existence result, for sufficiently smalldata, would be equivalent to a global result. More precisely any smooth initial data, small in thecorresponding critical norm, would be globally smooth. Such a well-posedness (WP) result would
be thus comparable with the so called - regularity results for nonlinear elliptic and parabolic
problems, which play such a fundamental role in the global regularity properties of generalsolutions. For quasilinear hyperbolic problems critical WP results have only been established in
the case of 1 + 1 dimensional systems, or spherically symmetric solutions of higher dimensional
problems, in which case the L2 - Sobolev norms can be replaced by bounded variation (BV)
type norms5. A particularly important example of this type is the critical BV well-posedness
result established by Christodoulou for spherically symmetric solutions of the Einstein equations
coupled with a scalar field, see [15]. The result played a crucial role in his famous Cosmic
Censorship results for the same model, see [16]. As well known, unfortunately, the BV-norms arecompletely inadequate in higher dimensions; the only norms which can propagate the regularity
properties of the data are necessarily L2 based.
The quest for optimal well-posedness in higher dimensions has been one of the major themes
in non-linear hyperbolic PDE’s in the last twenty years. Major advances have been made in theparticular case of semi-linear wave equations. In the case of geometric wave equations such as
Wave Maps and Yang-Mills, which possess a well understood null structure, well-posedness holds
true for all exponents larger than the corresponding critical exponent. For example, in the caseof Wave Maps defined from the Minkowski space Rn+1 to a complete Riemannian manifold, thecritical scaling exponents is sc = n/2 and well-posedness is known to hold all the way down tosc for all dimensions n ≥ 2. This critical well-posedness result, for s = n/2, plays a fundamental
1The original proof in [11] relied on representation formulas, following an approach pioneered by Sobolev,
see [67].
2Based only on energy estimates and classical Sobolev inequalities.
3The original proof in [21] [29] actually requires one more derivative for the uniqueness. The fact thatuniqueness holds at the same level of regularity has been obtained in [58].
4That is any past directed, in-extendable causal curve in M intersects Σ .
0
5Recall that the entire theory of shock waves for 1+1 systems of conservation laws is based on BV norms,
which are critical with respect to the scaling of the equations. Note also that these BV norms are not conserved
and that Glimm’s famous existence result [28] can be interpreted as a global well posedness result for initial datawith small BV norms.

1.1. GENERAL INTRODUCTION

5role in the recent, large data, global results of [77], [69], [70] and [44] for 2 + 1 dimensional
wave maps.
The role played by critical exponents for quasi-linear equations is much less understood. The
first well posedness results, on any (higher dimensional) quasilinear hyperbolic system, which go
beyond the classical Sobolev exponents, obtained in [2], [3], and [79], [78] and [35], do not take
into account the specific (null) structure of the equations. Yet the presence of such structure
was crucial in the derivation of the optimal results mentioned above, for geometric semilinear
equations. In the case of the Einstein equations it is not at all clear what such structure shouldbe, if there is one at all. Indeed, the only specific structural condition, known for (EVE),
discovered in [46] under the name of the weak null condition, is not at all adequate for improved
well posedness results, see remark 1.3. It is known however, see [45], that without such astructure one cannot have well posedeness for exponents6 s ≤ 2. Yet (EVE) are of fundamental
importance and as such it is not unreasonable to expect that such a structure must exist.
Even assuming such a structure, a result of well-posedness for the Einstein equations at,or near, the critical regularity sc = 3/2 is not only completely out of reach but may in fact bewrong. This is due to the presence of a different scaling connected to the geometry of boundaries
of causal domains. It is because of this more subtle scaling that we need at least L2 -bounds forthe curvature to derive a lower bound on the radius of injectivity of null hypersurfaces and thus
control their local regularity properties. This imposes a crucial obstacle to well posedness belows = 2. Indeed, as we will show in the next subsection, any such result would require, crucially,bilinear and even trilinear estimates for solutions to wave equations of the form g φ = F .
Such estimates, however, depend on Fourier integral representations, with a phase function u
which solves the eikonal equation gαβ ∂α u∂β u = 0. Thus the much needed bilinear estimates
depend, ultimately, on the regularity properties of the level hypersurfaces of the phase u whichare, of course, null. The catastrophic breakdown of the regularity of these null hypersurfaces,in the absence of a lower bound for the injectivity radius, would make these Fourier integral
representations entirely useless.
These considerations lead one to conclude that, the following conjecture, proposed in [31],is most probably sharp in so far as the minimal number of derivatives in L2 is concerned:
Conjecture[Bounded L2 Curvature Conjecture (BCC)] The Einstein- vacuum equations admit local Cauchy developments for initial data sets (Σ0 , g, k) with locally finite L2 curvature andlocally finite L2 norm of the first covariant derivatives of k 7.
Remark 1.2. It is important to emphasize here that the conjecture should be primarily interpreted as a continuation argument for the Einstein equations; that is the space-time constructedby evolution from smooth data can be smoothly continued, together with a time foliation, as
long as the curvature of the foliation and the first covariant derivatives of its second fundamental form remain L2 - bounded on the leaves of the foliation. In fact the conjecture implies thebreak-down criterion previously obtained in [37] and improved in [56], [82]. According to thatcriterion a vacuum space-time, endowed with a constant mean curvature (CMC) foliation Σt ,6Note that the dimension here is n = 3.
7As we shall see, from the precise theorem stated below, other weaker conditions, such as a lower bound on

the volume radius, are needed.

6

1. INTRODUCTION

can be extended, together with the foliation, as long as the L1t L∞ (Σt ) norm of the deformation
tensor of the future unit normal to the foliation remains bounded. It is straightforward to see,2
by standard energy estimates, that this condition implies bounds for the L∞t L (Σt ) norm of thespace-time curvature from which one can derive bounds for the induced curvature tensor R andthe first derivatives of the second fundamental form k. Thus, if we can ensure that the time ofexistence of a space-time foliated by Σt depends only on the L2 norms of R and first covariant
derivatives of k, we can extend the space-time indefinitely.
1.1.3. Brief history. The conjecture has its roots in the remarkable developments of thelast twenty years centered around the issue of optimal well-posedness for semilinear wave equations. The case of the Einstein equations turns out to be a lot more complicated due to thequasilinear character of the equations. To make the discussion more tangible it is worthwhile
to recall the form of the Einstein vacuum equations in the wave gauge. Assuming given coordinates xα , verifying g xα = 0, the metric coefficients gαβ = g(∂α , ∂β ), with respect to thesecoordinates, verify the system of quasilinear wave equations,g µν ∂µ ∂ν gαβ = Fαβ (g, ∂g)(1.3)

where Fαβ are quadratic functions of ∂g, i.e. the derivatives of g with respect to the coordinates
xα . We compare (1.3) with the semilinear wave equation,
φ = F (φ, ∂φ)(1.4)

with F quadratic in ∂φ. Using standard energy estimates, one can prove an estimate, roughly,of the form: Z t
kφ(t)ks . kφ(0)ks exp Csk∂φ(τ )kL∞ dτ .
0

The classical exponent s > 3/2 + 1 arises simply from the Sobolev embedding of H r , r > 3/2,
into L∞ . To go beyond the classical exponent, see [59], one has to replace Sobolev inequalities
with Strichartz estimates of, roughly, the following type,


Z t
1/2
Z t
2
. C k∂φ(0)kH 1+ε +kφ(τ )kH 1+εk∂φ(τ )kL∞ dτ00where ε > 0 can be chosen arbitrarily small. This leads to a gain of 1/2 derivatives, i.e. we canprove well-posedness for equations of type (1.4) for any exponent s > 2.
The same type of improvement in the case of quasilinear equations requires a highly nontrivial extension of such estimates for wave operators with non-smooth coefficients. The first
improved regularity results for quasilinear wave equations of the type,g µν (φ)∂µ ∂ν φ = F (φ, ∂φ)(1.5)

with g µν (φ) a non-linear perturbation of the Minkowski metric mµν , are due to [2], [3], and[79], [78] and [35]. The best known results for equations of type (1.3) were obtained in [36]and [66]. According to them one can lower the Sobolev exponent s > 5/2 in Theorem 1.1 tos > 2. It turns out, see [45], that these results are sharp in the general class of quasilinear wave
equations of type (1.3). To do better one needs to take into account the special structure of the1.1. GENERAL INTRODUCTION

7

Einstein equations and rely on a class of estimates which go beyond Strichartz, namely the socalled bilinear estimates8.
In the case of semilinear wave equations, such as Wave Maps, Maxwell-Klein-Gordon andYang-Mills, the first results which make use of bilinear estimates go back to [32], [33], [34]. In
the particular case of the Maxwell-Klein-Gordon and Yang-Mills equation the main observation
was that, after the choice of a special gauge (Coulomb gauge), the most dangerous nonlinear
terms exhibit a special, null structure for which one can apply the bilinear estimates derived in[32]. With the help of these estimates one was able to derive a well posedness result, in the flat
Minkowski space R1+3 , for the exponent s = sc + 1/2 = 1, where sc = 1/2 is the critical Sobolev
exponent in that case9.
To carry out a similar program in the case of the Einstein equations one would need, at thevery least, the following crucial ingredients:
A. Provide a coordinate condition, relative to which the Einstein vacuum equations verifies
an appropriate version of the null condition.
B. Provide an appropriate geometric framework for deriving bilinear estimates for the null
quadratic terms appearing in the previous step.
C. Construct an effective progressive wave representation ΦF (parametrix) for solutions
to the scalar linear wave equation g φ = F , derive appropriate bounds for both theparametrix and the corresponding error term E = F − g ΦF and use them to derive
the desired bilinear estimates.
Note that the last two steps are to be implemented using only hypothetical L2 bounds forthe space-time curvature tensor, consistent with the conjectured result.
To start with, it is not at all clear what should be the correct coordinate condition, or even
if there is one for that matter.
Remark 1.3. As mentioned above, the only known structural condition related to the classicalnull condition, called the weak null condition, tied to wave coordinates, fails the test. Indeed, thefollowing simple system in the Minkowski space R3+1 verifies the weak null condition and yet,according to [45], it is ill posed for s = 2.
φ = 0,ψ = φ · ∆φ.

Coordinate conditions, such as spatial harmonic10, also do not seem to work.
We rely instead on a Coulomb type condition, for orthonormal frames, adapted to a maximal
foliation. Such a gauge condition appears naturally if we adopt a Yang-Mills description of theEinstein field equations using Cartan’s formalism of moving frames11, see [14]. It is important
to note nevertheless that it is not all a priori clear that such a choice would do the job. Indeed,
8Note that no such result, i.e. well-posedness for s = 2, is presently known for either scalar equations of theform (1.5) or systems of the form (1.3).
9This corresponds precisely to the s = 2 exponent in the case of the Einstein-vacuum equations
10Maximal foliation together with spatial harmonic coordinates on the leaves of the foliation would be thecoordinate condition closest in spirit to the Coulomb gauge.
11We would like to thank L. Anderson for pointing out to us the possibility of using such a formalism as apotential bridge to [33] .

8

1. INTRODUCTION

the null form nature of the Yang-Mills equations in the Coulomb gauge is only revealed once wecommute the resulting equations with the projection operator P on the divergence free vectorfields. Such an operation is natural in that case, since P commutes with the flat d’Alembertian.
In the case of the Einstein equations, however, the corresponding commutator term [g , P] generates12 a whole host of new terms and it is quite a miracle that they can all be treated by anextended version of bilinear estimates. At an even more fundamental level, the flat Yang-Mills
equations possess natural energy estimates based on the time symmetry of the Minkowski space.
There are no such timelike Killing vectorfield in curved space. We have to rely instead on the
future unit normal to the maximal foliation Σt whose deformation tensor is non-trivial. This
leads to another class of nonlinear terms which have to be treated by a novel trilinear estimate.
We will make more comments concerning the implementations of all three crucial ingredientslater on, in the section 1.2.4.
Remark 1.4. In addition to the ingredients mentioned above, we also need a mechanism of
reducing the proof of the conjecture to small data, in an appropriate sense. Indeed, even in theflat case, the Coulomb gauge condition cannot be globally imposed for large data. In fact [34]relied on a cumbersome technical device based on local Coulomb gauges, defined on domain of
dependence of small balls. Here we rely instead on a variant of the gluing construction of [18],[19], see section 1.2.3.
1.2. Statement of the main results
1.2.1. Maximal foliations. In this section, we recall some well-known facts about maximal foliations (see for example the introduction in [17]). We assume the space-time (M, g) to befoliated by the level surfaces Σt of a time function t. Let T denote the unit normal to Σt , and let
k the the second fundamental form of Σt , i.e. kab = −g(Da T, eb ), where ea , a = 1, 2, 3 denotes
an arbitrary frame on Σt and Da T = Dea T . We assume that the Σt foliation is maximal, i.e.
we have:
trg k = 0

(1.6)

where g is the induced metric on Σt . The constraint equations on Σt for a maximal foliation are
given by:∇a kab = 0,

(1.7)

where ∇ denotes the induces covariant derivative on Σt , and
R = |k|2 .

(1.8)

Also, we denote by n the lapse of the t-foliation, i.e. n−2 = −g(Dt, Dt). n satisfies the following
elliptic equation on Σt :∆n = n|k|2 .

(1.9)

Finally, we recall the structure equations of the maximal foliation:∇0 kab = Ra 0 b 0 − n−1 ∇a ∇b n − kac kb c ,(1.10)∇a kbc − ∇b kac = Rc0ab(1.11)

12Note also that additional error terms are generated by projecting the equations on the components of theframe.

1.2. STATEMENT OF THE MAIN RESULTS

9and:Rab − kac k c b = Ra0b0 .

(1.12)

1.2.2. Main Theorem. We recall below the definition of the volume radius on a general
Riemannian manifold M .
Definition 1.5. Let Br (p) denote the geodesic ball of center p and radius r. The volume
radius rvol (p, r) at a point p ∈ M and scales ≤ r is defined by|Br0 (p)|
,
r ≤r
r3

rvol (p, r) = inf
0with |Br | the volume of Br relative to the metric on M . The volume radius rvol (M, r) of M onscales ≤ r is the infimum of rvol (p, r) over all points p ∈ M .
Our main result is the following:
Theorem 1.6 (Main theorem). Let (M, g) an asymptotically flat solution to the Einsteinvacuum equations (1.1) together with a maximal foliation by space-like hypersurfaces Σt defined
as level hypersurfaces of a time function t. Assume that the initial slice (Σ0 , g, k) is such that
the Ricci curvature Ric ∈ L2 (Σ0 ), ∇k ∈ L2 (Σ0 ), and Σ0 has a strictly positive volume radius on
scales ≤ 1, i.e. rvol (Σ0 , 1) > 0. Then, there exists a time
T = T (kRickL2 (Σ0 ) , k∇kkL2 (Σ0 ) , rvol (Σ0 , 1)) > 0
and a constant
C = C(kRickL2 (Σ0 ) , k∇kkL2 (Σ0 ) , rvol (Σ0 , 1)) > 0
such that the following control holds on 0 ≤ t ≤ T :kRkL∞[0,T ]

L2 (Σt ) ≤ C, k∇kkL∞
L2 (Σt ) ≤ C and[0,T ]

inf rvol (Σt , 1) ≥

0≤t≤T

1
.
C

Remark 1.7. Since the core of the main theorem is local in nature we do not need to bevery precise here with our asymptotic flatness assumption. We may thus assume the existenceof a coordinate system at infinity, relative to which the metric has two derivatives bounded in
L2 , with appropriate asymptotic decay. Note that such bounds could be deduced from weighted
L2 bounds assumptions for Ric and ∇k.
Remark 1.8. Note that the dependence on kRickL2 (Σ0 ) , k∇kkL2 (Σ0 ) in the main theorem
can be replaced by dependence on kRkL2 (Σ0 ) where R denotes the space-time curvature tensor.
Indeed this follows from the following well known L2 estimate (see section 8 in [37]).
Z1|∇k|2 + |k|4 ≤4Σ0

and the Gauss equation relating Ric to R.

Z
Σ0|R|2 .

(1.13)

10

1. INTRODUCTION

1.2.3. Reduction to small initial data. We first need an appropriate covering of Σ0 byharmonic coordinates. This is obtained using the following general result based on CheegerGromov convergence of Riemannian manifolds.
Theorem 1.9 ([1] or Theorem 5.4 in [57]). Given c1 > 0, c2 > 0, c3 > 0, there exists r0 > 0such that any 3-dimensional, complete, Riemannian manifold (M, g) with kRickL2 (M ) ≤ c1 andvolume radius at scales ≤ 1 bounded from below by c2 , i.e. rvol (M, 1) ≥ c2 , verifies the following
property:
Every geodesic ball Br (p) with p ∈ M and r ≤ r0 admits a system of harmonic coordinates
x = (x1 , x2 , x3 ) relative to which we have
(1 + c3 )−1 δij ≤ gij ≤ (1 + c3 )δij ,andZ|∂ 2 gij |2r(1.14)

p|g|dx ≤ c3 .

(1.15)Br (p)

We consider ε > 0 which will be chosen as a small universal constant. We apply theorem
1.9 to the Riemannian manifold Σ0 . Then, there exists a constant:r0 = r0 (kRickL2 (Σ0 ) , k∇kkL2 (Σ0 ) , rvol (Σ0 , 1), ε) > 0
such that every geodesic ball Br (p) with p ∈ Σ0 and r ≤ r0 admits a system of harmonic
coordinates x = (x1 , x2 , x3 ) relative to which we have:(1 + ε)−1 δij ≤ gij ≤ (1 + ε)δij ,andZ|∂ 2 gij |2rp|g|dx ≤ ε.

Br (p)

Now, by the asymptotic flatness of Σ0 , the complement of its end can be covered by theunion of a finite number of geodesic balls of radius r0 , where the number N0 of geodesic balls
required only depends on r0 . In particular, it is therefore enough to obtain the control of R, k
and rvol (Σt , 1) of Theorem 1.6 when one restricts to the domain of dependence of one such ball.
Let us denote this ball by Br0 . Next, we rescale the metric of this geodesic ball by:
!
ε2
ε2
,
, r0 ε > 0.
gλ (t, x) = g(λt, λx), λ = minkRk2L2 (Br ) k∇kk2L2 (Br )
0

13

0

Rλ , kλ and Brλ0 be the rescaled versions of R, k and Br0 . Then, in view of our choice forLet
λ, we have:kRλ kL2 (Brλ ) =0k∇kλ kL2 (Brλ ) =
0

and
2

k∂ gλ kL2 (Brλ ) =
0

√

√
√

λkRkL2 (Br0 ) ≤ ε,λk∇kkL2 (Br0 ) ≤ ε,r2

λk∂ gkL2 (Br0 ) ≤

λε
≤ ε.
r0

13Since in what follows there is no danger to confuse the Ricci curvature Ric with the scalar curvature R we

use the short hand R to denote the full curvature tensor Ric.

1.2. STATEMENT OF THE MAIN RESULTS

11

Note that Brλ0 is the rescaled version of Br0 . Thus, it is a geodesic ball for gλ of radius rλ0 ≥ 1ε ≥ 1.
Now, considering gλ on 0 ≤ t ≤ 1 is equivalent to considering g on 0 ≤ t ≤ λ. Thus, since r0 ,N0 and λ depend only on kRkL2 (Σ0 ) , k∇kkL2 (Σ0 ) , rvol (Σ0 , 1) and ε, Theorem 1.6 is equivalent
to the following theorem:
Theorem 1.10 (Main theorem, version 2). Let (M, g) an asymptotically flat solution to the
Einstein vacuum equations (1.1) together with a maximal foliation by space-like hypersurfaces
Σt defined as level hypersurfaces of a time function t. Let B a geodesic ball of radius one in Σ0 ,and let D its domain of dependence. Assume that the initial slice (Σ0 , g, k) is such that:1
kRkL2 (B) ≤ ε, k∇kkL2 (B) ≤ ε and rvol (B, 1) ≥ .
2
Let Bt = D ∩ Σt the slice of D at time t. Then, there exists a small universal constant ε0 > 0such that if 0 < ε < ε0 , then the following control holds on 0 ≤ t ≤ 1:
1
kRkL∞ L2 (Bt ) . ε, k∇kkL∞ L2 (Bt ) . ε and inf rvol (Bt , 1) ≥ .
[0,1]
[0,1]
0≤t≤1
4
Notation: In the statement of Theorem 1.10, and in the rest of the paper, the notation f1 . f2
for two real positive scalars f1 , f2 means that there exists a universal constant C > 0 such that:f1 ≤ Cf2 .

Theorem 1.10 is not yet in suitable form for our proof since some of our constructions will
be global in space and may not be carried out on a subregion B of Σ0 . Thus, we glue a smooth
asymptotically flat solution of the constraint equations (1.2) outside of B, where the gluing takes
place in an annulus just outside B. This can be achieved using the construction in [18], [19].
We finally get an asymptotically flat solution to the constraint equations, defined everywhereon Σ0 , which agrees with our original data set (Σ0 , g, k) inside B. We still denote this data setby (Σ0 , g, k). It satisfies the bounds:
1
kRkL2 (Σ0 ) ≤ 2ε, k∇kkL2 (Σ0 ) ≤ 2ε and rvol (Σ0 , 1) ≥ .
4
Remark 1.11. Notice that the gluing process in [18], [19] requires the kernel of a certain
linearized operator to be trivial. This is achieved by conveniently choosing the asymptoticallyflat solution to (1.2) that is glued outside of B to our original data set. This choice is always
possible since the metrics for which the kernel is nontrivial are non generic (see [6]).
Remark 1.12. Assuming only L2 bounds on R and ∇k is not enough to carry out theconstruction in [18], [19]. These gluing constructions should allow lower regularity, in fact evenunder H (3/2)+ assumptions, but it is not particularly essential for our purposes and we leave thisissue to further investigation.
Remark 1.13. Since kkk2L4 (Σ0 ) ≤ kRickL2 we deduce that kkkL2 (B) . ε1/2 on the geodesicball B of radius one. Furthermore, asymptotic flatness is compatible with a decay of |x|−2 atinfinity, and in particular with k in L2 (Σ0 ). So we may assume that the gluing process is suchthat the resulting k satisfies:
kkkL2 (Σ0 ) . ε.

12

1. INTRODUCTION

Finally, we have reduced Theorem 1.6 to the case of a small initial data set:
Theorem 1.14 (Main theorem, version 3). Let (M, g) an asymptotically flat solution to the
Einstein vacuum equations (1.1) together with a maximal foliation by space-like hypersurfaces
Σt defined as level hypersurfaces of a time function t. Assume that the initial slice (Σ0 , g, k) issuch that:1
kRkL2 (Σ0 ) ≤ ε, kkkL2 (Σ0 ) + k∇kkL2 (Σ0 ) ≤ ε and rvol (Σ0 , 1) ≥ .
2Then, there exists a small universal constant ε0 > 0 such that if 0 < ε < ε0 , then the following
hold on 0 ≤ t ≤ 1:
1
kRkL∞ L2 (Σt ) . ε, kkkL∞ L2 (Σt ) + k∇kkL∞ L2 (Σt ) . ε and inf rvol (Σt , 1) ≥ .
[0,1]
[0,1]
[0,1]
0≤t≤1
4
The rest of this paper is devoted to the proof of Theorem 1.14.
1.2.4. Strategy of the proof. The proof of Theorem 1.14 consists in three main steps.
Step A (Yang-Mills formalism) We first cast the Einstein-vacuum equations within a YangMills formalism. This relies on the Cartan formalism of moving frames. The idea is to give up
on a choice of coordinates and express instead the Einstein vacuum equations in terms of theconnection 1-forms associated to moving orthonormal frames, i.e. vectorfields eα , which verify,g(eα , eβ ) = mαβ = diag(−1, 1, 1, 1).
The connection 1-forms (they are to be interpreted as 1-forms with respect to the external indexµ with values in the Lie algebra of SO(3, 1)), defined by the formulas,(Aµ )αβ = g(Dµ eβ , eα )(1.16)Dµ Fµν + [Aµ , Fµν ] = 0(1.17)

verify the equations,where, denoting (Fµν )αβ := Rαβµν ,(Fµν )αβ = Dµ Aν − Dν Aµ − [Aµ , Aν ] αβ .

(1.18)

In other words we can interpret the curvature tensor as the curvature of the so(3, 1)-valued
connection 1-form A. Note also that the covariant derivatives are taken only with respect tothe external indices µ, ν and do not affect the internal indices α, β. We can rewrite (1.17) in
the form,
g Aν − Dν (Dµ Aµ ) = Jν (A, DA)(1.19)

where,
Jν = Dµ ([Aµ , Aν ]) − [Aµ , Fµν ].
Observe that the equations (1.17)-(1.18) look just like the Yang-Mills equations on a fixed
Lorentzian manifold (M, g) except, of course, that in our case A and g are not independent
but connected rather by (1.16), reflecting the quasilinear structure of the Einstein equations.
Just as in the case of [32], which establishes the well-posedness of the Yang-Mills equation in
Minkowski space in the energy norm (i.e. s = 1), we rely in an essential manner on a Coulomb
type gauge condition. More precisely, we take e0 to be the future unit normal to the Σt foliation

1.2. STATEMENT OF THE MAIN RESULTS

13

and choose e1 , e2 , e3 an orthonormal basis to Σt , in such a way that we have, essentially (seeprecise discussion in section 2.1.2), div A = ∇i Ai = 0, where A is the spatial component of A.
It turns out that A0 satisfies an elliptic equation while each component Ai = g(A, ei ), i = 1, 2, 3
verifies an equation of the form,
g Ai = −∂i (∂0 A0 ) + Aj ∂j Ai + Aj ∂i Aj + l.o.t.

(1.20)

with l.o.t. denoting nonlinear terms which can be treated by more elementary techniques (including non sharp Strichartz estimates).
Step B (Bilinear and trilinear estimates) To eliminate ∂i (∂0 A0 ) in (1.20), we need toproject (1.20) onto divergence free vectorfields with the help of a non-local operator which wedenote by P. In the case of the flat Yang-Mills equations, treated in [32], this leads to an
equation of the form,
Ai = P(Aj ∂j Ai ) + P(Aj ∂i Aj ) + l.o.t.
where both terms on the right can be handled by bilinear estimates. In our case we encounterhowever three fundamental differences with the flat situation of [32].
• To start with the operator P does not commute with g . It turns out, fortunately, thatthe terms generated by commutation can still be estimated by a larger class of bilinear
estimates which includes contractions with the curvature tensor, see section 2.2.5.
• All energy estimates used in [32] are based on the standard timelike Killing vectorfield∂t . In our case the corresponding vectorfield e0 = T ( the future unit normal to Σt )is not Killing. This leads to another class of trilinear error terms which we discuss insections 2.5 and 2.2.5.
• The main difference with [32] is that we now need bilinear and trilinear estimates forsolutions of wave equations on background metrics which possess only limited regularity.
This last item is a major problem, both conceptually and technically. On the conceptual side weneed to rely on a more geometric proof of bilinear estimates based on a plane wave representation
formula14 for solutions of scalar wave equations,g φ = 0.
The proof of the bilinear estimates rests on the representation formula15
Z Z ∞ω
φf (t, x) =eiλ u(t,x) f (λω)λ2 dλdωS2(1.21)

0where f represents schematically the initial data16, and where
equation17,

ω u is a solution of the eikonal

gαβ ∂α ω u ∂β ω u = 0,
14We follow the proof of the bilinear estimates outlined in [38] which differs substantially from that of [32]and is reminiscent of the null frame space strategy used by Tataru in his fundamental paper [80].
15(1.21) actually corresponds to the representation formula for a half-wave. The full representation formula
corresponds to the sum of two half-waves (see section 2.7)
16Here f is in fact at the level of the Fourier transform of the initial data and the norm kλf k 2 3 corresponds,
L (R )
roughly, to the H 1 norm of the data .
17In the flat Minkowski space ω u(t, x) = t ± x · ω.

14

1. INTRODUCTION

with appropriate initial conditions on Σ0 and dω the area element of the standard sphere in R3 .
Remark 1.15. Note that (1.21) is a parametrix for a scalar wave equation. The lack of agood parametrix for a covariant wave equation forces us to develop a strategy based on writing
the main equation in components relative to our frame. This scalarization procedure produces
several terms which are potentially dangerous, and, just like the commutator terms [, P], they
can be controlled by the use of an extended18 class of bilinear estimates.
Step C (Control of the parametrix) To prove the bilinear and trilinear estimates of Step
B, we need in particular to control the parametrix at initial time (i.e. restricted to the initialslice Σ0 )Z Z ∞
ωφf (0, x) =eiλ u(0,x) f (λω)λ2 dλdω(1.22)
S20

and the error term corresponding to (1.21)
Z Z ∞ω
Ef (t, x) = g φf (t, x) = i
eiλ u(t,x) (g ω u)f (λω)λ3 dλdωS2(1.23)

0i.e. φf is an exact solution of g only in flat space in which case g ω u = 0. Note that thescalar g ω u is a crucial invariant quantity19, whose L∞ norm controls the radius of injectivityof the level hypersurfaces of ω u. Without a lower bound on the radius of injectivity of thesenull hypersurfaces the parametrix representation (1.21) is useless.
To make sense of the parametrix and control the error term Ef we need the following four
sub steps:
C1 Make an appropriate choice for the equation satisfied by ω u(0, x) on Σ0 , and control
the geometry of the foliation of Σ0 by the level surfaces of ω u(0, x).
C2 Prove that the parametrix at t = 0 given by (1.22) is bounded in L(L2 (R3 ), L2 (Σ0 ))using the estimates for ω u(0, x) obtained in C1.
C3 Control the geometry of the foliation of M given by the level hypersurfaces of ω u.
C4 Prove that the error term (1.23) satisfies the estimate kEf kL2 (M) ≤ Ckλf kL2 (R3 ) using
the estimates for ω u and g ω u proved in C3.
As mentioned above, to achieve Step C3 and Step C4, we need, at the very least, to control
g ω u in L∞ . This issue was first addressed in the sequence of papers [39]–[41] where an L∞
bound for g ω u was established, depending only on the L2 norm of the curvature flux alongnull hypersurfaces. The proof, which required an interplay between both geometric and analytic
techniques, is absolutely sharp when measured with respect to the natural scaling associated toa null hypersurface20.
To obtain the L2 bound for the Fourier integral operator E defined in (1.23), we need, ofcourse, to go beyond uniform estimates for g ω u. The classical L2 bounds for Fourier integral
18Such as contractions between the Riemann curvature tensor and derivatives of solutions of scalar wave

equations.
19the trace of the null second fundamental form of the level surfaces of ω u, also known as its expansion.
20This is the reason why we cannot expect to control the radius of injectivity of null hypersurfaces with lessthan two derivatives in L2 for the metric. Note also that classically, this requires, at the very least, the control of
R in L∞ .

1.2. STATEMENT OF THE MAIN RESULTS

15

operators of the form (1.23), which is very wasteful in terms of the number of integration byparts which are needed, would require a lot more differentiability of this quantity than we haveavailable. To obtain satisfactory L2 bound both for the parametrix at initial time (1.22) andthe error term (1.23), within such restrictive regularity properties we need, in particular:• In Step C1 and Step C3, a precise control of derivatives of ω u and g ω u with respect
to both ω as well as with respect to various directional derivatives21. To get optimal
control we need, in particular, a very careful construction of the initial condition for ω u
on Σ0 and then sharp space-time estimates of Ricci coefficients, and their derivatives,
associated to the foliation induced by ω u.
• In Step C2 and Step C4, a careful decompositions of the Fourier integral operators(1.22) and (1.23) in both λ and ω, similar to the first and second dyadic decompositionin harmonic analysis, see [68], as well as a third decomposition, which in the caseof (1.23) is done with respect to the space-time variables relying on the geometric
Littlewood-Paley theory developed in [41].
Below, we make further comments on Steps C1-C4:(1) The choice of u(0, x, ω) on Σ0 in Step C1. Let us note that the typical choice u(0, x, ω) =x · ω in a given coordinate system would not work for us, since we don’t have enoughcontrol on the regularity of a given coordinate system within our framework. Instead,
we need to find a geometric definition of u(0, x, ω). A natural choice would beg u = 0 on Σ0
which by a simple computation turns out to be the following simple variant of theminimal surface equation22




∇u ∇u
∇u
=k
,
on Σ0 .
div|∇u||∇u| |∇u|
Unfortunately, this choice does not allow us to have enough control of the derivativesof u in the normal direction to the level surfaces of u. This forces us to look for analternate equation for u:




∇u
1
∇u ∇u
div
=1−
+k
,
on Σ0 .
|∇u||∇u||∇u| |∇u|This equation turns out to be parabolic in the normal direction to the level surfaces ofu, and allows us to obtain the desired regularity in Step C1. On closer inspection it isrelated with the well known mean curvature flow on Σ0 .
(2) How to achieve Step C3. The regularity obtained in Step C1, together with null transport equations tied to the eikonal equation, elliptic systems of Hodge type, see [40],the geometric Littlewood-Paley theory of [41], sharp trace theorems as in [39], andan extensive use of the structure of the Einstein equations, allows us to propagate theregularity on Σ0 to the space-time, thus achieving Step C3.
21Taking into account the different behavior in tangential and transversal directions with respect to the levelsurfaces of ω u.
22In the time symmetric case k = 0, this is exactly the minimal surface equation

16

1. INTRODUCTION(3) The regularity with respect to ω in Steps C1 and C3. The regularity with respect tox for u is clearly limited as a consequence of the fact that we only assume L2 bounds
on R. On the other hand, R is independent of the parameter ω, and one might inferthat u is smooth with respect to ω. Surprisingly, this is not at all the case. Indeed,
the regularity in x obtained for u in Steps C1 and C3 is better in directions tangent to
the level hypersurfaces of u. Now, the ω derivatives of the tangential directions havenon zero normal components. Thus, when differentiating the structure equations withrespect to ω, tangential derivatives to the level surfaces of u are transformed in nontangential derivatives which in turn severely limits the regularity in ω obtained in Steps
C1 and C3.
(4) How to achieve Steps C2 and C4. Let us note that the classical arguments for proving L2
bounds for Fourier operators are based either on a T T ∗ argument, or a T ∗ T argument,
which requires several integration by parts either with respect to x for T ∗ T , of withrespect to (λ, ω) for T T ∗ . Both methods would fail by far within the regularity for u
obtained in Step C1 and Step C3. This forces us to design a method which allows totake advantage both of the regularity in x and ω. This is achieved using in particular
the following ingredients:• geometric integrations by parts taking full advantage of the better regularity properties in directions tangent to the level hypersurfaces of u,• the standard first and second dyadic decomposition in frequency space, with respect to both size and angle (see [68]), an additional decomposition in physicalspace relying on the geometric Littlewood-Paley projections of [41] for Step C4,
as well as another decomposition involving frequency and angle for Step C2.
Even with these precautions, at several places in the proof, one encounters log-divergences
which have to be tackled by ad-hoc techniques taking full advantage of the structureof the Einstein equations.
Remark 1.16 ( (Sharp L4 (M) Strichartz estimate)). We note that the parametrix constructed in Step C needs to be also used to prove a sharp L4 (M) Strichartz estimate. Indeed theproof of several bilinear estimates of Step B reduces to the proof of a sharp L4 (M) Strichartz
estimate for the parametrix (1.21) with λ localized in a dyadic shell. We refer to Proposition
2.30 for a precise statement.
Remark 1.17. Note that apart from the results of Chapter 2 which require the projectionof various tensors on a frame, the computations and estimates in all the other chapters arecovariant.
1.2.5. Structure of the paper. In Chapter 2, we perform Step A and Step B, i.e. werecast the Einstein equations as a quasilinear Yang-Mills type system, we prove bilinear estimates, and we reduce the proof of Theorem 1.14 to Step C and Step D. Next, we perform Step
C on the control of the plane wave parametrix (1.21). More precisely, in Chapter 3, we perform
Step C4 on the control of the error term (1.23). Next, in Chapter 4, we perform Step C3 on thespace-time control of the optical function ω u. Then, in Chapter 5, we perform Step C2 on thecontrol of the parametrix at initial time (1.22). Finally, in Chapter 6, we perform Step C1 on
the control of the optical function ω u on the initial slice Σ0 .

1.2. STATEMENT OF THE MAIN RESULTS

17

Remark 1.18. Chapter 2 summarizes the results obtained in [43]. Chapter 3 summarizes
the results obtained in [76]. Chapter 4 summarizes the results obtained in [75]. Chapter 5
summarizes the results obtained in [74]. And Chapter 6 summarizes the results obtained in [73].
Remark 1.19. The structure of this overview is such that each part motivates the next one.
In particular, Chapter 2 relies on the control of the parametrix (1.21) (see Proposition 2.25 in
Chapter 2), and thus motivates Chapters 3, 4, 5 and 6 which precisely deal with the controlof that parametrix. Next, in order to control the error term (1.23) in Chapter 3, we rely on
estimates for the optical function ω u, which motivates Chapter 4 where these estimates areproved. In turn, the space-time estimates for ω u in Chapter 4 are obtained in particular usingtransport equations, and we need the corresponding control for ω u on the initial slice Σ0 , which
motivates Chapter 6. Finally, in order to control the parametrix at initial time (1.22) in Chapter
5, we rely on estimates for the function ω u on Σ0 , which motivates again Chapter 6.
1.2.6. Conclusion. Though this result falls short of the crucial goal of finding a scale invariant well-posedness criterion in GR, it is clearly optimal in terms of all currently available
ideas and techniques. Within our current understanding, a better result would require enhanced
bilinear estimates, which in turn would rely heavily on parametrices. On the other hand, parametrices are based on solutions to the eikonal equation whose control requires, at least, L2
bounds for the curvature tensor, as can be seen in many instances in our work. Thus, if we areto ultimately find a scale invariant well-posedness criterion, it is clear that an entirely new circle
of ideas is needed.
1.2.7. Acknowledgements. This work would be inconceivable without the extraordinary
advancements made on nonlinear wave equations in the last twenty years in which so many haveparticipated. We would like to single out the contributions of those who have affected this workin a more direct fashion, either through their papers or through relevant discussions, in various
stages of its long gestation. D. Christodoulou’s seminal work [16] on the weak cosmic censorshipconjecture had a direct motivating role on our program, starting with a series of papers between
one of the authors and M. Machedon, in which spacetime bilinear estimates were first introduced
and used to take advantage of the null structure of geometric semilinear equations such as Wave
Maps and Yang-Mills. The works of Bahouri- Chemin [2]-[3] and D.Tataru [78] were the first to
go below the classical Sobolev exponent s = 5/2, for any quasilinear system in higher dimensions.
This was, at the time, a major psychological and technical breakthrough which opened the way
for future developments. Another major breakthrough of the period, with direct influence on
our approach to bilinear estimates in curved spacetimes, is D. Tataru’s work [80] on critical
well posedness for Wave Maps, in which null frame spaces were first introduced. His joint workwith H. Smith [66] which, together with [36] is the first to reach optimal well-posedness without
bilinear estimates, has also influenced our approach on parametrices. The authors would also
like to acknowledge fruitful conversations with L. Anderson, and J. Sterbenz.

CHAPTER 2

Einstein vacuum equations as a Yang-Mills gauge theory
Recall Steps A, B, C and D introduced in section 1.2.4. In this chapter, we perform Step A
and Step B, i.e. we recast the Einstein equations as a quasilinear Yang-Mills type system andwe prove bilinear estimates. This allows us to reduce the proof of Theorem 1.14 to Step C andStep D. Here, we only outline the main ideas, and we refer to [43] for the details.
2.1. Yang-Mills formalism
2.1.1. Cartan formalism. Consider an Einstein vacuum space-time (M, g). We denotethe covariant differentiation by D. Let eα be an orthonormal frame on M, i.e.
g(eα , eβ ) = mαβ = diag(−1, 1, . . . , 1).
Consistent with the Cartan formalism we define the connection 1 form,(A)αβ (X) = g(DX eβ , eα )(2.1)

where X is an arbitrary vectorfield in T (M). Observe that,(A)αβ (X) = −(A)βα (X)i.e. the 1 -form Aµ dxµ takes values in the Lie algebra of SO(1, 3). We separate the internal
indices α, β from the external indices µ according to the following notation.
(Aµ )αβ := (A)αβ (∂µ ) = g(Dµ eβ , eα )(2.2)
The 
Riemann curvature tensor is defined by R(X, Y, U, V ) = g X, DU DV − DV DU −D[U,V ] Ywith X, Y, U, V arbitrary vectorfields in T (M). Thus, taking U = ∂µ , V = ∂ν ,coordinate vector-fields,
R(eα , eβ , ∂µ , ∂ν ) = ∂µ (Aν )αβ − ∂ν (Aµ )αβ + (Aν )α λ (Aµ )λβ − (Aµ )α λ (Aν )λβ .

(2.3)

Defining the Lie bracket,([Aµ , Aν ])αβ = (Aµ )α γ (Aν )γβ − (Aν )α γ (Aµ )γβ(2.4)

we obtain:Rαβµν = ∂µ (Aν )αβ − ∂ν (Aµ )αβ − ([Aµ , Aν ])αβ ,or, since ∂µ (Aν ) − ∂ν (Aµ ) = Dµ Aν − Dν Aµ
(Fµν )αβ = Rαβµν = Dµ Aν − Dν Aµ − [Aµ , Aν ] αβ .

(2.5)

where interpret F is the curvature of the connection A.
The usual covariant derivative of the Riemann curvature tensor can be expressed as follows:
Dσ Rαβµν = (A)Dσ Fµν := Dσ Fµν + [Aσ , Fµν ]
19

(2.6)

20

2. EINSTEIN VACUUM EQUATIONS AS A YANG-MILLS GAUGE THEORY

where we denote by (A)D the covariant derivative on the corresponding vector bundle. Moreprecisely if U = Uµ1 µ2 ...µk is any k-tensor on M with values on the Lie algebra of SO(3, 1),(A)

Dσ U = Dσ U + [Aσ , U].

(2.7)

Remark 2.1. Recall that in (Aµ )αβ , α, β are called the internal indices, while µ are called
the external indices. Now, the internal indices are mostly irrelevant in our work. Thus, fromnow on, we will drop them, except for rare instances where we will need to distinguish between
internal indices of the type ij and internal indices of the type 0i.
The Bianchi identities for Rαβµν take the form(A)

Dσ Fµν + (A)Dµ Fνσ + (A)Dν Fσµ = 0.

(2.8)

As it is well known the Einstein vacuum equations Rαβ = 0 imply Dµ Rαβµν = 0. Thus, in viewof equation (2.6),µ0 = (A)D Fµν = Dµ Fµν + [Aµ , Fµν ](2.9)

or, in view of (2.5) and the vanishing of the Ricci curvature of g,.
Aν − Dν (Dµ Aµ ) = Jν

(2.10)

Jν = Dµ ([Aµ , Aν ]) − [Aµ , Fµν ].

(2.11)

where

Using again the vanishing of the Ricci curvature it is easy to check,Dν Jν = 0.

(2.12)

Finally we recall the general formula of transition between two different orthonormal frames
eα and eeα on M, related by,eeα = Oγα eγwhere mαβ = Oγα Oδβ mγδ , i.e. O is a smooth map from M to the Lorentz group O(3, 1). In
other words, raising and lowering indices with respect to m,
Oαλ Oβλ = δαβ(2.13)e µ )αβ = g(Dµ eeβ , eeα ). Therefore,
Now, (Ae µ )αβ = Oγ Oδ (Aµ )γδ + ∂µ (Oγ ) Oδ mγδ
(A
α β
α
β

(2.14)

2.1.2. Compatible frames. Recall that our space-time is assumed to be foliated by thelevel surfaces Σt of a time function t, which are maximal, i.e. denoting by k the second fundamental form of Σt we have,trg k = 0(2.15)

where g is the induced metric on Σt . Let us choose e(0) = T , the future unit normal to theΣt foliation, and e(i) , i = 1, 2, 3 an orthonormal frame tangent to Σt . We call this a framecompatible with our Σt foliation. We consider the connection coefficients (2.2) with respect2.1. YANG-MILLS FORMALISM

21

to this frame. Thus, in particular, denoting by A0 , respectively Ai , the temporal and spatial
components of Aµ(Ai )0j= (Aj )0i = −kij ,(A0 )0i = −n

−1

∇i n

i, j = 1, 2, 3

i = 1, 2, 3

(2.16)(2.17)

where n denotes the lapse of the t-foliation, i.e. n−2 = −g(Dt, Dt). With this notation we notethat,∇l kij = ∇l (ki )j + kin (Al )j n = ∇l (Ai )0j + kin (Al )j n
where, as before, the notation ∇l (ki )j or ∇l (Ai )0j , is meant to suggest that the covariant differentiation affects only the external index i. Recalling from (1.7) that k verifies the constraint
equations,∇i kij = 0,
we derive,∇i (Ai )0j = ki m (Ai )mj .

(2.18)

Besides the choice of e0 we are still free to make a choice for the spatial elements of theframe e1 , e2 , e3 . In other words we consider frame transformations which keep e0 fixed, i.etransformations of the type,ẽi = Oij ej
with O in the orthogonal group O(3). We now have, according to (2.14),(Ãm )ij = Oik Ojl (Am )kl + ∂m (Oik )Ojl δklor, schematically,Ãm = OAm O−1 + (∂m O)O−1(2.19)

formula in which we understand that only the spatial internal indices are involved. We shall use
this freedom later to exhibit a frame e1 , e2 , e3 such that the corresponding connection A satisfies
the coulomb gauge condition ∇l (Al )ij = 0 (see Lemma 2.6).
2.1.3. Notations. We use greek indices to denote general indices on M which do not referto the particular frame (e0 , e1 , e2 , e3 ). The letters a, b, c, d will be used to denote general indiceson Σt which do not refer to the particular frame (e1 , e2 , e3 ). Finally, the letters i, j, l, m, n will
only denote indices relative to the frame (e1 , e2 , e3 ). Also, recall that D denotes the covariant
derivative on M, while ∇ denotes the induced covariant derivative on Σt . Furthermore, ∂will always refer to the derivative of a scalar quantity relative to one component of the frame(e0 , e1 , e2 , e3 ), while ∂ will always refer to the derivative of a scalar quantity relative to one
component of the the frame (e1 , e2 , e3 ), so that ∂ = (∂0 , ∂). For example, ∂A may be any termof the form ∂i (Aj ), ∂0 (A) may be any term of the form ∂0 (Aj ), ∂(A 0 ) may be any term of theform ∂j (A0 ), and ∂A = (∂A, ∂(A0 )) = ∂0 (A0 ), ∂(A0 ), ∂0 (A), ∂(A) . Note that we use brackets
such as (Aj ) to emphasize that we are dealing with su(3, 1) objects. Often, however, we will
simply drop them.
We introduce the curl operator curl defined for any su(3, 1)-valued triplet (ω1 , ω2 , ω3 ) offunctions on Σt as follows:(curl ω)i =∈i jl ∂j (ωl ),(2.20)

22

2. EINSTEIN VACUUM EQUATIONS AS A YANG-MILLS GAUGE THEORY

where ∈ijl is fully antisymmetric and such that ∈123 = 1. We also introduce the divergence
operator div defined for any su(3, 1)-valued tensor A on Σt as follows:
div A = ∇l (Al ) = ∂ l (Al ) + A2 .

(2.21)

Remark 2.2. The term A2 in (2.21) corresponds to a quadratic expression in components of
A, where the particular indices do not matter. In the rest of this part, we will adopt this schematic
notation for lower order terms (e.g. terms of the type A2 and A3 ) where the particular indices
do not matter.
Finally, A0 and Ai will always be understood as (A0 ) and (Ai ), while (A)α refers
to the tensorial wave equation. Also, ∆A0 will always refer to ∆(A0 ).
Remark 2.3. Since ∂0 and ∂j are not coordinate derivatives, note that the commutators[∂j , ∂0 ] and [∂j , ∂l ] do not vanish. In fart we have, schematically,[∂i , ∂j ]φ = A∂φ and [∂j , ∂0 ]φ = A∂φ,

(2.22)

for any scalar function φ on M.
2.1.4. Main equations for (A0 , A). Using the conventions above one can prove the following proposition.
Proposition 2.4. Consider an orthonormal frame eα compatible with a maximal Σt foliationof the space-time M with connection coefficients Aµ defined by (2.2), their decomposition A =(A0 , A) relative to the same frame eα , and Coulomb- like condition on the frame,
div A = A2 .
In such a frame the Einstein-vacuum equations take the form,∆A0 = A∂A + A∂(A0 ) + A3 ,(2.23)

Ai + ∂i (∂0 A0 ) = Aj ∂j Ai + Aj ∂i Aj + A0 ∂A + A∂(A0 ) + A3 .

(2.24)

Remark 2.5. It is extremely important to our strategy that we have reduced the covariant
wave equation (2.10) to the system of scalar equations (2.23) (2.24) (see remark 1.15).
2.2. Strategy of the proof of the main Theorem
In this section, we discuss the strategy of the proof of the main theorem after reduction tosmall initial data, i.e. Theorem 1.14.
2.2.1. The Uhlenbeck type lemma. In order to exhibit a frame e1 , e2 , e3 such thattogether with e0 = T we obtain a connection A satisfying our Coulomb type gauge on the slice
Σt , we will need the following result in the spirit of the Uhlenbeck lemma [81].
Lemma 2.6. Let (M, g) a 3 dimensional Riemannian asymptotically flat manifold. Let R
denote its curvature tensor and rvol (M, 1) its volume radius on scales ≤ 1. Let Ã a connection
on M corresponding to an orthonormal frame. Assume the following bounds:
kÃkL2 (M ) + k∇ÃkL2 (M ) + kRkL2 (M ) ≤ δ

1
and rvol (M, 1) ≥ ,4

2.2. STRATEGY OF THE PROOF OF THE MAIN THEOREM

23

where δ > 0 is a small enough constant. Assume also that Ã and ∇Ã belong to L2 (M ). Then,
there is another connection A on M satisfying he Coulomb like gauge condition, and such thatkÃkL2 (M ) + k∇ÃkL2 (M ) ≤ δ
Furthermore, if ∇2 Ã belongs to L2 (M ), then ∇2 A belongs to L2 (M ).
The proof of Lemma 2.6 is a straightforward adaptation, in a simpler situation, of [81].
2.2.2. Classical local existence. We rely on the following standard well-posedness result
for the Cauchy problem for the Einstein equations (1.1) in the maximal foliation.
Theorem 2.7 (Well-posedness for the Einstein equation in the maximal foliation). Let(Σ0 , g, k) be asymptotically flat and satisfying the constraint equations (1.2), with R, ∇R, ∇2 R,k, ∇k, ∇2 k and ∇3 k in L2 (Σ0 ), and rvol (Σ0 , 1) > 0. Then, there exists a unique asymptoticallyflat solution (M, g) to the Einstein vacuum equations (1.1) corresponding to this initial data set,
together with a maximal foliation by space-like hypersurfaces Σt defined as level hypersurfaces ofa time function t. Furthermore, there exists a timeT∗ = T∗ (k∇l RkL2 (Σ0 ) , 0 ≤ l ≤ 2, k∇j kkL2 (Σ0 ) , 0 ≤ j ≤ 3, rvol (Σ0 , 1)) > 0
such that the maximal foliation exists for on 0 ≤ t ≤ T∗ with a corresponding control in
2
2
2
3
L∞
[0,T∗ ] L (Σt ) for R, ∇R, ∇ R, k, ∇k, ∇ k and ∇ k.
Theorem 2.7 requires two more derivatives both for R and k with respect to the main
Theorem 1.6. Its proof is standard and relies solely on energy estimates (as opposed to Strichartz
or bilinear estimates). We refer the reader to [17] chapter 10 for a related statement.
Remark 2.8. In the proof of our main theorem the result above will be used only as anextension tool (see steps 1 and 3 below), only for very tiny values of the time interval.
2.2.3. Weakly regular null hypersurfaces. We shall be working with null hyper surfaces
in M verifying a set of reasonable assumptions, described below. These assumptions will beeasily verified by the level hyper surfaces Hu solutions u of the eikonal equation gµν ∂µ ∂ν = 0
discussed in section 2.7. The regularity of the eikonal equation is studied in detail in [75] (seealso Chapter 4).
Definition 2.9. Let H be a null hypersurface with future null normal L verifying g(L, T ) =−1. Let also N = L − T . We denote by ∇/ the induced connection along the 2-surfaces H ∩ Σt .
We say that H is weakly regular provided that,
kDLkL3 (H) + kDN kL3 (H) . 1,

(2.25)and the following Sobolev embedding holds for any scalar function f on H:
kf kL6 (H) . k∇
/ f kL2 (H) + kL(f )kL2 (H) + kf kL2 (H) .

(2.26)

2.2.4. Main bootstrap assumptions. Let M ≥ 1 a large enough constant to be chosen
later in terms only of universal constants. By choosing ε > 0 sufficiently small, we can also
ensure M ε is small enough. From now on, we assume the following bootstrap assumptions holdtrue on a fixed interval [0, T ∗ ], for some 0 < T ∗ ≤ 1. Note that H denotes an arbitrary weakly
regular null hypersurface with future normal L, normalized by the condition g(L, T ) = −1.

24

2. EINSTEIN VACUUM EQUATIONS AS A YANG-MILLS GAUGE THEORY• Bootstrap curvature assumptions
kRkL∞≤ M ε.
2t L (Σt )(2.27)

kR · LkL2 (H) ≤ M ε,

(2.28)

Also,

where R · L denotes any component of R such that at least one index is contractedwith L.
• Bootstrap assumptions for the connection We also assume that there exist A = (A0 , A)verifying our Coulomb type condition on [0, T ∗ ] , such that,kAkL∞+ k∂AkL∞+ kAkL2t L7 (Σt ) ≤ M ε,22t L (Σt )t L (Σt )(2.29)

and:kA0 kL∞+ k∂A0 kL∞+ kA0 kL2t L∞ (Σt ) + k∂A0 kL∞
2
2
3t L (Σt )t L (Σt )t L (Σt )+k∂∂A0 k ∞ 23

Lt L (Σt )≤ M ε.

(2.30)

Remark 2.10. Together with the estimates in [75] (see section 4.4 in that paper, and also
section 4.2.3), the bootstrap assumption (2.27) yields:
kkkL∞+ k∇kkL∞. M ε.
2
2
t L (Σt )t L (Σt )(2.31)

Furthermore, the bootstrap assumption (2.28) together with the estimates in [75] (see section 4.2
in that paper, and also section 4.2.2) yields:
1
inf rvol (Σt , 1) ≥ .
t4
In addition we make the following bilinear estimates assumptions for A and R.:• Bilinear assumptions I. Assume,kAj ∂j AkL2 (M) . M 3 ε2 .

(2.32)(2.33)

Also, let B = (−∆)−1 curl A (see (2.60) and the accompanying explanations). Then,
we have:kAj ∂j (∂B)kL2 (M) . M 3 ε2 ,(2.34)

kR· · j 0 ∂ j BkL2 (M) . M 3 ε2 .

(2.35)

and:Finally, for any weakly regular null hypersurface H and any smooth scalar function φon M,
kkj · ∂ j φkL2 (M) . M 2 ε supk∇/ φkL2 (H) ,(2.36)

kAj ∂j φkL2 (M) . M 2 ε supk∇/ φkL2 (H) ,(2.37)

H

andH

where the supremum is taken over all null hypersurfaces H.

2.2. STRATEGY OF THE PROOF OF THE MAIN THEOREM

25• Bilinear assumptions II. We assume,1k(−∆)− 2 (Qij (A, A))kL2 (M) . M 3 ε2 ,(2.38)

where the bilinear form Qij is given by Qij (φ, ψ) = ∂i φ∂j ψ − ∂j φ∂i ψ. Furthermore, we
also have:1k(−∆)− 2 (∂(Al )∂l A)kL2 (M) . M 3 ε2 .

(2.39)• Non-sharp Strichartz assumption
kAkL2t L7 (Σt ) . M 2 ε.

(2.40)

and, for B = (−∆)−1 curl A, (see (2.60) and the accompanying explanations).
k∂BkL2t L7 (Σt ) . M 2 ε.

(2.41)

Remark 2.11. Note that the Strichartz estimate for kAkL2t L7 (Σt ) is far from being sharp.
Nevertheless, this estimate will be sufficient for the proof as it will only be used to deal withlower order terms.
Finally we also need a trilinear bootstrap assumption. For this we need to introduce theBell Robinson tensor,Qαβγδ = Rα λ γ σ Rβ λ δ σ + ∗ Rα λ γ σ ∗ Rβ λ δ σ• Trilinear bootstrap assumption. We assume the following,Z
Qijγδ k ij eγ0 eδ0 . M 4 ε3 .

(2.42)(2.43)M

Let us conclude this section by remarking that the bootstrap assumptions are verified for a
sufficiently small final value T ∗ .
Proposition 2.12. The above bootstrap assumptions are verified on 0 ≤ t ≤ T ∗ for a
sufficiently small T ∗ > 0.
The only challenge in the proof of Proposition 2.12 is to show the existence of the desiredconnection A using in particular the Uhlenbeck type Lemma 2.6. All other estimates follow
trivially from our initial bounds and the local existence theorem above, for sufficiently small T ∗ .
We refer to Proposition 4.6 in [43].
2.2.5. Proof of the bounded L2 curvature conjecture. In the following two propositions, we state the improvement of our bootstrap assumptions.
Proposition 2.13. Let us assume that all bootstrap assumptions of the previous section holdfor 0 ≤ t ≤ T ∗ . If ε > 0 is sufficiently small, then the following improved estimates hold true on
0 ≤ t ≤ T ∗:3kRkL∞
. ε + M 2 ε 2 + M 3 ε2 ,2t L (Σt )3

kR · LkL2 (H) . ε + M 2 ε 2 + M 3 ε2 ,3kAkL∞
+ k∂Ai kL∞. ε + M 2 ε 2 + M 3 ε2 ,22t L (Σt )t L (Σt )(2.44)(2.45)(2.46)

26

2. EINSTEIN VACUUM EQUATIONS AS A YANG-MILLS GAUGE THEORYkA0 kL∞+ k∂A0 kL∞+ kA0 kL2t L∞ (Σt )2
2t L (Σt )t L (Σt )+k∂A0 kL∞
+ k∂∂A0 k ∞ 23
3t L (Σt )Lt L (Σt )3. ε + M 2 ε 2 + M 3 ε2 ,(2.47)

Proposition 2.14. Let us assume that all bootstrap assumptions of the previous section holdfor 0 ≤ t ≤ T ∗ . If ε > 0 is sufficiently small, then the following improved estimates hold true on
0 ≤ t ≤ T ∗:kAj ∂j AkL2 (M) . M 2 ε2 ,(2.48)

kAj ∂j (∂B)kL2 (M) . M 2 ε2 ,(2.49)

kR· · j 0 ∂ j BkL2 (M) . M 2 ε2 .

(2.50)

andAlso, for any scalar function φ on M, we have:
kkj · ∂ j φkL2 (M) . M ε supk∇/ φkL2 (H) ,(2.51)

kAj ∂j φkL2 (M) . M ε supk∇/ φkL2 (H) ,(2.52)

H

andH

where the supremum is taken over all null hypersurfaces H. Finally, we have:1k(−∆)− 2 (Qij (A, A))kL2 (M) . M 2 ε2 ,1

(2.53)k(−∆)− 2 (∂Al ∂l A)kL2 (M) . M 2 ε2 .

(2.54)

kAkL2t L7 (Σt ) . M ε,

(2.55)k∂BkL2t L7 (Σt ) . M ε.

(2.56)

Qijγδ k ij eγ0 eδ0(2.57)andZM. M 3 ε3 .

The proof of Proposition 2.13 is postponed to section 2.6, while the proof of Proposition 2.14
is postponed to section 2.8. Next, let us show how Proposition 2.13 and Proposition 2.14 imply
our main theorem 1.14. We proceed, by the standard bootstrap method , along the following
steps:
Step 1. We show that all bootstrap assumptions are verified for a sufficiently small final value
T ∗.
Step2. Assuming that all bootstrap assumptions hold for fixed values of 0 < T ∗ ≤ 1 and M
sufficiently large we show that, for ε > 0 sufficiently small, we may improve on theconstant M in our bootstrap assumptions.
Step 3. Using the estimates derived in step 2 we can extend the time of existence T ∗ to T ∗ + δsuch that all the bootstrap assumptions remain true.

2.3. SIMPLE CONSEQUENCES OF THE BOOTSTRAP ASSUMPTIONS

27

Now, Step 1 follows form Proposition 2.12. Step 2 follows Proposition 2.13 and Proposition
2.14. Finally, using the improved estimates of Step 2 at t = T ∗ , and arguing as in the proof of
Proposition 2.12, we obtain Step 3. Thus, the bootstrap assumptions hold on 0 ≤ t ≤ 1 for auniversal constant M . In particular, this yields together with (2.31):
kRkL∞. ε and kkkL∞. ε for all 0 ≤ t ≤ 1.
2
2
t L (Σt )t L (Σt )(2.58)

In view of (2.32), we also obtain the following control on the volume radius:1
inf rvol (Σt , 1) ≥ .
4(2.59)

0≤t≤1

Finally, (2.58) and the control on the volume radius (2.59) yield the conclusion of Theorem 1.14.
Together with the reduction to small initial data performed in section 1.2.3, this concludes theproof of the main Theorem 1.6.
The rest of the chapter deals with the proofs of propositions 2.13 and 2.14. The core of theproofs is to control A, the spatial part of the connection A. As explained in the introduction
we need to project our equation for the spatial components A onto divergence free vectorfields.
This is needed for two reasons, to eliminate the term ∂i (∂0 A0 ) on the left hand side of (2.24)
and to obtain, on the right hand side, terms which exhibit the crucial null structure we need to
implement our proof. Rather than work with the projection P, which is too complicated, we
introduce instead the new variable,
B = (−∆)−1 curl (A)(2.60)

for which we derive a suitable wave equation. Since we have (see Lemma 2.18) A = curl (B)+l.o.t
it suffices to obtain estimates for B which lead us to an improvement of the bootstrap assumption(2.29) on A.
2.3. Simple consequences of the bootstrap assumptions
2.3.1. Sobolev embeddings and elliptic estimates on Σt . The bootstrap assumption(2.27) on R and the estimate for k (2.31) together with the estimates in [75] (see section 4.4 inthat paper) yield the following lapse estimates:kn − 1kL∞ (M) + k∇nkL∞ (M) + k∇2 nkL∞+ k∇2 nkL∞
2
3
t L (Σt )t L (Σt )3+k∇(∂0 n)kL∞+ k∇ nk ∞ 23
3t L (Σt )Lt L (Σt )2+ k∇ (∂0 (n))k ∞ 32

Lt L (Σt )(2.61). M ε,

where ∇ denotes the induced covariant derivative on Σt .
Remark 2.15. Recall from (2.17) that (A0 )0i = −n−1 ∇i n. Thus, the estimates (2.61) forn could in principle be deduced from the bootstrap assumptions (2.30) for A0 . However, noticethat ∇n ∈ L∞ (M) in view of (2.61), while A0 is only in L2t L∞ (Σt ) according to (2.30). This
improvement for the components (A0 )0i of A0 will turn out to be crucial (see remark 2.22). Its
proof is given in section 4.4 of [75] (see also the discussion in section 4.2.3).
Next, we record the following Sobolev embeddings and elliptic estimates on Σt derived under
the assumptions (2.28) and (2.27) in [75] (see sections 3.5 and 4.2 in that paper).

28

2. EINSTEIN VACUUM EQUATIONS AS A YANG-MILLS GAUGE THEORY

Lemma 2.16 (Calculus inequalities on Σt ). Assume that the assumptions (2.28) and (2.27)
hold, and assume that the volume radius at scales ≤ 1 on Σ0 is bounded from below by a universal
constant. Then, the Sobolev embedding on Σt holds for any tensor F
kF kL6 (Σt ) . k∇F kL2 (Σt ) ,(2.62)

1

Also, we define the operator (−∆)− 2 acting on tensors on Σt as:
Z +∞
3
1
− 21

(−∆) F =τ − 4 U (τ )F dτ,1Γ 4 0
where Γ is the Gamma function, and where U (τ )F is defined using the heat flow on Σt :(∂τ − ∆)U (τ )F = 0, U (0)F = F.
We have the following Bochner estimates:
1k∇(−∆)− 2 kL(L2 (Σt )) . 1 and k∇2 (−∆)−1 kL(L2 (Σt )) . 1,

(2.63)

where L(L2 (Σt )) denotes the set of bounded linear operators on L2 (Σt ). (2.63) together with the
Sobolev embedding (2.62) yields:
1k(−∆)− 2 F kL2 (Σt ) . kF k 65

L (Σt ).

(2.64)

2.3.2. Elliptic estimates for B. Here we record simple estimates for B, based the bootstrap assumptions (2.29) (2.30) for A and A0 and standard elliptic estimates such as the Bochnerand Sobolev inequalities on Σt , see (2.63) and (2.64).
Proposition 2.17. Let Bi = (−∆)−1 (curl (A)i ). Then, we have, for each component of B:k∂BkL∞+ k∂ 2 BkL∞
+ k∂(∂0 B)kL∞. M ε.
2
2
2
t L (Σt )t L (Σt )t L (Σt )(2.65)

2.3.3. Decomposition for A. Recall that B = (−∆)−1 (curl (A)). We indicate below how
to recover A from B:
Lemma 2.18. We have the following estimate:A = curl (B) + E
where E satisfies:k∂EkL∞+ k∂ 2 Ek ∞ 23
3t L (Σt )Lt L (Σt )+ kEkL2t L∞ (Σt ) . M 2 ε2 .

Proof. We have, symbolically,A = (−∆)−1 curl (curl (A) + (−∆)−1 (A∂A + A3 ).
from which,A = curl (B) − (−∆)−1 [∆, curl ]B + (−∆)−1 (A∂A + A3 )The rest of the proof uses elliptic estimates on Σt , the bootstrap assumptions for A and R, andthe bootstrap assumption (2.40). We refer to [43] for the details.
2.4. ESTIMATES FOR B

29

2.4. Estimates for B
We outline the proof of two important propositions concerning estimates for curl A andB, with B = ∆−1 curl (A). The proofs makes use of the special structure of various bilinear
expressions and thus is based not only on the bootstrap assumptions for A0 , A, k and R but
also some of our bilinear bootstrap assumptions.
We record first the following straightforward commutation lemma, see [43].
Lemma 2.19. Let φ a so(3, 1) scalar function on M. We have, schematically,∂j (φ) − (∂j (φ)) = 2(Aλ )j µ ∂λ ∂µ φ + ∂0 (A0 )∂φ + A2 ∂φ.

(2.66)We also have:[, ∆]φ = −4k ab ∇a ∇b (∂0 φ) + 4n−1 ∇b n∇b (∂0 (∂0 φ)) − 2∇0 k ab ∇a ∇b φ
+ F

(1)

2

∂ φ+F

(2)

(2.67)∂φ,F (1) = ∂A0 + A2 ,F (2) = ∂∂A0 + A∂A + A3 ,where ∇a and ∇b denote induced covariant derivatives on Σt applied to the scalars φ, ∂0 φ and∂0 (∂0 φ).
The estimates for curl A and B are given by the following propositions.
Proposition 2.20. We have3
X

1k(−∆)− 2 (curl Ai )kL2 (M) . M 2 ε2 .

(2.68)

i=1

Proposition 2.21 (Estimates for B). The components Bi = (−∆)−1 (curl (A)i ) verify thefollowing estimate,
3XkBi kL2 (M) + k∂ Bi kL2 (M) . M 2 ε2 .

(2.69)

i=1

We also have,3Xk∂0 ∂0 Bi kL2 (M) . M ε.

(2.70)

i=1

The proof of Proposition 2.20 and Proposition 2.21 are similar in spirit. We give below a
short outline of the proof of Proposition 2.21 which is slightly more difficult.
Proof. In what follows we outline the main steps in the proof of space-time estimates(2.68), (2.69) for B and ∂02 B. We have:(Bi ) = [, (−∆)−1 ](curl (A)i ) + (−∆)−1 ((curl (A)i ))= −(−∆)−1 [, ∆](−∆)−1 (curl (A)i ) + (−∆)−1 ((curl (A)i ))= −(−∆)−1 [, ∆](Bi ) + (−∆)−1 ((curl (A)i )).

30

2. EINSTEIN VACUUM EQUATIONS AS A YANG-MILLS GAUGE THEORY

Thus, using the L2 boundedness of ∂(−∆)1/2 and result of proposition 2.20, we obtain:1k∂(Bi )kL2 (M) . k(−∆)− 2 [, ∆](Bi )kL2 (M) + M 3 ε2 ,It remains to estimate k(−∆)
to write,− 12(2.71)[, ∆](Bi )kL2 (M) . We rely on the commutator formula (2.67)[, ∆]B = −4k ab ∇a ∇b (∂0 B) + 4n−1 ∇b n∇b (∂0 (∂0 B)) − 2∇0 k ab ∇a ∇b B
(1)

F

(1)

2

+ F ∂ B+F
= ∂(A0 ) + A2 ,(2)

(2.72)∂B

F (2) = ∂∂(A0 ) + A∂A + A3 .
with B any component (Bl ), l = 1, 2, 3. It is easy to check that,kF (1) kL∞+ kF (2) k ∞ 23
3t L (Σt )Lt L (Σt ). M ε,

We write,1k(−∆)− 2 [, ∆](Bl )kL2 (M) . N1 + N2 + N3 + N4

(2.73)

1

N1 = k(−∆)− 2 [k ab ∇a ∇b (∂0 (Bl ))]kL2 (M)1N2 = k(−∆)− 2 [n−1 ∇b n∇b (∂0 (∂0 (Bl )))]kL2 (M)1N3 = k(−∆)− 2 [∇0 k ab ∇a ∇b (Bl )]kL2 (M)N4 = kF (1) kL∞k∂ 2 (Bl )kL2 (M) + kF (2) k ∞ 32
3t L (Σt )Lt L (Σt )k∂(Bl )kL∞
6t L (Σt )Using the estimates (2.65) we easily infer that
N4 . M ε 2 .

(2.74)

To estimate N1 we proceed as follows, using the constraint equations (1.7) for k,k ab ∇a ∇b (∂0 (Bl )) = ∇a [k ab ∇b (∂0 (Bl ))]Together with the Bochner inequality on Σt (2.63) and the bilinear assumption (2.34) , weobtain:N1 . kk ab ∂b (∂0 (Bl ))]kL2 (M) . M 3 ε2 .

(2.75)

To estimate N2 we write,n−1 ∇b n∇b (∂0 (∂0 (Bl ))) = ∇b [n−1 ∇b n∂0 (∂0 (Bl ))] − (n−1 ∆n − n−2 |∇n|2 )∂0 (∂0 (Bl )).
Together with the estimates (2.61) for the lapse n and the Sobolev embedding on Σt (2.64), thisyields:N2 . kn−1 ∇b n∂0 (∂0 (Bl ))kL2 (M) + k(n−1 ∆n − n−2 |∇n|2 )∂0 (∂0 (Bl ))k 2 56

Lt L (Σt ). (k∇nkL∞ + kn

−1∆n − n. M εk∂0 (∂0 (Bl ))kL2 (M) .

−22|∇n| kL∞)k∂0 (∂0 (Bl ))kL2 (M)3t L (Σt )(2.76)

Remark 2.22. Note that there is no room in the estimate (2.76). Indeed the sharp estimatek∇nkL∞ (M) . M ε given by (2.61) is crucial as emphasized in remark 2.15.

2.5. ENERGY ESTIMATE FOR THE WAVE EQUATION ON A CURVED BACKGROUND

31

Finally, we consider N3 . Recall from (1.10) that the second fundamental form satisfies thefollowing equation:∇0 kab = Eab − n−1 ∇a ∇b n − kac kb c = Eab + l.o.t.

(2.77)

where E is the 2-tensor on Σt defined as Eab = Ra 0 b 0 . In view of the estimates (2.31) for k and(2.61) for n,
k l.o.t.kL∞. k∇2 nkL∞
+ kkk2L∞ L6 (Σt ) . M ε.
33t L (Σt )t L (Σt )tThus instead of estimating ∇0 k ab ∇a ∇b B in the definition of N3 it suffices to estimate the term
E ab ∇a ∇b B = ∇a (E ab ∇b B) − ∇a E ab ∇b B
Using the maximal foliation assumption, the Bianchi identities and the symmetries of R, we
can write, schematically, ∇a Eab = AR and therefore, together with the bootstrap assumptions(2.29) for A and (2.30) for A0 , and the bootstrap assumption (2.27) for R yields:k∇a Eab k ∞ 23

Lt L (Σt ). kAkL∞kRkL∞. M 2 ε2 .
6
2t L (Σt )t L (Σt )(2.78)We thus have,∇0 kab = ∇a (E ab ∇b B) + l.o.t.
Now, in view of the bilinear assumption (2.35),k(−∆)1/2 ∇a (E ab ∇b B)kL2 (M) . kR0a0b ∇b BkL2 (M) . M 3 ε2Hence, putting all the above together we infer that,
N3 . M 3 ε2 .
Together with (2.73), (2.75), (2.76) and (2.74), we derive,k∂ BkL2 (M) . M 3 ε2 + M εk∂0 (∂0 B)kL2 (M) .

(2.79)To close the estimate for k∂ BkL2 (M) it remains to control the right-hand side of (2.79). Thisis achieved relying in particular on the following formula∂0 (∂0 B) = −B + ∆B + n−1 ∇n · ∇B.

2.5. Energy estimate for the wave equation on a curved background
Recall that e0 = T , the future unit normal to the Σt foliation. Let π be the deformation
tensor of e0 , that is the symmetric 2-tensor on M defined as:παβ = Dα Tβ + Dβ Tα .
In view of the definition of the second fundamental form k and the lapse n, we have:
πab = −2kab , πa0 = π0a = n−1 ∇a n, π00 = 0.

(2.80)In what follows H denotes an arbitrary weakly regular null hypersurface1 with future normal L
verifying g(L, T ) = −1. We denote by ∇/ the induced connection on the 2-surfaces H ∩ Σt .
1i.e. it satisfies assumptions (2.25) and (2.26)

32

2. EINSTEIN VACUUM EQUATIONS AS A YANG-MILLS GAUGE THEORY

Lemma 2.23. Let F a scalar function on M, and let φ0 and φ1 two scalar functions on Σ0 .
Let φ the solution of the following wave equation on M:
φ = F,(2.81)
φ|Σ0 = φ0 , ∂0 (φ)|Σ0 = φ1 .
Let E0 , E1 denote the energy quantities,
E0 [φ] := k∂φkL∞+ sup(k∇/ φkL2 (H) + kL(φ)kL2 (H) )2t L (Σt )H

E1 [φ] := k∂(∂φ)kL∞+ k∂0 (∂0 φ)kL2 (M) + sup k∇/ (∂φ)kL2 (H) + kL(∂φ)kL2 (H)2t L (Σt )

Hwhere the supremum is taken over all weakly regular null hypersurfaces H (satisfying assumptions(2.25) and (2.26)). The following estimates hold true, provided that εM 2 is sufficiently small,
E0 . k∇φ0 kL2 (Σ0 ) + kφ1 kL2 (Σ0 ) + kF kL2 (M) ,(2.82)

2

E1 . k∇ φ0 kL2 (Σ0 ) + k∇φ1 kL2 (Σ0 ) + k∇F kL2 (M) .

(2.83)

Proof. We introduce the energy momentum tensor Qαβ on M given by:1Qαβ = Qαβ [φ] = ∂α φ∂β φ − gαβ (gµν ∂µ φ∂ν φ) .
2
In view of the equation (2.81) satisfied by φ, we have:Dα Qαβ = F ∂β φ.
Now, consider the divergence of the 1-tensor Pα = Qαβ eβ0 = Qα0 ,1
Dα Pα = F ∂0 φ + Qαβ π αβ ,2where π is the deformation tensor of e0 . Integrating over well-chosen regions of M, we easily
obtain:Z
Z
2
2
F ∂0 φdM +Qαβ π αβ dM(2.84)
E0 . k∇φ0 kL2 (Σ0 ) + kφ1 kL2 (Σ0 ) +
M
M
Z
2
2
Qαβ π αβ dM .
. k∇φ0 kL2 (Σ0 ) + kφ1 kL2 (Σ0 ) + kF kL2 (M) k∂0 φkL2 (M) +M

Next, we deal with the last term in the right-hand side of (2.84). In view of (2.80) and our
maximal foliation assumption, we have:Z
Z
Z
Qαβ π αβ dM = −2∂a φ∂b φk ab dM + n−1 ∇a n∂a φ∂0 φdM.
M

M

Together with the bilinear bootstrap assumptions (2.36) and the estimates (2.61) for the lapsen, this yields:Z
Qαβ π αβ dM . kka · ∂ a φkL2 (M) k∂φkL2 (M) + k∇nkL∞ (M) k∂φk2L2 (M)M2. M ε supk∇/ φkL2 (H) k∂φkL2 (M) + M εk∂φk2L2 (M) ,Hwhich together with (2.84) concludes the proof of the (2.82). Though more technical the proofof (2.83) follows the same ideas, and we refer to [43] for the details.
2.6. IMPROVEMENT OF THE BOOTSTRAP ASSUMPTIONS (PART 1)

33

2.6. Improvement of the bootstrap assumptions (part 1)
In this section, we discuss the proof of Proposition 2.13. More precisely, we derive estimatesfor R, A0 and A which allow us to improve the basic bootstrap assumptions (2.27), (2.28), (2.29)
and (2.30).
2.6.1. Curvature estimates. We derive the curvature estimates using the Bell-Robinson
tensor,Qαβγδ = Rα λ γ σ Rβ λ δ σ + ∗ Rα λ γ σ ∗ Rβ λ δ σLetPα = Qαβγδ eβ0 eγ0 eδ0 .
Then, we have:Dα Pα = 3Qαβγδ π αβ eγ0 eδ0 ,(2.85)

where π is the deformation tensor of e0 . We introduce the Riemannian metric,
hαβ = gαβ + 2(e0 )α (e0 )β(2.86)

and use it to define the following space-time norm for tensors U :00|U |2 = Uα1 ···αk Uα01 ···α0k hα1 α1 · · · hαk αk .
Given two space-time tensors U, V we denote by U · V a given contraction between the two
tensors and by |U · V | the norm of the contraction according to the above definition.
Let H be a weakly regular null hypersurface with future normal L, g(L, T ) = −1. Integrating(2.85) on a well-chosen, causal, space-time region, we have:Z
Z
Z
Z|R|2 +|R · L|2 . kRk2L2 (Σ0 ) +Qαβγδ π αβ eγ0 eδ0 . ε2 +Qαβγδ π αβ eγ0 eδ0 .
Σt

H

M

M

We need to estimate the term in the right-hand side of the previous inequality. Note that since
π00 = 0, π0j = n−1 ∇j n, and πij = kij , the bootstrap assumption (2.27) for R, and the estimates(2.61) for n yield:
Z
Z
Z
2
2
2
2
|R| +
|R · L| . ε + k∇nkL∞ kRkL∞ L2 (Σt ) +Qijγδ k ij eγ0 eδ0
t
Σt
H
M
Z
. ε2 + (M ε)3 +Qijγδ k ij eγ0 eδ0 .
M

The term Rin the right-hand side of the previous inequality is dangerous. Schematically it
has the form M kR2 . Typically this term is estimated by:Z
kR2 . kkkL2t L∞ (Σt ) kRk2L∞ L2 (Σt ) ,tMwhich requires a Strichartz estimate for k which is false even in flat space. It is for this reason
that we need the trilinear bootstrap assumption (2.43). Using it we derive,
Z
Z
2
|R| +
|R · L|2 . ε2 + M 4 ε3 .
(2.87)
ΣtH

which, for small ε, improves the bootstrap assumptions (2.27) and (2.28).

34

2. EINSTEIN VACUUM EQUATIONS AS A YANG-MILLS GAUGE THEORY

2.6.2. Improvement of the bootstrap assumption for A0 . Recall (2.23)∆A0 = A∂A + A∂A0 + A3 .

(2.88)

Then, using (2.88), elliptic estimates on Σt , and commuting (2.88) with ∂t in order to control∂t A0 , we are able to obtain the improved estimate (2.47) (see [43] for the details).
2.6.3. Improvement of the bootstrap assumption for A. Using the estimates for Bi
derived in Lemma 2.21, the estimates for B on the initial slice Σ0 , and the energy estimate (2.83)
derived in Lemma 2.23, we have:k∂ 2 BkL∞. ε + M 2 ε2 .
2t L (Σt )(2.89)

Using then (2.89) with Lemma 2.18, we obtain:k∂AkL∞. k∂ 2 BkL∞
+ k∂EkL∞. ε + M 2 ε2 .
222t L (Σt )t L (Σt )t L (Σt )(2.90)

which proves corresponding estimate in (2.46).
To estimate ∂0 A. we recall that, ∂0 (Aj ) = ∂j (A0 ) + R0j·· . Thus, we have:k∂0 AkL∞. k∂A0 kL∞+ kRkL∞,22
2t L (Σt )t L (Σt )t L (Σt )which together with the improved estimates for R and A0 yields:
3k∂0 AkL∞. ε + (M ε) 2 .
2t L (Σt )(2.91)

2.7. Parametrix for the wave equation
Let u± two families, indexed by ω ∈ S2 , of scalar functions on the space-time M satisfying
the Eikonal equation for each ω ∈ S2 . We also denote ω u± (t, x) = u± (t, x, ω). We have thefreedom of choosing ω u± on the initial slice Σ0 , and in order for the results in [74], [76] toapply, we need to initialize ω u± on Σ0 as in [73] (see also Chapter 6).
Let H ω u± denote the corresponding null level hypersurfaces. Let ω L± its normal. ω L± isnull, and we fix it by imposing g( ω L± , T ) = −1. Let the vectorfield tangent to Σt ω N ± be
defined such as to satisfy:ωL± = ±T + ω N ± .
We pick ( ω e± )A , A = 1, 2 vectorfields in Σt such that together with ω N ± we obtain an orthonormal basis of Σt . Finally, we denote by ∇/ ± derivatives in the directions ( ω e± )A , A = 1, 2.
Remark 2.24. Note that H ω u± satisfy assumptions (2.25) and (2.26) from the results in[75] (see Theorem 2.15 and section 3.4 in that paper).
For any pair of functions f± on R3 , we define the following scalar function on M:
Z Z ∞
Z Z ∞
ω
iλ ω u+ (t,x)
2ψ[f+ , f− ](t, x) =ef+ (λω)λ dλdω +eiλ u− (t,x) f− (λω)λ2 dλdω.
S2

0

S2

0

We appeal to the following result from [74] [76] (see also Chapters 3 and 5):
Theorem 2.25. Let φ0 and φ1 two scalar functions on Σ0 . Then, there is a unique pair offunctions (f+ , f− ) such that:ψ[f+ , f− ]|Σ0 = φ0 and ∂0 (ψ[f+ , f− ])|Σ0 = φ1 .

2.7. PARAMETRIX FOR THE WAVE EQUATION

35

Furthermore, f± satisfy the following estimates:
kλf+ kL2 (R3 ) + kλf− kL2 (R3 ) . k∇φ0 kL2 (Σ0 ) + kφ1 kL2 (Σ0 ) ,and:
kλ2 f+ kL2 (R3 ) + kλ2 f− kL2 (R3 ) . k∇2 φ0 kL2 (Σ0 ) + k∇φ1 kL2 (Σ0 ) .
Finally, ψ[f+ , f− ] satisfies the following estimates:kψ[f+ , f− ]kL2 (M) . M ε(k∇φ0 kL2 (Σ0 ) + kφ1 kL2 (Σ0 ) ),and:k∂ψ[f+ , f− ]kL2 (M) . M ε(k∇2 φ0 kL2 (Σ0 ) + k∇φ1 kL2 (Σ0 ) ).
Remark 2.26. The existence of f± and the first two estimates of Theorem 2.25 are provedin [74] (see also Chapter 5), while the last two estimates in Theorem 2.25 are proved in [76](see also Chapter 3).
We associate to any pair of functions φ0 , φ1 on Σ0 the function Ψom [φ0 , φ1 ] defined for(t, x) ∈ M as:Ψom [φ0 , φ1 ] = ψ[f+ , f− ]where (f+ , f− ) is defined in view of Theorem 2.25 as the unique pair of functions associated to(φ0 , φ1 ). In particular, we obtain:
kλf+ kL2 (R3 ) + kλf− kL2 (R3 ) . k∇φ0 kL2 (Σ0 ) + kφ1 kL2 (Σ0 ) ,kλ2 f+ kL2 (R3 ) + kλ2 f− kL2 (R3 ) . k∇2 φ0 kL2 (Σ0 ) + k∇φ1 kL2 (Σ0 ) ,kΨom [φ0 , φ1 ]kL2 (M) . M ε(k∇φ0 kL2 (Σ0 ) + kφ1 kL2 (Σ0 ) ),(2.92)k∂Ψom [φ0 , φ1 ]kL2 (M) . M ε(k∇2 φ0 kL2 (Σ0 ) + k∇φ1 kL2 (Σ0 ) ).

(2.93)

and:Next, let ω,s u± two families, indexed by ω ∈ S2 and s ∈ R, of scalar functions on thespace-time M satisfying the Eikonal equation for each ω ∈ S2 and s ∈ R. We have the freedomof choosing ω,s u± on the slice Σs , and in order for the results in [74] [76] to apply, we needto initialize ω,s u± on Σs as in [73]. Note that the families ω u± correspond to ω,s u with the
choice s = 0. For any pair of functions f± on R3 , and for any s ∈ R, we define the following
scalar function on M:
Z Z ∞
Z Z ∞
ω,s
iλ ω,s u+ (t,x)
2ef+ (λω)λ dλdω +eiλ u− (t,x) f− (λω)λ2 dλdω.
ψs [f+ , f− ](t, x, s) =
S2

0

S2

0

We have the following straightforward corollary of Theorem 2.25:
Corollary 2.27. Let s ∈ R. Let φ0 and φ1 two scalar functions on Σs . Then, there is aunique pair of functions (f+ , f− ) such that:ψs [f+ , f− ]|Σs = φ0 and ∂0 (ψs [f+ , f− ])|Σs = φ1 .
Furthermore, f± satisfy the following estimates:
kλf+ kL2 (R3 ) + kλf− kL2 (R3 ) . k∇φ0 kL2 (Σs ) + kφ1 kL2 (Σs ) ,and:kλ2 f+ kL2 (R3 ) + kλ2 f− kL2 (R3 ) . k∇2 φ0 kL2 (Σs ) + k∇φ1 kL2 (Σs ) .

36

2. EINSTEIN VACUUM EQUATIONS AS A YANG-MILLS GAUGE THEORY

Finally, ψs [f+ , f− ] satisfies the following estimates:kψs [f+ , f− ]kL2 (M) . M ε(k∇φ0 kL2 (Σs ) + kφ1 kL2 (Σs ) ),and:k∂ψs [f+ , f− ]kL2 (M) . M ε(k∇2 φ0 kL2 (Σs ) + k∇φ1 kL2 (Σs ) ).
Next, for any s ∈ R, we associate to any function F on Σs the function Ψ(t, s)F defined for(t, x) ∈ M as:Ψ(t, s)F = ψs [f+ , f− ](t)where (f+ , f− ) is defined in view of Corollary 2.27 as the unique pair of functions associated tothe choice (φ0 , φ1 ) = (0, F ). In particular, we obtain:
kλf+ kL2 (R3 ) + kλf− kL2 (R3 ) . kF kL2 (Σs ) ,kλ2 f+ kL2 (R3 ) + kλ2 f− kL2 (R3 ) . k∇F kL2 (Σs ) ,kΨ(t, s)F kL2 (M) . M εkF kL2 (Σs ) ,(2.94)k∂Ψ(t, s)F kL2 (M) . M εk∇F kL2 (Σs ) .

(2.95)

and:Now, we are in position to construct an exact parametrix for the wave equation (2.81):
Theorem 2.28 (Representation formula). Let F a scalar function on M, and let φ0 and φ1
two scalar functions on Σ0 . Let φ the solution of the wave equation (2.81) on M. Then, there
is a sequence φ(j) , j ≥ 0, of scalar functions approximations of φ and a sequence F (j) , j ≥ 0, of
scalar functions on M, with of the form:
Z tφ(0) = Ψom [φ0 , φ1 ] +Ψ(t, s)F (0) (s, .)ds,
F (0) = F0and for all j ≥ 1:
φ

(j)

Z t
=

Ψ(t, s)F (j) (s, .)ds,

0such that,
φ=

+∞
X

φ(j) ,

j=0

and such that φ(j) and F (j) satisfy the following estimates:k∂φ(j) kL∞+ kF (j) kL2 (M) . (M ε)j (k∇φ0 kL2 (Σ0 ) + kφ1 kL2 (Σ0 ) + kF kL2 (M) ),2t L (Σt )and:k∂∂φ(j) kL∞+ k∂F (j) kL2 (M) . (M ε)j (k∇2 φ0 kL2 (Σ0 ) + k∇φ1 kL2 (Σ0 ) + k∂F kL2 (M) ),2t L (Σt )2.8. IMPROVEMENT OF THE BOOTSTRAP ASSUMPTIONS (PART 2)

37

Proof. Let us define:F(0)= F and φ(0)

Z t= Ψom [φ0 , φ1 ] +Ψ(t, s)F (0) (s, .)ds.

0Then, we define iteratively for j ≥ 1:
F (j) = −φ(j−1) + F (j−1) and φ(j) =Z tΨ(t, s)F (j) (s, .)ds.

0

The proof follows from the estimates (2.92), (2.93), (2.94) and (2.95), together with the energy
estimates for the wave equation of Lemma 2.23. We refer to [43] for the details.

2.8. Improvement of the bootstrap assumptions (part 2)
The goal of this section and next section is to prove Proposition 2.14. This requires inparticular to write B using the representation formula of Theorem 2.28. In this section we
derive the improved bilinear estimate (2.48), (2.49), (2.50), (2.51) and (2.52) of Proposition
2.14. We also derive the improved trilinear estimate (2.57).
2.8.1. Improvement of the bilinear bootstrap assumptions I. In this section, wegive the main ideas on the how we derive the improved bilinear estimate (2.48), (2.49), (2.50),(2.51) and (2.52) of Proposition 2.14. These bilinear estimates all involve the norm in L2 (M)of quantities of the type:C(U, ∂φ),
where C(U, ∂φ) denotes a contraction with respect to one index between a tensor U and ∂φ,with φ being a scalar function which is solution to the wave equation (2.81) with F, φ0 and φ1
satisfying the estimate:k∇2 φ0 kL2 (Σ0 ) + k∇φ1 kL2 (Σ0 ) + k∂F kL2 (M) . M ε.
In particular, we may use the parametrix constructed in Lemma 2.28 for φ:
φ=

+∞
X

φ(j) ,

j=0

with:
φ

(0)

Z t
Ψ(t, s)F (s, .)ds,= Ψom [φ0 , φ1 ] +0and for all j ≥ 1:
φ

(j)

Z t
=

Ψ(t, s)F (j) (s, .)ds.

0Thus, we need to estimate the norm in L2 (M) of contractions of quantities of the type:+∞ Z tX
C(U, ∂(Ψom [φ0 , φ1 ])) +C(U, ∂(Ψ(t, s)F (j) (s, .)))ds.
j=00

After using the definition of Ψom and Ψ(t, s), and the estimates for F (j) provided by Lemma
2.28, this reduces to estimating:
Z Z ∞
Z Z ∞
ω
ω
C(U, ∂(eiλ u+ (t,x) ))f+ (λω)λ2 dλdω +C(U, ∂(eiλ u− (t,x) ))f− (λω)λ2 dλdω,
S2

0

S2

0

38

2. EINSTEIN VACUUM EQUATIONS AS A YANG-MILLS GAUGE THEORY

where f± in view of Theorem 2.25 and the estimates for F, φ0 and φ1 satisfies:
kλ2 f± kL2 (R3 ) . M ε.
Since both half waves parametrices are estimated in the same way, the bilinear estimates (2.33),(2.34), (2.35), (2.36) and (2.37) all estimate the norm in L2 (M) of contractions of quantities of
the type:
Z Z ∞ωC(U, ∂(eiλ u(t,x) ))f (λω)λ2 dλdω,S20where f satisfies:
kλ2 f kL2 (R3 ) . M ε.
ω

(2.96)

ω

Furthermore we observe that ∂j (eiλ u ) = iλeiλ u ∂j ( ω u), and that the gradient of ω u on Σt is∇ ωugiven by: ∇( ω u) = ω b−1 ω N , with ω b = |∇( ω u)|−1 is the null lapse, and ω N = |∇ω u| is theunit normal to H ω u ∩ Σt along Σt . Thus, the bilinear estimates (2.33), (2.34), (2.35), (2.36) and(2.37) all reduce to L2 (M)-estimates of expressions of the form:Z Z ∞C[U, f ] :=S2eiλ

ω u(t,x) ω

b−1 C(U, ω N )f (λω)λ3 dλdω,(2.97)

0where f satisfies (2.96). To estimate C[U, f ] we follow the strategy of [38].
kC[U, f ]kL2 (M)

Z +∞
Z
ω −1
ω
iλ ω u(t,x)
3.
b C(U, N )ef (λω)λ dλ
S2

Z
.

S2

0

.

sup k b
ω∈S2


.

ω −1


kL∞ (M)sup k ω b−1 kL∞ (M)sup kC(U,
ω∈S2



ω∈S2

dω
L2 (M)k ω b−1 kL∞ (M) kC(U, ω N )kL∞ω u L2 (H ω u )

(2.98)

ω

Z +∞

eiλ

ω u(t,x)f (λω)λ3 dλ

dω
L2ω u

0

 Z
N )kL∞ω u L2 (H ω u )

S2



3

kλ f (λω)kL2 dωλ

sup kC(U, ω N )kL∞ω u L2 (H ω u ) kλ2 f kL2 (R3 ) ,ω∈S2Now, since ω u has been initialized on Σ0 as in [73], and satisfies the Eikonal equation on M,
the results in [75] (see Theorem 2.15 in that paper, and also (4.42)) under the assumption ofTheorem 1.14 imply:
sup k ω b−1 kL∞ (M) . 1.
ω∈S2

Together with the fact that f satisfies (2.96), and with (2.98), we finally obtain:Z Z ∞ω
eiλ u(t,x) ω b−1 C(U, ω N )f (λω)λ3 dλdω
S2

0


. Mε

sup kC(U,ω∈S2L2 (M)ωN )kL∞ω u L2 (H ω u ) .

(2.99)

2.8. IMPROVEMENT OF THE BOOTSTRAP ASSUMPTIONS (PART 2)

39

It remains to estimate the right-hand side of (2.99) for the contractions appearing in the bilinear
estimates (2.33), (2.34), (2.35), (2.36) and (2.37). Since all the estimates in the proof will be
uniform in ω, we drop the index ω to ease the notations.
Remark 2.29. In the proof of bilinear estimates (2.48), (2.49), (2.50), (2.51) and (2.52),
the tensor U appearing in the expression C(U, N ) is either R or derivatives of solutions φ of a ascalar wave equation. In view of the bootstrap assumption (2.28) for the curvature flux, as well as
the first energy estimate for the wave equation in Lemma 2.23, we can control kC(U, N )kL∞2
u L (Hu )2
as long as we can show that C(U, N ) can be expressed in terms of, R · L, ∇/ φ and L(φ).
2.8.1.1. Proof of (2.48). Since A = curl (B) + E in view of Lemma 2.18 and bootstrap
assumption (2.29), we have:kAj ∂j (A)kL2 (M) . k(curl (B))j ∂j (A)kL2 (M) + kEkL2t L∞ (Σt ) k∂AkL∞
2
t L (Σt )j(2.100)2 2. k(curl (B)) ∂j (A)kL2 (M) + M ε ,To estimate k(curl (B))j ∂j (A)kL2 (M) we write, (curl (B))j ∂j (A) =∈jmn ∂m (Bn )∂j (A). We are
now ready to apply the representation theorem 2.28 to B. Indeed, according to Lemma 2.21,
and proposition 2.17, we have
B = F,k∂F kL2 (M) . M 2 ε2(2.101)k∂B(0)kL2 (Σ0 ) + k∂ 2 B(0)kL2 (Σ0 ) + k∂(∂0 B(0))kL2 (Σ0 ) . M ε.
We are thus in a position to apply the reduction discussed in the subsection above and reduce
our desired bilinear estimate to an estimate for,C(U, N ) = ∈jm· Nm ∂j (A)Now, we decompose ∂j on the orthonormal frame N, fA , A = 1, 2 of Σt , where we recall thatfA , A = 1, 2 denotes an orthonormal basis of Hu ∩ Σt . We have schematically:∂j = N j N + ∇/,(2.102)where ∇/ denotes derivatives which are tangent to Hu ∩ Σt . Thus, we have:∈jm· Nm ∂j (A) =∈jm· Nm Nj ∂N (A) + ∇/ (A) = ∇/ (A),
where we used the antisymmetry of ∈jm· in the last equality. Therefore, we obtain in this case:kC(U, N )kL∞
. k∇
/ (A)kL∞.
2
2
u L (Hu )u L (Hu )It remains to estimate k∇/ (A)kL∞. Since A = curl (B) + E we have, using Lemma 2.18
2u L (Hu )again, followed by Proposition 2.21 and Lemma 2.23:k∇/ (A)kL∞. k∇/ (∂B)kL∞+ k∇/ (E)kL∞2
2
2
u L (Hu )u L (Hu )u L (Hu ). k∇/ (∂B)kL∞+ k∂EkL∞+ k∂ 2 Ek ∞ 32
23
u L (Hu )t L (Σt )Lt L (Σt ). k∇/ (∂B)kL∞+ M ε,2u L (Hu ). M ε.
2In other words, our goal is to check that the term C(U, N ) does not involve the dangerous terms of the type
α and Lφ, where L is the vectorfield defined as L = 2T − L, and α is the two tensor on Σt ∩ Hu defined by
αAB = R L A L B .

40

2. EINSTEIN VACUUM EQUATIONS AS A YANG-MILLS GAUGE THEORY

Therefore,kAj ∂j (A)kL2 (M) . k(curl (B))j ∂j (A)kL2 (M) + M 2 ε2 . M 2 ε2 ,as desired.
2.8.1.2. Proof of (2.49). The proof of (2.49) is similar to the one of (2.48) in view of Lemma
2.18.
2.8.1.3. Proof of (2.50). Since B satisfies a wave equation in view of Lemma 2.21, thequantity C(U, N ) is in this case3,
1
C(U, N ) = Nj R0 j · · = R0 N · · = RL L · ·2which together with the bootstrap assumption for the curvature flux (2.28) improves the bilinear
estimate (2.35).
2.8.1.4. Proof of (2.51). We have kj · = Aj and A = curl (B) + E in view of Lemma 2.18.
Arguing as in (2.100), we reduce the proof to the estimate of:k(curl B)j ∂j φkL2 (M) .
Then, the proof proceeds as the one of (2.48).
2.8.1.5. Proof of (2.52). The proof of (2.52) proceeds as in (2.51).
2.8.2. Improvement of the trilinear estimate. In this section,
we shall derive theR
improved trilinear estimate (2.57). To estimate the trilinear quantity M Qijγδ k ij eγ0 eδ0 . we firstwrite, according to Lemma 2.18, A = curl (B) + E by. Arguing as in (2.100), we reduce theproof of (2.57) to an estimate for:Z
Q· jγδ (curl (B))j eγ0 eδ0 .
M

Making use of the wave equation (2.101) for B we argue as in the beginning of section 2.8.1 toreduce the proof to an estimate of the following:Z Z Z ∞
ωeiλ u(t,x) ω b−1 (∈jm· ω N m Qj · · · ) f (λω)λ3 dλdωdM
M

S2

0

where f satisfies:
kλ2 f kL2 (R3 ) . M ε.
Arguing exactly as in (2.98) (2.99), we can estimate the latter integral by the quantity supω∈S2 k∈jm·
Nm Qj · · · kL2ω L1 (H ω u ) M ε. In other words,uZ
Qijγδ k ij eγ0 eδ0 . sup k∈jm· Nm Qj · · · kL2ω L1 (H ω u ) M ε + M 3 ε3 .
(2.103)Mω∈S2u

Next, we estimate the right-hand side of (2.103). Since all the estimates in the proof will
be uniform in ω, we drop the index ω to ease the notations. The formula (2.42) for the BellRobinson tensor Q yields:
Qj··· = Rj λ · ·R· λ · · + dual
1
1
= − Rj L ·· R· L · · − Rj L ·· R· L · · + Rj A ·· R· A · · + dual,
22
3Use also L = T + N , L = T − N and the symmetries of R

2.9. IMPROVEMENT OF THE BOOTSTRAP ASSUMPTIONS (PART 3)

41

where we used the frame L, L, fA , A = 1, 2 in the last equality. Thus, we have schematically:∈jm· Nm Qj · · · = R(R · L+ ∈jm· Nm Rj A ·· )Decomposing ej with respect to the orthonormal frame N, fB , B = 1, 2, we note that:∈jm· Nm RjA·· =∈jm· Nj Nm RN A·· + ∈jm· (fB )j Nm RBA·· = RBA·· .
On the other hand, decomposing RBA·· further and using the symmetries of R, one easily checks
that RBA·· must contain at least one L so that it is of the type R·L. Thus, we have schematically:∈jm· Nm Qj · · · = R(R · L).

(2.104)Thus, in view of (2.103), making use of the bootstrap assumptions (2.27) on R and (2.28) on
the curvature flux, we deduce,Z
Qijγδ k ij eγ0 eδ0 . (M ε)3 + M εkRRL kL2u L1 (Hu )M. (M ε)3 + M εkRkL2 (M) kRL kL∞
2
u L (Hu ). M 3 ε3In other words,
Z
M

Qijγδ k ij eγ0 eδ0 . (M ε)3 .

(2.105)which yields the desired improvement of the trilinear estimate (2.43).
2.9. Improvement of the bootstrap assumptions (part 3)
In this section, we conclude the proof of Proposition 2.14. More precisely, we give the mainideas in the improvement of the bilinear bootstrap assumptions II. We start with a discussionof the sharp L4 (M)- Strichartz estimate.
2.9.1. The sharp Strichartz L4 (M) estimate. To a function f on R3 and a family ω uindexed by ω ∈ S2 of scalar functions on the space-time M satisfying the Eikonal equation foreach ω ∈ S2 , we associate a half wave parametrix:
Z Z ∞ω
eiλ u(t,x) f (λω)λ2 dλdω.
S20

Let an integer p and a smooth cut-off function ψ on (0, +∞) supported in a shell. We call a halfwave parametrix localized at frequencies of size λ ∼ 2p the following Fourier integral operator:
Z Z ∞ω
eiλ u(t,x) ψ(2−p λ)f (λω)λ2 dλdω.
S20

We have the following L4 (M) Strichartz estimate localized in frequency for a half parametrix:
Proposition 2.30 (L4 (M) Strichartz estimate localized in frequency). Let f a function
on R3 , let p ∈ N, and let ψ a smooth function on (0, +∞) compactly supported in the shell
1/2 ≤ λ ≤ 2. Let ω u a family indexed by ω ∈ S2 of scalar functions on the space-time M

42

2. EINSTEIN VACUUM EQUATIONS AS A YANG-MILLS GAUGE THEORY

satisfying the Eikonal equation for each ω ∈ S2 and initialized on the initial slice Σ0 as in [73].
Let φp the scalar function on M defined by the following oscillatory integral:Z Z ∞ωeiλ u(t,x) ψ(2−p λ)f (λω)λ2 dλdω.
φp (t, x) =S2

0
4
Then, we have the following L (M) Strichartz estimates for φp :p

kφp kL4 (M) . 2 2 kψ(2−p λ)f kL2 (R3 ) ,(2.106)3p
2k∂φp kL4 (M) . 2 kψ(2−p λ)f kL2 (R3 ) ,(2.107)5p2(2.108)k∂ 2 φp kL4 (M) . 2 kψ(2−p λ)f kL2 (R3 ) .

Note that this Strichartz estimate is sharp. Proposition 2.30 is not yet proved (see also Step
D in section 1.2.4), and will be the subject of a forthcoming paper.
2.9.2. Improvement of the non sharp Strichartz estimates. Here, we derive theimproved non sharp Strichartz estimates (2.55) and (2.56). In view of Lemma 2.18, (2.55) easily
follows from (2.56), so we focus on the later improved estimate.
Corollary 2.31. B satisfies the following Strichartz estimate:k∂BkL2t L7 (Σt ) . M ε.
Proof. Recall (2.101) which allows us to apply the representation formula of Theorem
2.28 to B. By a straightforward reduction the proof then reduces to the following non-sharp
Strichartz estimate for a half wave parametrix:
Z Z ∞
iλ ω u(t,x)
2∂ef (λω)λ dλdω. kλ2 f kL2 (R3 ) .
(2.109)S2

L2t L7 (Σt )0Then, the proof of Corollary 2.31 follows in particular from the sharp Strichartz estimate of
Proposition 2.30. We refer to [43] for the details.

2.9.3. Improvement of the bilinear bootstrap assumptions II. In this section, we
sketch the proofs of the improved bilinear estimates (2.53) and (2.54) of Proposition 2.14. Basedon the decomposition A = curl (B) + E of Lemma 2.18 it is easy to show that that the proof ofthe bilinear estimates (2.38) and (2.39) reduces to:1k(−∆)− 2 (Qij (∂B, ∂B))kL2 (M) . M 2 ε2 .

(2.110)Decomposing B according to Theorem 2.28,
1k(−∆)− 2 (Qij (∂B, ∂B))kL2 (M) ≤

+∞
X

1

k(−∆)− 2 (Qij (φ(m) , φ(n) ))kL2 (M) .

(2.111)m,n=0Thus it suffices to prove for all m, n ≥ 0:1k(−∆)− 2 (Qij (φ(m) , φ(n) ))kL2 (M) . (M ε)m+1 (M ε)n+1 .

(2.112)The estimates in (2.112) are analogous for all m, n, so it suffices to prove (2.112) in the case(m, n) = (0, 0). In view of the definition of φ(0) , the estimates for B on the initial slice Σ0 ,

2.9. IMPROVEMENT OF THE BOOTSTRAP ASSUMPTIONS (PART 3)

43estimate (2.69) for ∂B, and the definition of Ψom and Ψ(t, s), (2.112) reduces to the following
bilinear estimate for half wave parametrices:1(−∆)− 2 (Qij (φf1 , φf2 )
with,

Z Z ∞
φf =
S2

L2 (M)eiλ. kλf1 kL2 (R3 ) kλf2 kL2 (R3 )ω u(t,x)(2.113)f (λω)λ2 dλdω.

0

We then decompose f1 , f2 with respect to frequency and reduce the desired estimate to L4 (M)Strichartz estimate localized in frequency of Proposition 2.30, see details in [43].
This concludes the proof of Proposition 2.14.

CHAPTER 3

Control of the error term
In this chapter, we consider the Fourier integral operator E given by (1.23) in which corresponds to the error term of a plane wave type parametrix. Recall that E is given by:
Z Z +∞
Ef (t, x) =eiλu(t,x,ω) g u(t, x, ω)f (λω)λ3 dλdω, (t, x) ∈ M,
S2

0

where u(., ., ω) is a solution to the eikonal equation gαβ ∂α u∂β u = 0 on M such that u(0, x, ω) ∼x · ω when |x| → +∞ on Σ0 (see section 3.1.1). The goal of this chapter is to outline the mainideas allowing us to obtain the control for the error term E in [76].
3.1. Geometric set-up and main results
3.1.1. Geometry of the foliation of M by u. Recall that u is a solution to the eikonal
equation gαβ ∂α u∂β u = 0 on M depending on a extra parameter ω ∈ S2 . The level hypersufacesu(t, x, ω) = u of the optical function u are denoted by Hu . Let L0 denote the space-time gradientof u, i.e.:
L0 = −gαβ ∂β u∂α .
(3.1)
Using the fact that u satisfies the eikonal equation, we obtain:
DL0 L0 = 0,

(3.2)

which implies that L0 is the geodesic null generator of Hu .
We foliate the space-time M by space-like hypersurfaces Σt defined as level hypersurfacesof a time function t and we denote by T the unit, future oriented, normal to Σt . We have:T (u) = ±|∇u|P3

where |∇u|2 = i=1 |ei (u)|2 relative to an orthonormal frame ei on Σt . Since the sign of T (u)is irrelevant, we choose by convention:
T (u) = |∇u|.

(3.3)

We denote by Pt,u the surfaces of intersection between Σt and Hu .
Definition 3.1 (Canonical null pair ).
L = bL0 = T + N,

L = 2T − L = T − N(3.4)

where L0 is the space-time gradient of u (3.1), b is the lapse of the null foliation (or shortly null
lapse)
b−1 = − < L0 , T >= T (u),
(3.5)
45

46

3. CONTROL OF THE ERROR TERM

and N is a unit normal, along Σt , to the surfaces Pt,u . Since u satisfies the eikonal equationgαβ ∂α u∂β u = 0 on M, this yields L0 (u) = 0 and thus L(u) = 0. In view of the definition of Land (3.3), we obtain:
∇u
N =−
.
(3.6)
|∇u|
Definition 3.2. A null frame e1 , e2 , e3 , e4 at a point p ∈ Pt,u consists, in addition to thenull pair e3 = L, e4 = L, of arbitrary orthonormal vectors e1 , e2 tangent to Pt,u .
Definition 3.3 (Second fundamental form). Let e1 , e2 , e3 , e4 be a null frame on Pt,u as
above. The second fundamental form on Pt,u associated to our canonical null pair is given byχAB =< DA e4 , eB > .
We decompose χ into its trace and traceless component.
1trχ = gAB χAB , χ
bAB = χAB − trχgAB .
2
Recall that trχ satisfies a transport equation called the Raychaudhuri equation:1
χ|2 + · · ·L(trχ) + (trχ)2 = −|b
2

(3.7)

(see precise equation in (4.19)).
We conclude this section with the identification of the symbol g u of the error term. We
have (see for example [75] for a proof):
g u = b−1 trχ.

(3.8)

Thus, we may rewrite the error term E as:
Z Z +∞
Ef (t, x) =eiλu(t,x,ω) b−1 (t, x, ω)trχ(t, x, ω)f (λω)λ3 dλdω.
S2

(3.9)

0

3.1.2. Some norms. We define some norms on H. For any 1 ≤ p ≤ +∞ and for any
tensor F on Hu , we have:
!1
Z
Z
1

kF kLp (Hu ) =pp|F | dµt,u

dt
0

,

Pt,u

where dµt,u denotes the area element of Pt,u .
Let x0 a coordinate system on P0,u . By transporting this coordinate system along the nullgeodesics generated by L, we obtain a coordinate system (t, x0 ) of H. We define the following
norms:
Z 1
 21
0 2
|F (t, x )| dt
kF kL∞0 L2t = sup,xx0 ∈P0,ukF kL2 0 L∞
=
t
x

0

sup |F (t, x0 ))|
0≤t≤1.
L2 (P0,u )3.1. GEOMETRIC SET-UP AND MAIN RESULTS

47

3.1.3. Estimates for the space-time foliation. In this section, we collect the estimatesthat are needed to follow the discussion of the control of the error term contained in this chapter.
An outline of the proof of these estimates will be given in Chapter 4 (see [75] for the complete
proof).
We start with the regularity in (t, x) of the lapse b and the second fundamental for χ. We
need:ktrχkL∞ (M) + k∇trχkL∞0 L2t + kb − 1kL∞ (M) + k∇bkL∞+ kb
χkL∞0 L2t + k∇
/χ
bkL∞
. ε.
2
2
u L (Hu )u L (Hu )xx(3.10)
Remark 3.4. In this section, all estimates hold for any ω ∈ S2 with the constant in theright-hand side being independent of ω. Thus, one may take the supremum in ω everywhere. To
ease the notations, we do not explicitly write down this supremum.
We also need an estimate for two derivatives of trχ with respect to ∇Njk∇N Pj (∇N trχ)kL2 (Hu ) . ε2j + 2 2 µ(u),(3.11)

where µ in a function satisfying:kµkL2 (R) . ε.
Next, we consider the regularity with respect to ω. We have:k∂ω bkL∞ (M) . ε,

(3.12)|N (t, x, ω) − N (t, x, ω 0 )| ' |ω − ω 0 |, ∀(t, x) ∈ M, ω, ω 0 ∈ S2 ,(3.13)k∂ω N kL∞ (M) . 1.

(3.14)

andFurthermore, we have the following decomposition for χb:χb = χ1 + χ2 ,(3.15)

where χ1 and χ2 are two symmetric traceless Pt,u -tangent 2-tensors satisfying in particular, forany 2 ≤ p < +∞:kχ1 kLpt L∞0 + k∂ω χ2 kL6− (Hu ) . ε.
(3.16)xRemark 3.5. The point of decomposition (3.15) is that χ1 has a better regularity with respect
to (t, x) than χb, while χ2 has a better regularity with respect to ω than χb (see explanation insection 4.4.1).
Finally, we need to compare quantities evaluated at two angles ω and ν in S2 satisfyingj|ω − ν| . 2− 2 . We have the following decomposition for N (t, x, ω) − N (t, x, ν):j

2 2 (N (t, x, ω) − N (t, x, ν)) = F1j (t, x, ν) + F2j (t, x, ω, ν)
where the tensor F1j does not depend on ω and satisfies:
kF1j kL∞ . 1,
and the tensor F2j satisfies:j

kF2j kL∞. 2− 2 .
2u L (Hu )(3.17)

48

3. CONTROL OF THE ERROR TERM

2
Here L∞u L (Hu ) is defined with respect to u = u(t, x, ω). We also have the following decomposition for trχ:trχ(t, x, ω) = f1j (t, x, ν) + f2j (t, x, ω, ν)(3.18)

where the scalar f1j does not depend on ω and satisfies:
kf1j kL∞ . ε,and where the scalar f2j satisfies:j

kf2j kL∞. ε2− 2 .
2u L (Hu )3.1.4. Main result. The main result of this chapter is the following.
Theorem 3.6. Let u be a function on M × S2 satisfying suitable assumptions (we refer to[76] for the complete set of assumptions, and to section 3.1.3 for some typical assumptions).
Let E the Fourier integral operator with phase u(t, x, ω) and symbol g u:
Z Z +∞
Ef (t, x) =eiλu(t,x,ω) b−1 (t, x, ω)trχ(t, x, ω)f (λω)λ3 dλdω.
(3.19)
S2

0

Then, E satisfies the estimate:kEf kL2 (M) . εkλf kL2 (R3 ) .

(3.20)

3.1.5. Geometric Littlewood-Paley projections on the 2-surfaces Pt,u . Throughout
the paper, we will use the geometric Littlewood-Paley projections on 2-surfaces (Pt,u in our case)constructed in [41]. In that paper, the following properties are proved
Theorem 3.7. The LP-projections Pj verify the following properties:i) Lp -boundedness For any 1 ≤ p ≤ ∞, and any interval I ⊂ Z,
kPI F kLp (Pt,u ) . kF kLp (Pt,u )ii)(3.21)

Bessel inequality
XkPj F k2L2 (Pt,u ) . kF k2L2 (Pt,u )

j

iii)

Finite band property

For any 1 ≤ p ≤ ∞.

k∆
/ Pj F kLp (Pt,u ) . 22j kF kLp (Pt,u )kPj F kLp (Pt,u ). 2−2j k∆/ F kLp (Pt,u ) .

(3.22)In addition, the L2 estimatesk∇/ Pj F kL2 (Pt,u ) . 2j kF kL2 (Pt,u )kPj F kL2 (Pt,u ). 2−j k∇
/ F kL2 (Pt,u )hold together with the dual estimatekPj ∇/ F kL2 (Pt,u ) . 2j kF kL2 (Pt,u )iv)

Weak Bernstein inequality

For any 2 ≤ p < ∞(1− p2 )jkPj F kLp (Pt,u ) . (2+ 1)kF kL2 (Pt,u ) ,kP<0 F kLp (Pt,u ) . kF kL2 (Pt,u )(3.23)

3.2. CONTROL OF THE ERROR TERM

49

together with the dual estimates(1− p2 )jkPj F kL2 (Pt,u ) . (2+ 1)kF kLp0 (Pt,u ) ,kP<0 F kL2 (Pt,u ) . kF kLp0 (Pt,u )
3.2. Control of the error term
3.2.1. The basic computation. We start the proof of Theorem 3.6 with the following
instructive computation:
Z
Z +∞
−1
eiλu f (λω)λ2 dλkEf kL2 (M) ≤b(t, x, ω) trχ(t, x, ω)
dω
2
S2
0
Z +∞ L (M)Z(3.24)dωeiλu f (λω)λ2 dλ≤kb(t, x, ω)−1 trχ(t, x, ω)kL∞2
u L (Hu )
S2

0

L2u

2

≤ εkλ f kL2 (R3 ) ,where we have used Plancherel with respect to λ, Cauchy-Schwarz with respect to ω, the estimates (3.10) for b and trχ. (3.24) misses the conclusion (3.20) of Theorem 3.6 by a powerof λ. Now, assume for a moment that we may replace a power of λ by a derivative onb(t, x, ω)−1 trχ(t, x, ω). Then, the same computation yields:
Z Z +∞∇(b(t, x, ω)−1 trχ(t, x, ω))eiλu f (λω)λdλdω2L2 (M)Z +∞Z S 0(3.25)eiλu f (λω)λ2 dλdω≤k∇(b(t, x, ω)−1 trχ(t, x, ω))kL∞
2
u L (Hu )S20≤ εkλf kL2 (R3 ) ,L2uwhere we used the fact thatk∇(b(t, x, ω)−1 trχ(t, x, ω))kL∞.ε2u L (Hu )(3.26)

in view of (3.10). Note that the estimate provided by (3.25) is consistent with the control ofthe error term (3.20). This suggests a strategy which consists in making integrations by parts
to trade powers of λ against derivatives of the symbol b(t, x, ω)−1 trχ(t, x, ω).
3.2.2. Structure of the proof of Theorem 3.6. The proof of Theorem 3.6 proceeds inthree steps. We first localize in frequencies of size λ ∼ 2j . We then localize the angle ω inpatches on the sphere S2 of diameter 2−j/2 . Finally, we estimate the diagonal terms.
3.2.2.1. Step 1: decomposition in frequency. For the first step, we introduce ϕ and ψ twosmooth compactly supported functions on R such that:X
ϕ(λ) +
ψ(2−j λ) = 1 for all λ ∈ R.
(3.27)
j≥0

We use (3.27) to decompose Ef as follows:
Ef (t, x) =Xj≥−1

Ej f (t, x),(3.28)

50

3. CONTROL OF THE ERROR TERM

where for j ≥ 0:
Z Z +∞
Ej f (t, x) =S2

andeiλu b(t, x, ω)−1 trχ(t, x, ω)ψ(2−j λ)f (λω)λ2 dλdω,(3.29)0Z Z +∞
E−1 f (t, x) =S2

eiλu b(t, x, ω)−1 trχ(t, x, ω)ϕ(λ)f (λω)λ2 dλdω.

(3.30)

0This decomposition is classical and is known as the first dyadic decomposition (see [68]). The
goal of this first step is to prove the following proposition:
Proposition 3.8. The decomposition (3.28) satisfies an almost orthogonality property, fromwhich it follows that:X(3.31)
kEj f k2L2 (M) + ε2 kf k2L2 (R3 ) .
kEf k2L2 (M) .
j≥−1

A discussion of the proof of Proposition 3.8 is postponed to section 3.3.
3.2.2.2. Step 2: decomposition in angle. Proposition 3.8 enables us to estimate kEj f kL2 (M)instead of kEf kL2 (M) . The analog of computation (3.24) for kEj f kL2 (M) yields:
kEj f kL2 (M) ≤ εkλψ(2−j λ)f kL2 (σ) . ε2j kψ(2−j λ)f kL2 (R3 ) ,(3.32)

which misses the wanted estimate by a power of 2j . We thus need to perform a second dyadic
decomposition (see [68]). We introduce a smooth partition of unity on the sphere S2 :X
ηjν (ω) = 1 for all ω ∈ S2 ,(3.33)
ν∈Γjwhere Γ is a lattice on S2 of size 2− 2 , where the support of ηjν is a patch on S2 of diameter∼ 2−j/2 . We use (3.33) to decompose Ej f as follows:X
Ej f (t, x) =Ejν f (t, x),(3.34)
ν∈Γ

where:
Ejν f (t, x) =We also define:Z Z +∞
S2

eiλu b(t, x, ω)−1 trχ(t, x, ω)ψ(2−j λ)ηjν (ω)f (λω)λ2 dλdω.

(3.35)

0γ−1 = kϕ(λ)f kL2 (R3 ) , γj = kψ(2−j λ)f kL2 (R3 ) , j ≥ 0,γjν = kψ(2−j λ)ηjν (ω)f kL2 (R3 ) , j ≥ 0, ν ∈ Γ,

(3.36)

which satisfy:
kf k2L2 (R3 ) =

X

γj2 =

j≥−1

X X

(γjν )2 .

(3.37)

j≥−1 ν∈ΓThe goal of this second step is to prove the following proposition:
Proposition 3.9. The decomposition (3.34) satisfies an almost orthogonality property, fromwhich it follows thatX
kEj f k2L2 (M) .
kEjν f k2L2 (M) + ε2 γj2 .
(3.38)
ν∈Γ

A discussion of the proof of Proposition 3.9 is postponed to section 3.5.

3.2. CONTROL OF THE ERROR TERM

51

3.2.2.3. Step 3: control of the diagonal term. Proposition 3.9 allows us to estimate kEjν f kL2 (M)instead of kEj f kL2 (M) . The analog of computation (3.24) for kEjν f kL2 (M) yields:
kEjν f kL2 (M)
Z +∞
Z
−1
eiλu ψ(2−j λ)ηjν (ω)f (λω)λ2 dλkb(t, x, ω) trχ(t, x, ω)kL∞≤2
u L (Hu )S20(3.39)dω
L2uq≤ 2j ε vol(supp(ηjν ))kλψ(2−j λ)ηjν (ω)f kL2 (R3 ). ε2j/2 γjν ,where the termqvol(supp(ηjν )) comes from the fact that we apply Cauchy-Schwarz in ω. Notethat we have used in (3.39) the fact that the support of ηjν is 2 dimensional and has diameter2−j/2 so that:qvol(supp(ηjν )) . 2−j/2 .
(3.40)
Now, (3.39) still misses the wanted estimate by a power of 2j/2 . Nevertheless, using more refinedtechniques, we are able to estimate the diagonal term:
Proposition 3.10. The diagonal term Ejν f satisfies the following estimate:kEjν f kL2 (M) . εγjν .

(3.41)A discussion of the proof of Proposition 3.10 is postponed to section 3.4.
Remark 3.11. Note that Proposition 3.9 together with Proposition 3.10 yields the estimate:
kEj f kL2 (M) . εγj .

(3.42)

Now, since the proof of Proposition 3.9 and the proof of Proposition 3.10 do not depend on the
proof of Proposition 3.8, we are allowed to use the conclusion of Proposition 3.9 and Proposition
3.10 in the proof of Proposition 3.8. In particular, the estimate (3.42) will be used for the proofof Proposition 3.8. In the same spirit, since the proof of Proposition 3.10 does not depend on the
proof of Proposition 3.9, we are allowed to use the conclusion of Proposition 3.10 in the proofof Proposition 3.9.
Convention. In the rest of this chapter, we will use several integration by parts. In turn,these integration by parts will each generate a large number of terms. For the sake of simplicity,
we will only discuss few typical terms. We will constantly use the notation ”+ · · · ” in various
identities and estimates in order to refer to the additional terms. That is not to say that these
additional terms are lower order or estimated in the same way, but simply that the typical terms
that we exhibit allow for a simple exposition of the main ideas of the proof. We refer the reader
to [76] for a complete proof which contains the control of the typical terms discussed here aswell as the numerous additional terms.

52

3. CONTROL OF THE ERROR TERM

3.2.2.4. Proof of Theorem 3.6. Proposition 3.8, 3.9 and 3.10 immediately yield the proof of
Theorem 3.6. Indeed, (3.31), (3.37), (3.38) and (3.41) imply:X
kEj f k2L2 (M) + ε2 kf k2L2 (R3 )kEf k2L2 (M) .
j≥−1
X X

kEjν f k2L2 (M) + ε2

.

j≥−1
ν∈Γ
X
X
2

.ε

(γjν )2 + ε2

j≥−1 ν∈ΓX

γj2 + ε2 kf k2L2 (R3 )X j≥−1γj2 + ε2 kf k2L2 (R3 )(3.43)

j≥−1. ε2 kf k2L2 (R3 ) ,which is the conclusion of Theorem 3.6.
The rest of this chapter is dedicated to a discussion of the proof of Propositions 3.8, 3.9 and3.10. The details of the proofs being very involved, we only give a very sketchy summary of themain ideas. We refer the reader to [76] for the details.
3.3. Almost orthogonality in frequency
We have to prove (3.31):
XkEf k2L2 (M) .

kEj f k2L2 (M) + ε2 kf k2L2 (R3 ) .

(3.44)

j≥−1

This will result from the following inequality using Shur’s Lemma:Z|j−k|
Ej f (t, x)Ek f (t, x)dM . ε2 2− 4 γj γk for |j − k| > 2.

(3.45)

M

In turn, (3.45) will follow from integrations by parts in u.
3.3.1. A first integration by parts. From now on, we focus on proving (3.45). We may
assume j ≥ k + 3. We have:Z
Ej f (t, x)Ek f (t, x)dM
(3.46)M
Z Z +∞ Z Z +∞ Ziλu−iλ0 u0
−1
0
−1
0
=
e
b(t, x, ω) trχ(t, x, ω)b(x, ω ) trχ(t, x, ω )dM
S2

0
−j

×ψ(2

S2

0
2

M
−k 0

λ)f (λω)λ ψ(2λ )f (λ0 ω 0 )(λ0 )2 dλdωdλ0 dω 0 .

We consider the coordinate system (t, u, x0 ) on M, and we would like to integrate by parts
with respect to ∂u in (3.46). Since ∇u = b−1 N and ∇u0 = b0 −1 N 0 , we have:0 0eiλu−iλ u = −

i

∂u (e
λ − λ0 bb0 g(N, N 0 )iλu−iλ0 u0),(3.47)

where we use the notation u for u(t, x, ω), b for b(t, x, ω), N for N (t, x, ω), u0 for u(t, x, ω 0 ), b0
for b(t, x, ω 0 ) and N 0 for N (x, ω 0 ). We will also use the notation trχ for trχ(t, x, ω) and trχ0 for3.3. ALMOST ORTHOGONALITY IN FREQUENCY

trχ(t, x, ω 0 ). Using (3.47), we obtain:ZZiλu−iλ0 u0 0ebb dM = ib−1 ∂u trχb0 −1 trχ0dM
λ − λ0 bb0 g(N, N 0 )MZ−1 trχ∂ (b0 −1 trχ0 )0 0bu
eiλu−iλ u+idM + · · · ,b0λ − λ b0 g(N, N 0 )
M

M

53

0 0

eiλu−iλ u(3.48)

where the additional terms in (3.48) arise when ∂u falls on the volume element of M or on the
denominator in the right-hand side of (3.47). Note that:
λ0 b
1
λ0 b
0
g(N,
N
)
≤
≤ + O(ε) < 1,
0
0
λb
λ b
2
where we used the estimate (3.10) satisfied by b and b0 and the fact that j ≥ k + 3 so that
λ0 ≤ λ/2. Thus, we may expand the fraction in (3.48):!p
X 1 λ0 b0 g(N, N 0 )1b=.
(3.49)λλ
λ − λ0 bb0 g(N, N 0 )p≥0Remark 3.12. The expansion (3.49) generates quantities of the typeZ +∞eiλu ψ(2−j λ)f (λω)(2−j λ)p λ2 dλ.
0where p ∈ Z. For simplicity, we omit the index p and denote them byZ +∞Fj (u) =eiλu ψ(2−j λ)f (λω)λ2 dλ.

(3.50)

0since they are essentially equivalent. Note that Plancherel yields:
kFj kL2ω,u ≤ kψ(2−j λ)f (λω)λkL2 (R3 ) . 2j γj .

(3.51)

Also, using Cauchy-Schwarz in λ, we have3jj

≤ 2 2 kψ(2−j λ)f (λω)λkL2 (R3 ) . 2 2 γj .
kFj kL2ω L∞
u(3.46), (3.48) and (3.49) imply:Z
Ej f (t, x)Ek f (t, x)dM

(3.52)(3.53)

M

= 2

−j

Z

Z

+2−j

 Z

b0 −1 trχ0 Fk (u0 )dω 0∇N trχFj (u)dω
M

S2

Z

Z

M


dM

S2

S2

 Z

−1
0
0
0
0
∇(b trχ )Fk (u )dω dM + · · · ,trχN Fj (u)dω ·S2where we only kept the first term in the expansion (3.49) in order to simplify the exposition1.
Remark 3.13. The second term in the right-hand side of (3.53) is easier because the derivative falls on the low frequency term. This is why we estimate this term directly while the otherterm requires a more elaborate treatment which is explained in section 3.3.2.
1note that in the last term in the right-hand side of (3.53), we wrote ∇ (b0 −1 trχ0 ) as N · ∇(b0 −1 trχ0 )N

54

3. CONTROL OF THE ERROR TERM

We estimate the second term in the right-hand side of (3.53). We have:
 Z

Z Z
−j
−1
0
0
0
0
trχN Fj (u)dω ·2∇(b trχ )Fk (u )dω dM
S2

M

. 2−j

(3.54)

S2

Z

Z

∇(b0

trχN Fj (u)dωS2L2 (M)We have the following analog of (3.42):
Z
trχN Fj (u)dωS2−1

trχ0 )Fk (u0 )dω 0

S2.
L2 (M). εγj .

(3.55)

L2 (M)

Indeed, one can show that the symbol trχ satisfies regularity assumptions which are at least as
good as b−1 trχ (see [75], and also section 6.1.5), so that the proof of (3.42) may adapted in astraightforward manner to obtain (3.55).
Next, we consider the second term in the right-hand side of (3.54). Then proceeding as in
the basic computation (3.24), and using the estimate (3.26), we obtainZ−1∇(b0 trχ0 )Fk (u0 )dω 0. εkλψ(2−k λ)f kL2 (R3 ) . ε2k γk .
(3.56)S2

L2 (M)

Together with (3.54) and (3.55), we finally obtain:

 Z
Z Z
−j
−1
0
0
0
0
∇(b trχ )Fk (u )dω dM . 2−j+k γk γj ,2trχN Fj (u)dω ·M(3.57)S2

S2

which is consistent with (3.45).
Remark 3.14. Estimating the first term in the right-hand side of (3.53) in the same way
would only yield:

 Z
Z Z
−j
−1
0
0
0
0
(3.58)b trχ Fk (u )dω dM . ε2 γj γk ,2
∇N trχFj (u)dω
M

S2

S2

which is not sufficient to obtain (3.45).
3.3.2. A more precise estimate. In this section, we estimate the first term the righthand side of (3.53). Using the geometric Littlewood-Paley projections on the 2-surfaces Pt,u , wedecompose ∇N trχ as:∇N trχ = P≤ j+k (∇N trχ) + P> j+k (∇N trχ).
2

2

In turn, this yields a decomposition for the first term in the right-hand side of (3.53):
 Z

Z Z
−j
−1
0
0
0
0
b trχ Fk (u )dω dM = A1 + A2 ,2
∇N trχFj (u)dωMS2(3.59)

S2

where:−jZ

Z

A1 = 2A2 = 2−j

ZM ZS2
MS2


P> j+k (∇N trχ)Fj (u)dω Ek f (t, x)dM,
2
P≤ j+k (∇N trχ)Fj (u)dω Ek f (t, x)dM.
2(3.60)

3.3. ALMOST ORTHOGONALITY IN FREQUENCY

55

We first estimate the easier term A1 . The definition of Pl implies Pl = 2−2l ∆/ Pl , and thusX
P> j+k (∇N trχ) =Pl (∇N trχ)2

l> j+k2=X

2−2l ∆/ Pl (∇N trχ),l> j+k2

which yields the following decomposition for A1 :
X
A1 =
A1,l

(3.61)

l> j+k2where A1,l is given by:
−j−2l

Z

Z


∆
/ Pl (∇N trχ)Fj (u)dω Ek f (t, x)dM.

A1,l = 2

M

S2

Integrating by parts ∆/ on Pt,u and using the fact that ∇/ Fj,−1 (u) = 0, we obtain:!
Z Z
Z
A1,l = −2−j−2l∇/ Pl (∇N trχ)∇/ (Ek f (t, x)b)dµt,u Fj (u)du dt dω + · · · ,
S2

t,u

Pt,u

where the additional term corresponds to the case where the derivative falls on the volume
element of M. Next, we apply Cauchy-Schwartz to the integral on M and obtain:Z|A1,l | ≤ 2−j−2lk∇/ Pl (∇N trχ)Fj (u)kL2 (M) k∇/ Ek kL2 (M) dω
(3.62)
S2
Z
−j−2l
k∇
/ Pl (∇N trχ)kL∞
kFj (u)kL2u k∇/ Ek kL2 (M) dω. 2
2u L (Hu )2
ZSk∇N trχkL∞
kFj (u)kL2u k∇/ Ek kL2 (M) dω. 2−j−l2u L (Hu )2
ZS
kFj (u)kL2u k∇Ek kL2 (M) dω,. ε2−j−lS2where we used the finite band property for Pl and the estimates (3.10) for trχ. In view of (3.62),
we also need to estimate k∇Ek kL2 (M) . We have:
Z Z +∞∇Ek f (t, x) =eiλu ∇(b−1 trχ)ψ(2−k λ)f (λω)λ2 dλdωS2 0Z Z
(3.63)+∞+i2k
eiλu b−1 trχ∇uψ(2−k λ)(2−k λ)f (λω)λ2 dλdω.
S20Using the basic computation (3.24) for the first term together with the estimate (3.26), and (3.42)
for the second term together with the fact that trχL satisfies the same regularity assumptions
than b−1 trχ, we obtain:k∇Ek kL2 (M) . ε2k γk .
(3.64)(3.62), (3.51), and (3.64) yield:|A1,l | . ε2 2−l+k ε2 γj γk .

56

3. CONTROL OF THE ERROR TERM

Together with (3.61), this yields:




j−k
 X −l  k 2|A1 | . 
2  ε2 ε γj γk . ε2 2− 2 γj γk ,(3.65)

l> j+k2

which is consistent with (3.45).
3.3.3. A second integration by parts in u. To estimate A2 , we perform a second integration by parts relying again on (3.47). This leads to:
Z Z−2j∇N P≤ j+k (∇N trχ)Fj (u)Ek f (t, x)dM + · · · ,(3.66)
A2 = 2M

2

S2

where we only keep the worst term, which is the one containing two derivatives of trχ. It is at
this stage that we need the estimate (3.11) for ∇N Pl (∇N trχ) which we recall now. We have:lk∇N Pl (∇N trχ)kL2 (Hu ) . ε2l + 2 2 µ(u),(3.67)

where µ in a function satisfying:kµkL2 (R) . ε.
In view of the estimate (3.67), we have:∇N P≤ j+k (∇N trχ)2

L2 (Hu ).

Xk∇N Pl (∇N trχ)kL2 (Hu )l≤ j+k2.

Xl(2l ε + 2 2 µ(u))

l≤ j+k2
j+kj+k. ε2 2 + 2 4 µ(u).
In view of (3.66), this yields after applying Cauchy-Schwartz:
Z−2j∇N P≤ j+k (∇N trχ) 2|A2 | . 2 kEk f kL2 (M)
−2j

Z

j+k

L (Hu )

2S2dω + · · ·Fj (u)(3.68)

L2u

j+k(ε2 2 + ε2 4 µ(u))Fj (u) 2 dω + · · ·
Lu
S2


Z
Z
j+k
j+k
−2j
. 2 εγk ε2 2
kFj (u)kL2u dω + 2 4kµkL2 (R) kFj (u)kL∞dω + · · ·
u

. 2

εγk

S2

. 2

− j−k
4

S2

ε2 γ k γ j + · · · ,where we used (3.42) for Ek f , Cauchy-Schwarz in ω, and the estimates (3.51) and (3.52) forFj (u).

3.4. CONTROL OF THE DIAGONAL TERM

57

3.3.4. End of the proof of Proposition 3.8. In view of (3.53), (3.57), (3.59), (3.65) and(3.68), we obtain:Z|j−k|
Ej f (t, x)Ek f (t, x)dM . ε2 2− 4 γj γk for |j − k| > 2.
(3.69)
M

Finally, (3.69) together with Shur’s Lemma yields:X
kEj f k2L2 (M) + ε2 kf k2L2 (R3 ) .
kEf k2L2 (M) .

(3.70)

j≥−1

This concludes the proof of Proposition 3.8.

3.4. Control of the diagonal term
Since the orthogonality argument in angle is the core of this chapter, we choose to deal firstwith the control of the diagonal term in this section. We will then proceed with the orthogonality
argument in angle in the rest of the chapter.
In order to control the diagonal term, we have to prove (3.41):
kEjν f kL2 (M) . εγjν .

(3.71)b−1 (t, x, ω)trχ(t, x, ω)Fj (u)ηjν (ω)dω,(3.72)

Recall that Ejν is given by:
Ejν f (t, x) =Z
S2

where Fj (u) is defined by:Z +∞Fj (u) =eiλu ψ(2−j λ)f (λω)λ2 dλ.

(3.73)

0

The proof of the estimate (3.71) will proceed in four steps:
Step 1. We first consider a decomposition roughly of the type:
Z

ν
−1
ν
Ej f (t, x) = b (t, x, ν)trχ(t, x, ν)Fj (u)ηj (ω)dω + · · · ,S2

so that we have to prove estimate (3.71) with b−1 trχ replaced by 1.
Step 2. That estimate is obtained by considering the transport equation along Lν :
Z

ν
Lν
Fj (u)ηj (ω)dω = · · · .
S2

Step 3. A certain term in the transport equation of Step 2 needs to be estimated using anenergy estimate for the wave equation.
Step 4. We conclude the proof using the estimates obtained in Step 2 and Step 3.

58

3. CONTROL OF THE ERROR TERM

3.4.1. Step 1: freezing the ω dependance in b−1 trχ. In view of the estimate (3.12)
for ∂ω b, the estimate (3.10) for b, and the decomposition (3.18) for trχ, we have:b−1 (t, x, ω)trχ(t, x, ω) = f1j (t, x, ν) + f2j (t, x, ν, ω),(3.74)

where f1j only depends on (t, x, ν) and satisfies:
kf1j kL∞ . ε,

(3.75)

and where f2j satisfies:j. 2− 2 ε,
kf2j kL∞2
u L (Hu )(3.76)

with u = u(., ω). (3.74) yields the following decomposition for the diagonal term:ZZ
Fj (u)ηjν (ω)dω +Fj (u)f2j (t, x, ω, ν)ηjν (ω)dω,
Ejν f (t, x) = f1j (t, x, ν)
S2

S2

which implies:
Ejν f (t, x) L2 (M)ZjFj (u)ηjν (ω)dω. kf1 kL∞ (M)
S2

Z
. ε

S2

Fj (u)ηjν (ω)dω(3.77)Z+
L2 (M)S2η ν (ω)dωkFj (u)kL2u kf2j kL∞
2
u L (Hu ) j+ εγjν ,L2 (M)

where we used in the last inequality the estimates (3.75) and (3.76), Cauchy-Schwarz in ω, thesize of the patch, and the estimate (3.51) for Fj (u).
Remark 3.15. The point of the decomposition (3.74) is to allow us to replace in the diagonalterm (3.72) the symbol b−1 trχ with 1. An obvious way to achieve this is to write the following
decomposition:b−1 trχ(t, x, ω) = b−1 trχ(t, x, ν) + (b−1 trχ(t, x, ω) − b−1 trχ(t, x, ν)).

(3.78)

The first term clearly satisfies (3.75) in view of the estimate (3.10) for b and trχ. On the other2
hand, we obtain in [75] (see also (4.49)) the estimate ∂ω trχ ∈ L∞
u L (Hu ) which together withthe estimate (3.12) for ∂ω b yields:k∂ω (b−1 trχ)kL∞. ε.
2u L (Hu )(3.79)

Now, we have:b−1 trχ(t, x, ω) − b−1 trχ(t, x, ν) = (ω − ν)Z 1∂ω (b−1 trχ)(t, x, ωσ )dσ02∞ 2
which together with (3.79) is not enough to conclude since L∞
u L (Hu ) and Luσ L (Huσ ) are notcomparable. We refer the reader to [75] where the decomposition (3.74) as well as several others
are proved (see also the discussion in section 4.5).

The following proposition allows us to estimate the right-hand side of (3.77).

3.4. CONTROL OF THE DIAGONAL TERM

Proposition 3.16. We have the following bound:Z
Fj (u)ηjν (ω)dωS2L2uν ,xν L∞t59. γjν .

(3.80)

Remark 3.17. In order to control the diagonal term, it suffices to have a bound of the L2 (M)norm for the left-hand side of (3.80). The improvement to a bound for the L2uν ,xν L∞t norm willbe crucial when proving the almost orthogonality in angle.
Assuming the result of the proposition, estimates (3.80) and (3.77) yield:
Ejν f (t, x) L2 (M) . εγjν ,which together with (3.72) and (3.77) implies:
kEjν f kL2 (M) . εγjν .
which is the wanted estimate (3.71). This concludes the proof of Proposition 3.10.
3.4.2. Step 2: A transport equation in the Lν direction. We still need to prove
Proposition 3.16. Note that it suffices to show:Zν. γjν .
(3.81)LνFj (u)ηj (ω)dωS2L2 (M)

Now, since the space-time gradient of u is given by b−1 L, we have:
Z

Z


ν
j
Lν
b−1 g L(t, x, ω), L(t, x, ν) Fj (u)ηjν (ω)dω,Fj (u)ηj (ω)dω = 2
S2

S2

where Fj has been defined in (3.50). In view of (3.82), we have:
Z

ν
Lν
Fj (u)ηj (ω)dωS2Z= 2j b−1 (t, x, ν)g(L(t, x, ω), L(t, x, ν))Fj (u)ηjν (ω)dω
2
S
Z
j
−1
+2
(b (t, x, ω) − b−1 (t, x, ν))g(L(t, x, ω), L(t, x, ν))Fj (u)ηjν (ω)dω.

(3.82)

S2

Next, we estimate the second term in the right-hand side of (3.82). We have:g L(t, x, ω), L(t, x, ν) = g(N (t, x, ω) − N (t, x, ν), N (t, x, ω) − N (t, x, ν)).

(3.83)

Thus, the estimate (3.14) for ∂ω N and the size of the patch yields:
kg(L(t, x, ω), L(t, x, ν))kL∞ (Hu ) . 2−j ,(3.84)

which implies:Z(b−1 (t, x, ω) − b−1 (t, x, ν))g(L(t, x, ω), L(t, x, ν))Fj (u)ηjν (ω)dω
S2

Z
.
. 2

(3.85)
L2 (M)

kb−1 (t, x, ω) − b−1 (t, x, ν)kL∞
kg(L(t, x, ω), L(t, x, ν))kL∞ kFj (u)kL2u ηjν (ω)dω2u L (Hu )S2−jεγjν ,

60

3. CONTROL OF THE ERROR TERM

where we used in the last inequality (3.84), the estimate (3.12) for ∂ω b, Cauchy-Schwarz in ω,
the size of the patch, and (3.51) for Fj (u). (3.82) together with (3.85) and the estimate (3.12)
for ∂ω b yields:
ZνFj (u)ηj (ω)dωLν
(3.86)
S2

Z

. 2j

S2

L2 (M)g(L(t, x, ω), L(t, x, ν))Fj (u)ηjν (ω)dω+ εγjν .
L2 (M)

Next, we estimate the right-hand side of (3.86). Using (3.83), the decomposition (3.17) for
N − N 0 , and arguing as in (3.77), we obtain:ZZνg(L(t, x, ω), L(t, x, ν))Fj (u)ηj (ω)dω.
(ω − ν)2 Fj (u)ηjν (ω)dω+ 2−j γjν .
S2L2 (M)

Together with (3.86), this implies:
Z

ν
Lν
Fj (u)ηj (ω)dω
S2Z.

L2 (M)S2

S2

L2 (M)

j(2 2 (ω − ν))2 Fj (u)ηjν (ω)dω+ γjν .

(3.87)

L2 (M)

Finally, we need to estimate the first term in the right-hand side of (3.87). We will rely on
the energy estimate for the wave equation.
3.4.3. Step3: The energy estimate for the wave equation. Recall from (3.8) that:g u = b−1 trχ.
Thus, we have:
 Z
Z
j
g
(ω − ν)2 Fj (u)ηjν (ω)dω =b−1 (t, x, ω)trχ(t, x, ω)(2 2 (ω − ν))2 Fj (u)ηjν (ω)dω. (3.88)
S2

S2

Arguing as in (3.77), we may replace b−1 trχ by 1:Zjb−1 (t, x, ω)trχ(t, x, ω)(2 2 (ω − ν))2 Fj (u)ηjν (ω)dω
S2

Z
. ε

S2

L2 (M)
j(2 2 (ω − ν))2 Fj (u)ηjν (ω)dω+ εγjν
L2 (M)

which together with (3.88) implies:
Z

2
ν
g
(ω − ν) Fj (u)ηj (ω)dω
S2

Z
. ε

S2

(3.89)
L2 (M)

j(2 2 (ω − ν))2 Fj (u)ηjν (ω)dω+ εγjν .
L2 (M)Let φ be the scalar function in the left-hand side of (3.88), i.e.:Z(ω − ν)2 Fj (u)ηjν (ω)dω.
φ=
S2

3.4. CONTROL OF THE DIAGONAL TERM

61

Then, using the energy estimate for the wave equation (2.84) we obtain:kDφk2L∞ L2 (Σt ) . k∇φ(0, .)k2L2 (Σ0 ) + kT φ(0, .)k2L2 (Σ0 ) + kg φkL2 (M) kDφkL2 (M)tZ+Qαβ π αβ dM ,(3.90)
M

where π is the deformation tensor of T
παβ = Dα Tβ + Dβ Tα .
and Qαβ on M is the energy momentum tensor associated to φ1Qαβ = Qαβ [φ] = ∂α φ∂β φ − gαβ (gµν ∂µ φ∂ν φ) .
2
The control of the parametrix at initial time in [74] (see also Chapter 5) yieldsk∇φ(0, .)kL2 (Σ0 ) + kT φ(0, .)kL2 (Σ0 ) . γjν .

(3.91)

Next, we consider the last term in the right-hand side of (3.90). From the maximal foliation
assumption, π is traceless, so thatQαβ π αβ = π αβ ∂α φ∂β φ
Z Z
2j
= 2
πN N 0 (ω − ν)2 (ω 0 − ν)2 Fj (u)Fj (u0 )ηjν (ω)ηjν (ω 0 )dωdω 0
S2

S2

Using (3.14), one obtains
Qαβ π αβ = 22j πNν Nν φ2 + · · · ,and thusQαβ π αβ ' πNν Nν (Dφ)2 + · · · .

(3.92)

It turns out that we have a trace estimate for πNν Nν (see details in [76]):kπNν Nν kL∞

L2uν ,x0 ν t.εwhich together with (3.92) implies
kQαβ π αβ kL2 (M) . kπNν Nν kL∞uν ,x0 ν

kDφkL2 (M)L2t kDφkL2uν ,xν L∞
t(3.93). ε kDφkL2u ,x L∞ kDφkL2 (M) .
ννtFinally, (3.89)-(3.93) implies
kDφkL2 (M) . εkDφkL2uν ,xν L∞+ γjν ,twhich is equivalent toZj(2 2 (ω − ν))2 Fj (u)ηjν (ω)dωS2Z.εL2 (M)

S2

j(2 2 (ω − ν))2 Fj (u)ηjν (ω)dωL2uν ,xν L∞t+ γjν .
(3.94)

In view of (3.94) and (3.87), we obtain:ZZjFj (u)ηjν (ω)dω. ε(2 2 (ω − ν))2 Fj (u)ηjν (ω)dωLν
S2

L2 (M)

S2

L2uν ,xν L∞t+ γjν ,

62

3. CONTROL OF THE ERROR TERM

and thus:ZS2

Fj (u)ηjν (ω)dωL2uν ,xν L∞t. γjν ,

which is the desired estimate (3.80).
3.5. Almost orthogonality in angle
We have to prove (3.38):
kEj f k2L2 (M) .

X

kEjν f k2L2 (M) + ε2 γj2 .

(3.95)

ν∈ΓThis will result from an estimate for:Z
M

0

Ejν f (t, x)Ejν f (t, x)dM .

(3.96)Let us introduce integration by parts first with respect to tangential directions, and thenwith respect to L.
3.5.1. Integration by parts.
3.5.1.1. Integration by parts in tangential directions. By definition of ∇/ , we have ∇/h =∇h − (∇N h)N for any function h on Σt . In particular, we have ∇/ (u) = 0 and ∇/ (u0 ) = b0 −1 N 0 −b0 −1 g(N 0 , N )N . Now, since g(N 0 − g(N, N 0 )N, N 0 ) = 1 − g(N 0 , N )2 and ∇u0 = b0 −1 N 0 , wededuce:
i
0 0
0 0
0
∇ 0
(eiλu−iλ u )(3.97)
eiλu−iλ u =λ0 g(N 0 − g(N, N 0 )N, ∇u0 ) N −g(N,N )N
ib0
0 0
=
∇N 0 −g(N,N 0 )N (eiλu−iλ u ),002
λ (1 − g(N , N ) )where we have used the fact that N 0 − g(N, N 0 )N is a tangent vector with respect of the levelsurfaces of u. We consider an oscillatory integral of the following form:
Z Z
0h(t, x)Fj (u)Fj (u0 )ηjν (ω)ηjν (ω 0 )dωdω 0 dM,
M

S2 ×S2where h is a scalar function on M. Integrating by parts once using (3.97) yields:
Z Z
0h(t, x)Fj (u)Fj (u0 )ηjν (ω)ηjν (ω 0 )dωdω 0 dM
M S2 ×S2
Z Z
b0((N 0 − g(N, N 0 )N )(h) + · · · )= −i2−j
0 )2
1
−
g(N,
N
2
2
M S ×S
0×Fj (u)Fj (u0 )ηjν (ω)ηjν (ω 0 )dωdω 0 dM,
where we only kept the term where the derivative falls on h, and neglected for the simplicity ofthe exposition the terms when the derivative falls on the denominator of the right-hand side of(3.97) or on the volume element of M. In view of (3.13), we have:
N 0 − g(N, N 0 )N ∼ N 0 − N ∼ |ω − ω 0 | ∼ |ν − ν 0 |,(3.98)

3.5. ALMOST ORTHOGONALITY IN ANGLE

63

and:
1 − g(N, N 0 ) =g(N − N 0 , N − N 0 )∼ |ω − ω 0 |2 ∼ |ν − ν 0 |2 ,2and we thus obtain:
Z Z

0h(t, x)Fj (u)Fj (u0 )ηjν (ω)ηjν (ω 0 )dωdω 0 dM
22
M S ×S
Z Z
1
0
= i j
b0 ∇
/ (h)Fj (u)Fj (u0 )ηjν (ω)ηjν (ω 0 )dωdω 0 dM + · · · .
02 |ν − ν | M S2 ×S2(3.99)(3.100)Remark 3.18. In the formula (3.100), we neglect two types of terms for the simplicity of the
exposition. First, we neglect the term when the derivative falls on the denominator of the righthand side of (3.97) or on the volume element of M. Next, make the following approximation:1
N 0 − g(N, N 0 )N
∼ j
.
0
2
1 − g(N, N )2 |ν − ν 0 |In the actual proof, we use (3.99) to derive the following expansion:p  0qX
01
1
N − Nν
N − Nν
,=cpq
1 − g(N, N 0 )2|Nν − Nν 0 |2|Nν − Nν 0 ||Nν − Nν 0 |(3.101)p,q≥0

for some explicit real coefficients cpq such that the seriesX
cpq xp y q
p,q≥0

has radius of convergence 1. Then, (3.100) corresponds to the first term in the expansion (3.101)
with the additional simplification which consists in replacing |Nν − Nν 0 | with |ν − ν 0 | again inview of (3.99). While these approximations greatly simplify the exposition, they still allow us toexhibit typical terms in the proof of the almost orthogonality in angle.
3.5.1.2. Integration by parts in L. Next, we also introduce integrations by parts with respect
to L. Since L(u) = 0 and L(u0 ) = b0 −1 g(L, L0 ), we have:0 0eiλu−iλ u =ib00 0L(eiλu−iλ u ).
00
λ g(L, L )We consider an oscillatory integral of the following form:
Z Z
0h(t, x)Fj (u)Fj (u0 )ηjν (ω)ηjν (ω 0 )dωdω 0 dM,
M

S2 ×S2where h is a scalar function on M. Integrating by parts once using (3.97) yields:
Z Z
0h(t, x)Fj (u)Fj (u0 )ηjν (ω)ηjν (ω 0 )dωdω 0 dM
M S2 ×S2
Z Z
b00(L(h) + · · · )Fj (u)Fj (u0 )ηjν (ω)ηjν (ω 0 )dωdω 0 dM,= −i2−j0
M S2 ×S2 g(L, L )(3.102)

64

3. CONTROL OF THE ERROR TERM

where we only kept the term where the derivative falls on h, and neglected for the simplicity ofthe exposition the term when the derivative falls on the denominator of the right-hand side of(3.102) or on the volume element. Using the fact that:g(L, L0 ) = −1 + g(N, N 0 )(3.103)

together with (3.99), and keeping only the first term in the expansion (3.101), with the additional
simplification which consists in replacing |Nν − Nν 0 | with |ν − ν 0 |, we obtain:
Z Z
0h(t, x)Fj (u)Fj (u0 )ηjν (ω)ηjν (ω 0 )dωdω 0 dM
(3.104)
M S2 ×S2
Z Z
1
0
= i j
b0 L(h)Fj (u)Fj (u0 )ηjν (ω)ηjν (ω 0 )dωdω 0 dM + · · · .
02
2 |ν − ν | M S2 ×S2
3.5.2. Presence of a log-loss. Let us explain why proceeding directly by integration byparts in (3.96) results in a log-loss. Let us define Ej,ν,ν 0 as:Z
0Ej,ν,ν 0 =Ejν f (t, x)Ejν f (t, x)dM.
M

We have:Z
Ej,ν,ν 0

Z +∞ Z +∞ Z=eiλu−iλ0 u0 −1

b

0 −1

trχb



0

trχ dM

S2 ×S2 0
0
M
2
ν
ν0
0
−j
×ηj (ω)ηj (ω )ψ(2 λ)ψ(2−j λ0 )f (λω)f (λ0 ω 0 )λ2 λ0 dλdλ0 dωdω 0 .

We integrate by parts tangentially using (3.100). Consider the term where the tangential derivative falls on trχ, which is of the form:
Z
Z +∞ Z +∞ Z10iλu−iλ0 u0 −1
0 −1
e
b ∇
/ trχb trχ dM
2j |ν − ν 0 |S2 ×S2 00M02

×ηjν (ω)ηjν (ω 0 )ψ(2−j λ)ψ(2−j λ0 )f (λω)f (λ0 ω 0 )λ2 λ0 dλdλ0 dωdω 0 .
Since L∇/ trχ is the only derivative of ∇/ trχ for which we have an estimate, our next integration
by parts must be with respect to L, that is we use (3.104). Consider the term where the Lderivative falls on trχ0 , which is of the form:
Z
Z +∞ Z +∞ Z1iλu−iλ0 u0 −1
0 −1
0
e
b
∇
/
trχb
L(trχ
)dM
(3.105)
22j |ν − ν 0 |3
S2 ×S2 00M02

×ηjν (ω)ηjν (ω 0 )ψ(2−j λ)ψ(2−j λ0 )f (λω)f (λ0 ω 0 )λ2 λ0 dλdλ0 dωdω 0 .
Now, note in view of (3.103), (3.99) and the estimate (3.14) for ∂ω N , that:g(L, L0 ) ∼ |ν − ν 0 |2 , g(L, e0A ) = g(L − L0 , e0A ) ∼ |ν − ν 0 | and g(L, L0 ) = −2 + g(L, L0 ) ∼ −2.
Thus, decomposing L on the frame L0 , N 0 , e0A , we obtain:
L ∼ L0 + |ν − ν 0 |∇/ 0 + |ν − ν 0 |2 N 0 .

(3.106)

3.5. ALMOST ORTHOGONALITY IN ANGLE

65

Together with (3.105), we finally obtain the sum of three terms:
Ej,ν,ν 0= Ej,ν,ν 0 [1] + Ej,ν,ν 0 [2] + Ej,ν,ν 0 [3]Z
Z +∞ Z +∞ Z1iλu−iλ0 u0 −1
0 −1 00=eb ∇/ trχb L (trχ )dM
22j |ν − ν 0 |3 S2 ×S2 0
0
M
0

2

0

2

0

2

(3.107)×ηjν (ω)ηjν (ω 0 )ψ(2−j λ)ψ(2−j λ0 )f (λω)f (λ0 ω 0 )λ2 λ0 dλdλ0 dωdω 0
ZZ +∞ Z +∞ Z1iλu−iλ0 u0 −1
0 −1 00eb ∇/ trχb ∇/ (trχ )dM+ 2j
2 |ν − ν 0 |2 S2 ×S2 00M
×ηjν (ω)ηjν (ω 0 )ψ(2−j λ)ψ(2−j λ0 )f (λω)f (λ0 ω 0 )λ2 λ0 dλdλ0 dωdω 0
ZZ +∞ Z +∞ Z1iλu−iλ0 u0 −1
0 −1 00+ 2jeb ∇/ trχb N (trχ )dM
2 |ν − ν 0 | S2 ×S2 00M
×ηjν (ω)ηjν (ω 0 )ψ(2−j λ)ψ(2−j λ0 )f (λω)f (λ0 ω 0 )λ2 λ0 dλdλ0 dωdω 0 .
We consider the second term in the right-hand side of (3.107) which is of the form:

Z Z
1
−1
ν
Ej,ν,ν 0 [2] =
b
∇
/
trχF
(u)η
(ω)dω
j
j
22j |ν − ν 0 |2 M
S2

Z
0
−1 0
/ trχ0 Fj (u0 )ηjν (ω 0 )dω 0 dM.
×
b0 ∇
S2

We claim that such a term leads to a log-loss. Indeed, we have:
Ej,ν,ν 0 [2].

1
22j |ν − ν 0 |2

Z

1

Z

S2

Z

b−1 ∇
/ trχFj (u)ηjν (ω)dωS2L2 (M)

b0

−1

0

∇
/ 0 trχ0 Fj (u0 )ηjν (ω 0 )dω 0
L2 (M)

kb−1 ∇/ trχFj (u)kL2 (M) ηjν (ω)dω

Z
0
0
ν0
0
0
0 −1 0/ trχ Fj (u )kL2 (M) ηj (ω )dω
×
kb ∇
S2
Z

1
ν
−1 ∞
/ trχkL∞kFj (u)kL2u ηj (ω)dω.
kb kL k∇2u L (Hu )22j |ν − ν 0 |2S2
Z
0 −1 ∞
0
0
0
ν0
0
0
×
kb kL k∇/ trχ kL∞kFj (u )kL2 0 ηj (ω )dω2
u L (Hu ).

.

22j |ν − ν 0 |2S20ε2 γjν γjνj(2 2 |ν − ν 0 |)2

S2

u

,

(3.108)where we used in the last inequality Cauchy-Schwartz in ω and ω 0 which gains the square rootof the volume of the patch, the estimates (3.10) for b and trχ, and the estimate (3.51) for Fj (u)and Fj (u0 ). This leads to a log-loss since we have:
X
1
sup
∼ j.
(3.109)(2j/2 |ν − ν 0 |)2
ν
0
ν

66

3. CONTROL OF THE ERROR TERM

Indeed, note that ν 0 runs on a lattice on S2 of basic size 2−j/2 so that (3.109) corresponds tothe sum
X
1
∼ j.
|l|2
j/2
2
l∈Z , 1≤|l|≤2

3.5.3. Strategy of the proof of Proposition 3.9. Let us explain informally the strategyof the proof. As we noticed in the previous section, the second term in (3.107) contains a logloss. Let us start by showing that the first and the third term in the right-hand side of (3.107)
do not contain a log-loss.
3.5.3.1. Control of the first term in the right-hand side of (3.107). We have

Z Z
1
−1
ν
b ∇
/ trχFj (u)ηj (ω)dω
Ej,ν,ν 0 [1] =22j |ν − ν 0 |3 M
S2

Z
0
0 ν0
0
0
0 −1 0×b L (trχ )Fj (u )ηj (ω )dω dM.
S2

In view of the Raychaudhuri equation (3.7), we have:
L0 (trχ0 ) = −|bχ0 |2 + · · · ,where we keep only the worst term. Thus, we obtain

 Z
Z Z
1
0 ν0
0
0
−1
ν
0 −1 0 2
Ej,ν,ν 0 [1] = 2j
|b
χ
|
F
(u
)η
(ω
)dω
dM.
b
∇
/
trχF
(u)η
(ω)dω
b
j
j
j
j
2 |ν − ν 0 |3 M
S2
S2
Let us decompose:
−1

−1

b0 |b
χ0 |2 = bν−1
bν 0 χ
b + (b0 |bχ0 |2 − b−1bν 0 χ
b),(3.110)0 χ
ν0 χ
and let us assume for the moment that we can control the second term in (3.110). Then, we are
led to control:

Z

Z Z
1
−1
−1
ν
0 ν0
0
0
bν 0
b χ
b∇
/ trχFj (u)ηj (ω)dω bν 0 χ
Fj (u )ηj (ω )dω dM.
22j |ν − ν 0 |3 M
S2S2
We have:

Z

Z Z
1
−1
−1
ν
0 ν0
0
0
bν 0
b χ
b∇
/ trχFj (u)ηj (ω)dω bν 0 χ
Fj (u )ηj (ω )dω dM
22j |ν − ν 0 |3 M
S2
S2
Z
Z
1
0
−1
−1
ν
0
b
χ
b
Fj (u0 )ηjν (ω 0 )dω 0
.
b
χ
b
∇
/
trχF
(u)η
(ω)dω
0
j
ν
j
ν
2j
0
3
2 |ν − ν |S2S2
L2 (M)
L2 (M)
Z1.
kb−1 χb∇/ trχFj (u)kL2 (M) ηjν (ω)dω
22j |ν − ν 0 |3
2
S
Z
0
−1
Fj (u0 )ηjν (ω 0 )dω 0× bν 0 χbν 0 L∞L2uν 0 ,x0 ν 0

0

.
.

εγjν
22j |ν − ν 0 |3Z

0ε2 γjν γjνj(2 2 |ν − ν 0 |)3

S2

,

t

S2

∞
0 Lt
ν 0 ,x ν 0

L2u

kb−1 kL∞ (M) kb
χkL∞0 L2t k∇/ trχkL∞0 L2t kFj (u)kL2u ηjν (ω)dωxx(3.111)

3.5. ALMOST ORTHOGONALITY IN ANGLE

67

where we used the estimate (3.10) for trχ, χb and b, the estimate (3.80), Cauchy-Schwartz in ω,
the size of the patch, and the estimate (3.51) for Fj (u). Note that the right-hand side of (3.111)
does not contain a log-loss since:
X
1
sup
. 1.
(3.112)j/2(2 |ν − ν 0 |)3ν0νRemark 3.19. While the estimate obtained in (3.111) is correct, one has to modify slightly
the method leading to it. Indeed, χb does not have enough regularity with respect to ω to be able
to handle the second term in the decomposition (3.110). The way to overcome this is to makeuse of the decomposition (3.15) for χb:χb = χ1 + χ2 .
Then, we exploit the fact that, in view of the estimate (3.16), χ1 has better regularity than χbwith respect to (t, x), while χ2 has better regularity than χb with respect to ω. We refer to [76]for more details.
3.5.3.2. Control of the third term in the right-hand side of (3.107). We have

Z Z
1
−1
ν
Ej,ν,ν 0 [3] =b ∇/ trχFj (u)ηj (ω)dω
22j |ν − ν 0 | M
S2

Z
0
−1
×
b0 N 0 (trχ0 )Fj (u0 )ηjν (ω 0 )dω 0 dM.
S2Now, we have:

 Z
Z Z
1
0
0 ν0
0
0
−1
ν
0 −1 0b ∇/ trχFj (u)ηj (ω)dωb N trχ Fj (u )ηj (ω )dω dM
22j |ν − ν 0 | M
S2
S2
Z
Z
1
0
−1
−1
ν
.
b ∇
/ trχFj (u)ηj (ω)dωb0 N 0 trχ0 Fj (u0 )ηjν (ω 0 )dω 0
2j
0
2 |ν − ν | S2
S2
L2 (M)
L2 (M)
Z1.
kb−1 ∇/ trχFj (u)kL2 (M) ηjν (ω)dω2j
2 |ν − ν 0 |
2
S

Z
−1
0
0
0
0
ν0
0
0
×
kb N trχ Fj (u )kL2 (M) ηj (ω )dω
S2
Z

1
ν
−1 ∞
.
kb kL k∇/ trχkL∞kFj (u)kL2u ηj (ω)dω2u L (Hu )22j |ν − ν 0 |S2
Z
0 −1 ∞
0
0
0
ν0
0
0
×
kb kL kN trχ kL∞
kFj (u )kL2 0 ηj (ω )dω2
u L (Hu ).

S202
ε γjν γjνjj

2 2 (2 2 |ν − ν 0 |)u,

(3.113)where we used in the last inequality Cauchy-Schwartz in ω and ω 0 which gains the square rootof the volume of the patch, the estimates (3.10) for b and trχ, and the estimate (3.51) for Fj (u)and Fj (u0 ). Note that the right-hand side of (3.113) does not contain a log-loss since we have:
X
1
sup
. 1.
(3.114)jjν
2 (2 2 |ν − ν 0 |)
2
0
ν

68

3. CONTROL OF THE ERROR TERM

3.5.3.3. A decomposition for Ejν f . To remove the log-loss exhibited in (3.108) (3.109), werely on a decomposition of trχ using the geometric Littlewood-Paley projections Pj . We have:X
Pl trχ
trχ = P≤j/2 (trχ) +l>j/2

which in turn yields the following decomposition for Ejν f :X ν,lEj f (t, x),
Ejν f (t, x) =(3.115)l≥j/2

where:

Zj2(3.116)b(t, x, ω)−1 P≤j/2 trχ(t, x, ω)Fj (u)ηjν (ω)dω.

(3.117)

In order to prove almost orthogonality in angle, i.e. (3.95), we will estimate:
XZ
0Ejν,l f (t, x)Ejν ,m f (t, x)dM .

(3.118)Ejν,l f (t, x) =

S2

and:
ν,j/2
Ej
f (t, x) =b(t, x, ω)−1 Pl trχ(t, x, ω)Fj (u)ηjν (ω)dω, ∀l >
Z

l,m

S2

M

3.5.3.4. The mechanism to remove the log-loss. In order to explain the mechanism whichallows us to remove the log-loss, let us assume for convenience that m ≤ l in (3.118). Then,
notice first from (3.107), (3.109), (3.112) and (3.114) that the only term in the right-hand side of(3.107) which contains a log-loss is the second one, i.e the term which contains only tangential
derivatives. In order to remove the log-loss, our goal will be to always put more tangential
derivatives on the lowest frequency, i.e. Pm trχ0 (as opposed to the higher frequency Pl trχ).
This is achieved as follows (see [76] for the details):(1) Integrate by parts with respect to L using (3.104).
(2) One term corresponds to the case where the L derivative falls on the largest frequency
Pl trχ, while the other term corresponds to the case where L falls on the lowest frequency
Pm trχ0 . For the second term, decompose the L derivative on the frame L0 , N 0 , e0A as in(3.106).
(3) Notice that the terms involving L, L0 or N 0 are estimated in the spirit of (3.111) and(3.113), and should in principle contain no log-loss in view of (3.112) and (3.114).
(4) Finally, the last term is the one containing the ∇/ 0 derivative. This term is the only one
which contains the log-loss exhibited in (3.109). Now, we have achieved our goal sinceafter integration by parts, the tangential derivative fell on Pm trχ0 which is the lowest
frequency.
Remark 3.20. Due to the decomposition (3.115), we now not only need to obtain summabilityin (ν, ν 0 ), but also in (l, m). This creates additional difficulties, in particular when estimating
the terms Ej,ν,ν 0 [1] and Ej,ν,ν 0 [3] in (3.107). We refer to [76] for more details.

CHAPTER 4

Control of the space-time foliation
The goal of this chapter is to prove the estimates on the control of the space-time foliationby the optical function u which are needed for the proof of Theorem 3.6 (see section 3.1.3), i.e.
for the control of the error term. Here, we outline the main ideas and we refer to [75] for thedetails.
4.1. Geometric set-up and main results
4.1.1. Geometry of the foliation of M by u. Recall from section 1.2.1 that the spacetime M is foliated by space-like hypersurfaces Σt defined as level hypersurfaces of a time functiont, where T denotes the unit, future oriented, normal to Σt and k its second fundamental form.
Recall also that u is a solution to the eikonal equation gαβ ∂α u∂β u = 0 on M depending on
a extra parameter ω ∈ S2 . The level hypersufaces u(t, x, ω) = u of the optical function u are
denoted by Hu . Let L0 denote the space-time gradient of u, i.e.:
L0 = −gαβ ∂β u∂α .

(4.1)

Using the fact that u satisfies the eikonal equation, we obtain:
DL0 L0 = 0,

(4.2)

which implies that L0 is the geodesic null generator of Hu .
We have:
T (u) = ±|∇u|P3
2
2
where |∇u| = i=1 |ei (u)| relative to an orthonormal frame ei on Σt . Since the sign of T (u)is irrelevant, we choose by convention:
T (u) = |∇u|.

(4.3)

We denote by Pt,u the surfaces of intersection between Σt and Hu . They play a fundamentalrole in our discussion.
Definition 4.1 (Canonical null pair ).
L = bL0 = T + N,

L = 2T − L = T − N(4.4)

where L0 is the space-time gradient of u (4.1), b is the lapse of the null foliation (or shortly null
lapse)
b−1 = − < L0 , T >= T (u),(4.5)
and N is a unit normal, along Σt , to the surfaces Pt,u . Since u satisfies the eikonal equationgαβ ∂α u∂β u = 0 on M, this yields L0 (u) = 0 and thus L(u) = 0. In view of the definition of L
69

70

4. CONTROL OF THE SPACE-TIME FOLIATION

and (4.3), we obtain:
N =−

∇u
.
|∇u|

(4.6)

Remark 4.2. u is prescribed on Σ0 as in [73]. For any (0, x) on Σ0 , L is defined as
L = T + N where T is the unit normal to Σ0 at (0, x) and N = −∇u/|∇u| at (0, x), and b
is defined as b−1 = |∇u|. Let κx (t) denote the null geodesic parametrized by t and such thatκx (0) = (0, x) and κ0x (0) = b−1 L. Then, we claim that
κ0x (t) = b(κx (t))−1 Lκx (t) for all t.

(4.7)

Indeed, L0 = b−1 L is the geodesic null generator of Hu (see (4.2)).
Definition 4.3. A null frame e1 , e2 , e3 , e4 at a point p ∈ Pt,u consists, in addition to thenull pair e3 = L, e4 = L, of arbitrary orthonormal vectors e1 , e2 tangent to Pt,u .
Definition 4.4 (Ricci coefficients). Let e1 , e2 , e3 , e4 be a null frame on Pt,u as above. The
following tensors on St,u
χAB =< DA e4 , eB >,χAB =< DA e3 , eB >,11
ζA = < D3 e4 , eA >, ζ A = < D4 e3 , eA >,221
ξ A = < D3 e3 , eA > .
2
are called the Ricci coefficients associated to our canonical null pair.
We decompose χ and χ into their trace and traceless components.

(4.8)

trχ = gAB χAB ,trχ = gAB χAB ,
(4.9)
1
1
(4.10)
χbAB = χAB − trχgAB , χb AB = χAB − trχgAB ,2
2Definition 4.5. The null components of the curvature tensor R of the space-time metric g
are given by:
αab = R(L, ea , L, eb ) ,1
R( L, L, L, L) ,41
βa =R(ea , L, L, L) ,2
where ? R denotes the Hodge dual of R.
ρ =1βa = R(ea , L, L, L),
2
1
σ = ? R( L, L, L, L)
4
αab = R( L, ea , L, eb )(4.11)(4.12)(4.13)

Observe that all tensors defined above are Pt,u -tangent.
Remark 4.6. Note that α is the only null component which does not contain a contractionof R with L. With the notation of Chapter 2 (see for instance (2.28)), we have:
R · L = (α, β, ρ, σ, β).

4.1. GEOMETRIC SET-UP AND MAIN RESULTS

71

Definition 4.7. We decompose the symmetric traceless 2 tensor k into the scalar δ, thePt,u -tangent 1-form , and the Pt,u -tangent symmetric 2-tensor η as follows: kN N = δ
kAN = A
(4.14)
kAB = ηAB .
The following Ricci equations can be easily derived from the definition of T , the fact that
L0 is geodesic (4.2), and the definition (4.8) of the Ricci coefficients (see [17] p. 171):DA e4 = χAB eB − A e4 ,DA e3 = χAB eB + A e3 ,D4 e4 = −δe4 ,D4 e3 = 2ζ A eA + δe3 ,D3 e4 = 2ζA eA + (δ + n−1

D3 e3 = 2ξ A eA − (δ + n∇N n)e4 ,(4.15)−1∇N n)e3 ,D4 e A = ∇/ 4 eA + ζ A e4 ,D3 eA = ∇/ 3 eA + ζA e3 + ξ A e4 ,1
1
DB eA = ∇/ B eA + χAB e3 + χAB e422
where, ∇/ 3, ∇/ 4 denote the projection on Pt,u of D3 and D4 , ∇/ denotes the induced covariant
derivative on Pt,u and δ,  are defined by:δ = δ − n−1 N (n), A = A − n−1 ∇A n.

(4.16)Also,χAB = −χAB − 2kAB ,ζ A = −A ,ξ A = A + n(4.17)−1∇/ A n − ζA .

4.1.2. Null structure equations. Below we write down our main structure equations (see[17] chapter 7 or [75] for a proof).
Proposition 4.8. The components trχ, χb, ζ and the lapse b verify the following equations1:
L(b) = −b δ,1L(trχ) + (trχ)2 = −|bχ|2 − δtrχ,2∇/ 4χ
b + trχb
χ = −δ χb − α,
1
1∇/ 4 ζA + (trχ)ζA = −(B + ζB )b
χAB − trχA − βA , .
2
2(4.18)(4.19)(4.20)(4.21)

Remark 4.9. Equation (4.19) is known as the Raychaudhuri equation in the relativity literature.
1which can be interpreted as transport equations along the null geodesics generated by L. Indeed observethat if a Pt,u -tangent tensor Π satisfies the homogeneous equation ∇/ 4 Π = 0 then Π is parallel transported alongnull geodesics.

72

4. CONTROL OF THE SPACE-TIME FOLIATION

To obtain estimates for χ, we may use the transport equations (4.19) (4.20). However, thisdoes not allow us to get enough regularity. Instead, we follow [17] [36] [39] and consider (4.19)
for trχ together with an elliptic system of Hodge type for χb.
Proposition 4.10. The expression (div/χb)A = ∇/ BχbAB verifies the following equation:
1
(div
/χ
b)A + χbAB B = (∇/ A trχ + A trχ) − βA .
(4.22)
2
See [17] chapter 7 or [75] for a proof.
Finally, we consider the control of ζ and Ltrχ. To this end, we follow again [36] [39]: wederive an elliptic system of Hodge type for ζ and a transport equation for Ltrχ.
Proposition 4.11. We have:1/ ζ + (δ + n−1 ∇N n)trχ − χb·χ
b + 2ζ · ζ + 2ρ.
(4.23)L(trχ) + trχtrχ = 2div
2
Also, the expressions div/ζ = ∇/ B ζB and curl/ ζ =∈AB ∇/ A ζB verify the following equations:


1
1
2
div
/ζ=
µ + trχtrχ + χb·χb − 2|ζ| − ρ,
(4.24)221curl/ζ=− χb∧χ
b + σ,(4.25)
2
where for F, G symmetric traceless Pt,u -tangent 2-tensors, we denote by F ∧ G the tensor F ∧GAB =∈AB FAC GBC . Finally, setting,µ = L(trχ) − δ + n−1 ∇N n trχ
(4.26)
we findb + ζ ⊗ζb − δχ
L(µ) + trχµ = 2(ζ − ζ) · ∇/ trχ − 2b
χ· ∇
/ ⊗ζ
b

− trχ 2div
/ ζ + 2ζ · ζ + 4( − ζ) · n−1 ∇/ n − 2δ(δ + n−1 ∇N n) + 4ρ

1
2
2
−1
2
− trχtrχ + 2|| + 3|b
χ| + 4bχ · ηb − 2|n N (n)| .
2(4.27)

See [36] or [75] for a proof.
4.1.3. Commutation formulas. We have the following useful commutation formulas (see[17] p. 159):
Lemma 4.12. Let UA be an m-covariant tensor tangent to the surfaces Pt,u . Then,∇/ B∇/ 4 UA − ∇/ 4∇/ B UA = χBC ∇/ C UA − n−1 ∇/ B n∇/ 4 UAX+(χAi B C − χBC Ai − ∈Ai C ∗ βB )UA1 ..Č..Am ,(4.28)i∇/ B∇/ 3 UA − ∇/ 3∇/ B UA = χBC ∇/ C UA − ξ B ∇/ 4 UA − b−1 ∇B b∇/ 3 UA +X(−χAi B ξ Ci+ χBC ξ A − χA B ζC + χBC ζAi + ∈Ai C ∗ β B )UA1 ..Č..Am ,ii(4.29)

4.1. GEOMETRIC SET-UP AND MAIN RESULTS

73

∇
/ 3∇
/ 4 UA − ∇/ 4∇/ 3 UA = −δ∇/ 3 UA + (δ + n−1 ∇N n)∇/ 4 UA + 2(ζB − ζ B )∇/ B UAX+ 2(ζ A ζC − ζ C ζAi + ∈Ai C ∗ σ)UA1 ..Č..Am .

(4.30)ii

4.1.4. Bianchi identities. In view of the formulas on p. 161 of [17], the Bianchi equations
for α, β, ρ, σ, β are:∇
/ L (β) = div/ α − δβ + (2 − ) · α∗(4.31)−1∗∇/ L (β) = ∇/ ρ + (∇/ σ) + 2b
χ · β + (δ + n ∇N n)β + ξ · α + 3(ζρ + ζσ)
1
L(ρ) = div/β − χ
b · α + ( − 2) · β
21/β − χ
L(ρ) = −divb · α + 2ξ · β + ( − 2ζ) · β
21 ∗L(σ) = −curl/β + χ
b α + (− + 2) ∗ β
2
1 ∗L(σ) = −curlb α − 2ξ ∗ β + ( − 2ζ) ∗ β/β − χ
2/ ρ + (∇/ σ)∗ + 2b
χ · β + δβ − 3(ζρ − ∗ ζσ)∇/ L (β) = −∇(4.32)(4.33)(4.34)(4.35)(4.36)(4.37)

4.1.5. Main results. We introduce the L2 curvature flux R relative to the time foliation:
1
2
R = kαk2L2 (Hu ) + kβk2L2 (Hu ) + kρk2L2 (Hu ) + kσk2L2 (Hu ) + kβk2L2 (Hu ) .
(4.38)
In view of Remark 4.6, we have R = kR · LkL2 (Hu ) . Thus, the we may rewrite the bootstrap
assumptions of Chapter 2 on R as:kRkL∞≤ M ε, sup R ≤ M ε.
2t L (Σt )u

To ease the notations, we drop the bootstrap constant M :kRkL∞≤ ε, sup R ≤ ε.
2t L (Σt )(4.39)u

The goal of this part is to control the geometry of the null hypersurfaces Hu of u up to timet = 1 when only assuming the smallness assumption (4.39).
Remark 4.13. In the rest of the chapter, all inequalities, except the ones of Theorem 4.16
below, hold for any u with the constant in the right-hand side being independent of u. Thus, one
may take the supremum in u in these inequalities. To ease the notations, we do not explicitly
write down the supremum in u in these estimates.
u is a solution to the eikonal equation gαβ ∂α u∂β u = 0 on M depending on a extra parameterω ∈ S2 . Now, for u to be uniquely defined, we need to prescribe it on Σ0 (i.e. at t = 0). Thisissue has been settled in [73] (see also Chapter 6). From now on, we assume that u is thesolution to the eikonal equation gαβ ∂α u∂β u = 0 on M which is prescribed on Σ0 as in [73].
Remark 4.14. In the rest of the chapter, all inequalities hold for any ω ∈ S2 with theconstant in the right-hand side being independent of ω. Thus, one may take the supremum in ω
everywhere. To ease the notations, we do not explicitly write down this supremum.

74

4. CONTROL OF THE SPACE-TIME FOLIATION

We define some norms on Hu . For any 1 ≤ p ≤ +∞ and for any tensor F on Hu , we have:
!1
Z
Z
1

kF kLp (Hu ) =p|F |p dµt,u

dt

,

Pt,u

0

where dµt,u denotes the area element of Pt,u . We also introduce the following norms:
N1 (F ) = kF kL2 (Hu ) + k∇
/ F kL2 (Hu ) + k∇
/ L F kL2 (Hu ) ,N2 (F ) = N1 (F ) + k∇
/ 2 F kL2 (Hu ) + k∇/∇
/ L F kL2 (Hu ) .
Let x0 a coordinate system on P0,u . By transporting this coordinate system along the nullgeodesics generated by L, we obtain a coordinate system (t, x0 ) of H. We define the following
norms:
Z 1
 12
0 2
kF kL∞0 L2t = sup|F (t, x )| dt,xx0 ∈P0,u0

sup |F (t, x0 ))|=kF kL2 0 L∞
t
x

.

0≤t≤1

L2 (P0,u )The following theorem investigates the regularity of u with respect to (t, x):
Theorem 4.15. Assume that u is the solution to the eikonal equation gαβ ∂α u∂β u = 0 on
M such that u is prescribed on Σ0 as in [73]. Assume also that the estimate (4.39) is satisfied.
Then, null geodesics generating Hu do not have conjugate points and distinct null geodesics do
not intersect. Furthermore, the following estimates are satisfied:2kn − 1kL∞ + k∇nkL∞
2 + k∇ nkL∞ L2 + k∇DT nkL∞ L2 . ε,
0
0
t
t
t L 0

(4.40)N1 (k) + k∇
/ L kL2 (Hu ) + k L(δ)kL2 (Hu ) + kkL∞0 L2t + kδkL∞0 L2t . ε,

(4.41)kb − 1kL∞ + N2 (b) + k L(b)kL2 0 L∞
. ε,t

(4.42)

ktrχkL∞ + k∇
+ k LtrχkL2 0 L∞
. ε,/ trχkL2 0 L∞tt

(4.43)kbχkL2 0 L∞+ N1 (bχ) + k∇/ Lχ
bkL2 (Hu ) . ε,t(4.44)kζkL2 0 L∞
+ N1 (ζ) . ε.
t(4.45)

x

x

x

x

x

x

x

x

x

x

We introduce the family of intrinsic Littlewood-Paley projections Pj which have been constructed in [41] using the heat flow on the surfaces Pt,u (see also section 3.1.5). This allows us
to state our second theorem which investigates the regularity of L Ltrχ and ∇/ L ζ.
Theorem 4.16. Assume that u is the solution to the eikonal equation gαβ ∂α u∂β u = 0 on M
such that u is prescribed on Σ0 as in [73]. Assume also that the assumption (4.39) is satisfied.
Then, there exists a function λ in L2 (R) such that for all j ≥ 0, we have:jkPj L LtrχkL2 (Hu ) . 2j ε + 2 2 λ(u),
and

j

kPj ∇
/ L (ζ)kL2 (Hu ) . ε + 2− 2 λ(u).

(4.46)(4.47)

The following theorem investigates the regularity with respect to the parameter ω ∈ S2 .

4.1. GEOMETRIC SET-UP AND MAIN RESULTS

75

Theorem 4.17. Assume that u is the solution to the eikonal equation gαβ ∂α u∂β u = 0 on
M such that u is prescribed on Σ0 as in [73]. Assume also that the estimate (4.39) is satisfied.
Then, we have the following estimates:k∂ω N kL∞ . 1,

(4.48)kD∂ω N kL2 0 L∞
+ k∂ω bkL∞ + k∇
/ ∂ω bkL2 0 L∞
+ k∂ω χkL2 0 L∞
+ k∂ω ζkL2 0 L∞
. ε.
t
t
t
t
x

x

x

(4.49)x

Furthermore, we have the following decomposition for χb:χb = χ1 + χ2 ,(4.50)

where χ1 and χ2 are two symmetric traceless Pt,u -tangent 2-tensors satisfying:
N1 (χ1 ) + k∇
/ L χ1 kL2 (Hu ) + k∂ω χ1 kL∞/ L χ2 kL2 (Hu ) + k∂ω χ2 kL∞
2 + N1 (χ2 ) + k∇
2 . ε (4.51)
t L 0
t L 0
x

x

and for any 2 ≤ p < +∞, we have:kχ1 kLpt L∞0 + k∂ω χ2 kLp L4− + k∂ω χ2 kL6− (Hu ) + k∇
/ ∂ω χ2 kL2 (Hu ) . ε,x(4.52)x0twhere for any real number a, a− = a − δ for any δ > 0.
Remark 4.18. Notice from (4.51) that χ1 and χ2 have at least the same regularity as χb.
Now, the point of the decomposition (4.50) is that both χ1 and χ2 have better regularity properties
than χ
b. Indeed, in view of (4.52), χ1 has better regularity with respect to (t, x) while χ2 hasbetter regularity with respect to ω.
Next, the following theorem contains estimates for second order derivatives with respect toω.
Theorem 4.19. Assume that u is the solution to the eikonal equation gαβ ∂α u∂β u = 0 on
M such that u is prescribed on Σ0 as in [73]. Assume also that the estimate (4.39) is satisfied.
Then, we have the following estimates:k∂ω2 N kL2 0 L∞
. 1,
t

(4.53)k∇/ L Π(∂ω2 N )kL2 (Hu ) . ε.

(4.54)

x

2
j
kPj ∇
/ L Π(∂ω2 N )kLpt L2 0 + kPj Π(∂ω2 χ)kL∞
2 + kPj Π(∂ω ζ)kLp L2 . 2 ε,
0
t L 0
t
x

x

(4.55)

xwhere p is any real number such that 2 ≤ p < +∞, and where Π denotes the projection on
Pt,u -tangent tensors.
Finally, we need to compare quantities evaluated at two angles ω and ν. The following
decompositions are used in sections 3.4 and 3.5
jTheorem 4.20. Let ω and ν in S2 such that |ω − ν| . 2− 2 . Let u = u(., ω), N = N (., ω)
and Nν = N (., ν). For any j ≥ 0, we have the following decomposition for N − Nν :j

2 2 (N − Nν ) = F1j + F2j
where the tensor F1j does not depend on ω and satisfies:
kF1j kL∞ . 1,

(4.56)

76

4. CONTROL OF THE SPACE-TIME FOLIATION

and where the tensor F2j satisfies:j. 2− 2 .
kF2j kL∞2
u L (Hu )We also have following decomposition for trχ:trχ = f1j + f2j(4.57)

where the scalar f1j does not depend on ω and satisfies:
kf1j kL∞ . ε,and where the scalar f2j satisfies:j

kf2j kL∞. ε2− 2 .
2u L (Hu )Let us conclude this section by mentioning several ingredients of [75] that have been omitted
here for the sake of simplicity:• estimates for the transport equations along L, and the elliptic systems of Hodge typeon Pt,u involved in the null structure equations• embeddings on Hu , Σt and Pt,u
• geometric Littlewood-Paley projections and Besov spaces on Σt• control of the Gauss curvature of Pt,u
• Bochner inequalities on Σt and Pt,u/ ], [DL , Pj ], [D L , Pj ],• estimates for various commutator terms of the type: [DL , ∇/ ], [D L , ∇...
4.2. Regularity of the foliation with respect to (t, x)
In this section, we outline the main ideas of the proof of Theorem 4.15. We assume thefollowing bootstrap assumptions:kn − 1kL∞ (Hu ) + kb − 1kL∞ (Hu ) ≤1,
10(4.58)

2k∇nkL∞
2 + k∇ nkL∞ L2 + k∇DT nkL∞ L2 + N2 (b) + k L(b)kL2 L∞ ≤ Dε,
0
0
0 t
t L 0
t
t

(4.59)N1 (k) + k∇
/ L kL2 (Hu ) + kD L δkL2 (Hu ) + kkL∞0 L2t + kδkL∞0 L2t ≤ Dε,(4.60)

ktrχkL∞ (Hu ) + k∇
/ trχkL2 0 L∞
+ k LtrχkL2 0 L∞
≤ Dε,
t
t

(4.61)+ N1 (bχ) + k∇/ Lχkb
χkL2 0 L∞
bkL2 (Hu ) ≤ Dε,t(4.62)kζkL2 0 L∞
+ N1 (ζ) ≤ Dε,t(4.63)

x

x

x

x

x

x

x

x

x

x

where D > 0 is a large enough constant. We will improve on these estimates.

4.2. REGULARITY OF THE FOLIATION WITH RESPECT TO (t, x)77

4.2.1. Non intersection of null geodesics on Hu . The control we obtain on the geometric quantities associated to our foliation is only valid as long as there are no conjugate points
and null geodesics do not intersect. The goal of this section is to prove that this holds at least
until t = 1. In addition to the bound (4.39) on the curvature tensor R of g, we make thefollowing regularity assumption on g. There exists a coordinate chart such that
kgkC 2 (M) ≤ M,

(4.64)

where M is a very large constant.
Remark 4.21. The assumption (4.64) is only used to prove the absence of caustic and thatnull geodesics do not intersect at least until t = 1, which is a qualitative property. On theother hand, we only rely on the bound (4.39) on R to prove the various quantitative bounds ofTheorems 4.15, 4.16, 4.17 and 4.19.
For (0, x) in Σ0 , recall the definition in Remark 4.2 of the null geodesic κx (t). For all
0 ≤ t ≤ 1, let Φt : Σ → Σt defined by Φt (0, x) = κx (t). We have Φ0 (0, x) = (0, x) on Σ0 . We
define t0 ≥ 0 as the supremum of 0 ≤ t ≤ 1 such that Φt is bijective from Σ0 to Σt .
Remark 4.22. As long as 0 ≤ t < t0 , there are no conjugate points and no distinct nullgeodesic intersections. Thus, we may assume that the u-foliation exists and satisfies the bounds(4.58)-(4.63) given by the bootstrap assumptions. Furthermore, we may assume the identity (4.7)
for the null geodesics κx (t).
Our goal is to show that we have in fact t0 = 1. We proceed in three steps (see [75] for thedetails):
Step 1. As noticed in Remark 4.22, the L∞ bound for trχ given by (4.61) holds for 0 ≤ t < t0 .
Furthermore, using the Raychaudhuri equation (4.19) and the bound (4.64), we obtain
the existence of a constant δ > 0 depending on M such that the L∞ bound for trχ
given by (4.61) holds for 0 ≤ t < t0 + δ. This control for trχ allows us to prove that
there are no conjugate points on 0 ≤ t < t0 + δ.
Step 2. Next, we prove that Σt = ∪u Pt,u for 0 ≤ t ≤ t0 + δ where δ > 0 is a constant depending
on M . This requires the bound (4.64), and the control it induces on forward andbackward light cones for small time intervals with a size depending on M .
Step 3. Assume now that 0 < t0 < 1. In view of Step 1 and Step 2, the only thing thatcan go wrong at t = t0 is that two distinct null geodesics intersect in Σt0 . Assume
by contradiction that this is indeed the case so that there exists (0, x1 ) 6= (0, x2 ) two
points in Σ0 such that κx1 (t0 ) = κx2 (t0 ) = (t0 , x0 ). Sinceκ0xj (t) = b(κxj (t) )−1 Lκxj (t) , j = 1, 2,
in view of Remark 4.2, the regularity of b and L yields κ0x1 (t0 ) = κ0x2 (t0 ). From theclassical uniqueness result for ODEs, we deduce that κx1 (t) = κx2 (t) for all t. In
particular, taking t = 0, we obtain (0, x1 ) = (0, x2 ) which yields a contradiction.
Finally, Steps 1, 2 and 3, yield t0 ≥ 1. In particular, we have:On 0 ≤ t ≤ 1, there are no conjugate points and no intersection of distinct
null geodesics. In particular, u exists on 0 ≤ t ≤ 1 and the bootstrap
assumptions (4.58)-(4.63) hold. Furthermore, Σt = ∪u Pt,u for all 0 ≤ t ≤ 1.

(4.65)

78

4. CONTROL OF THE SPACE-TIME FOLIATION

4.2.2. Lower bound on the volume radius of Σt . In this section, we prove the lower
bound on the volume radius of Σt given by the estimate (2.32). We use the global coordinate
system x0 = (x1 , x2 ) on P0,u which has been constructed in [73] (see also Proposition 6.13).
Transporting this coordinate system along the null geodesics generated by L, we obtain a coordinate system x0 of Pt,u , which in particular satisfies(1 − O(ε))|ξ|2 ≤ γAB (p)ξ A ξ B ≤ (1 + O(ε))|ξ|2 ,uniformly for all p ∈ Pt,u ,(4.66)

where γ is the metric induces by g on Pt,u . We denote by x0 this global coordinate system on
Pt,u .
Next, we obtain a global coordinate system on Σt as follows. First, recall from (4.65) thatΣt = ∪Pt,u so that u is defined on Σt . To any p ∈ Σt , we associate the coordinates (u(p), x0 (p))
where u(p) is the value of the optical function u at p, and x0 (p) are the coordinate of p in thecoordinate system of Pt,u . In this coordinate system, the metric gt on Σt (i.e. the restriction ofg on Σt ) takes the following form:
 −2

b
0
gt =
,
(4.67)
0γ
where γ is the induced metric on Pt,u . Together with the estimate (4.58) for b and (4.66) for γ,
we obtain the following lower bound on the volume radius of Σt at scales ≤ 1:
1
rvol (Σt , 1) ≥ ,4(4.68)

which is the estimate (2.32).
4.2.3. Estimates for the second fundamental form k and the lapse n. We firstestimate k on Σt . k satisfies the following symmetric Hodge system on Σt : curlkij = ∗ Rµiνj T µ T ν ,∇j kij = 0,(4.69)

trk = 0,lm
ij
where curlkij = 21 (∈lmi ∇l kmj + ∈j ∇l kmi ) and trk = g kij . Using an elliptic estimate for the
Hodge system (4.69), we easily obtain:. ε.
k∇kkL∞2t L (Σt )(4.70)

Recall from (1.9) that the lapse n satisfies the following elliptic equation on Σt :∆n = |k|2 n.

(4.71)

Using (4.71) and (4.70), together with elliptic estimates on Σt , we improve the estimate for n in
the bootstrap assumptions (4.58) (4.59). We also prove the following estimate which is needed
for the estimate (2.61)
k∇nkL∞ (M) . ε.
(4.72)
Using (4.71) and (4.70) together with the Sobolev embedding on the three dimensional rie3
mannian manifold Σt yields ∆n ∈ L∞
t L (Σt ). Together with elliptic estimates, this implies2∞3∇ n ∈ Lt L (Σt ), and thus ∇n misses to be in L∞ (M) by a log divergence. However, one canovercome this loss by exploiting the Besov improvement with respect to the Sobolev embedding

4.2. REGULARITY OF THE FOLIATION WITH RESPECT TO (t, x)79

on Σt . This requires to introduce a geometric Littlewood-Paley theory on Σt 2. We refer thereader to section 4.4 in [75] for the details.
Finally, we estimate k on Hu . To this end, we use the decomposition of k (4.14) in δ,  and η,and obtain a Hodge system for δ,  and η on Hu . This allows us to derive the following estimate
N1 (k) . N1 (η) + N1 () + N1 (δ) . ε.

(4.73)

Deriving an estimate for T (δ) and ∇/ T , together with (4.73), then yields
kD L δkL∞+ k∇/ L kL∞. ε.
2
2u L (Hu )u L (Hu )(4.74)

4.2.4. Time foliation versus geodesic foliation. While we work with a time foliation,
we recall that the estimates corresponding to the bootstrap assumptions on χ and ζ have alreadybeen proved in the context of a geodesic foliation in [39] [41] [40]. One may reprove theseestimates by adapting the proofs to the context of a time foliation. However, this would berather lengthy and we suggest a more elegant solution which consists in translating certain
estimates from the geodesic foliation to the time foliation, and in obtaining directly the rest of
the estimates. More precisely, we wish to obtain the L∞ bound from trχ, and the trace boundsfor χ
b and ζ by exploiting the corresponding estimates in the geodesic foliation. We will obtainthe trace bounds for δ and  by reducing to estimates in the geodesic foliation in section 4.2.5.
Finally, these trace bounds and the null structure equations will allow us to get all the remaining
estimates in section 4.2.6. We start by recalling some of the results obtained in the context ofthe geodesic foliation in [39] [41] [40].
4.2.4.1. The case of the geodesic foliation. Recall that L0 = −gαβ ∂β u∂α is the geodesic null
0 the level
generator of Hu . Let s denote its affine parameter, i.e. L0 (s) = 1. We denote by Ps,usurfaces of s in Hu .
0
Definition 4.23. A null frame e01 , e02 , e03 , e04 at a point p ∈ Ps,uconsists, in addition to
0
0
0
0
0
e4 = L , of arbitrary orthonormal vectors e1 , e2 tangent to Ps,u and the unique vectorfield e03 = L0
satisfying the relations:g(e03 , e04 ) = −2, g(e03 , e03 ) = 0, g(e03 , e01 ) = 0, g(e03 , e02 ) = 0.
Definition 4.24 (Ricci coefficients in the geodesic foliation). Let e01 , e02 , e03 , e04 be a null frame
0 as above. The following tensors on P 0
on Ps,us,uχ0AB =< De0A e04 , e0B >, χ0AB =< De0A e03 , e0B >,0 = 1 < D 0 e0 , e0 >ζA
eA 4 32(4.75)are called the Ricci coefficients associated to the geodesic foliation.
We decompose χ0 and χ0 into their trace and traceless components.
trχ0 = gAB χ0AB ,1
χb0AB = χ0AB − trχ0 gAB ,2

trχ0 = gAB χ0AB ,1
χb 0AB = χ0AB − trχ0 gAB .
2(4.76)(4.77)

2Note that we use a geometric construction based on the heat flow on Σ since we don’t have enough regularityt

for the metric in order to use a coordinate dependent Littlewood-Paley decomposition

80

4. CONTROL OF THE SPACE-TIME FOLIATION

Definition 4.25. The null components of the curvature tensor R of the space-time metricg in the geodesic foliation are given by:
10αab= R(L0 , e0a , L0 , e0b ) ,βa0 = R(e0a , L0 , L0 , L0 ),(4.78)
2
1
1
ρ0 =R( L0 , L0 , L0 , L0 ) ,σ 0 = ? R( L0 , L0 , L0 , L0 )(4.79)
441
α0 ab = R( L0 , e0a , L0 , e0b )(4.80)
β 0a =
R(e0a , L0 , L0 , L0 ) ,2
where ? R denotes the Hodge dual of R.
We now recall the main estimates obtained in [39] [41] [40]. We have:ktrχ0 kL∞ (Hu ) + kb
χ0 kL2 0 L∞
+ kζ 0 kL2 0 L∞.εss

(4.81)kχ0 kL2 0 L∞
+ N10 (χ0 ) + N10 (ζ 0 ) . ε,s

(4.82)

x

x

and
x

where the norm N10 is given by
N10 (F ) = kF kL2 (Hu ) + k∇
/ 0 F kL2 (Hu ) + k∇
/ L0 F kL2 (Hu ) .
Remark 4.26. Note that the norm L∞ (Hu ) does not depend on the particular foliation.
Now, this is also the case for the trace norm L2x0 L∞
s . Indeed, recall the definition of the nullgeodesic κx in Remark 4.2. Then, we have:
Z 1
Z 1
kF k2L∞ L2 = sup|F (κx (t))|2 dt = sup|F (κx (s))|2 n−1 b−1 ds ∼ kF k2L∞0 L2sx0t(0,x)∈Σ00(0,x)∈Σ0x0dt
where we used the fact that ds= n−1 b−1 and the fact that nb ∼ 1 by the bootstrap assumption(4.58).

In the next section, we will obtain the estimates corresponding to (4.81) in the time foliation.
For now, we conclude this section by recalling the definition and some properties of the Besov
0 -tangent tensors F on H , 0 ≤ a ≤ 1, we introduce
spaces constructed in [39] [41] [40]. For Ps,uu
the Besov norms:
X
0
kF kB0 a =2ja sup kPj0 F kL2 (Ps,u(4.83)
0 ) + sup kP<0 F kL2 (P 0 ) ,s,uj≥0kF kP 0 a

=

X

0≤s≤1

0≤s≤1

0
2ja kPj0 F kL2 (Hu ) + kP<0
F kL2 (Hu )(4.84)j≥0
0 . Using thewhere Pj0 are the geometric Littlewood-Paley projections on the 2-surfaces Ps,udefinition of these Besov spaces, we have (see [39] [41] [40])kχ0 kB0 0 . ε.

(4.85)

Furthermore, we have for scalar functions on Hu (see [39] section 5):
kf kL∞ (Hu ) . kf kB0 1 . kf kL∞/ 0 f kB0 0 .
2 + k∇
s L 0(4.86)

x

Finally, we have the following version of the sharp classical trace theorem (see Corollary 4.21 in[75] for a proof).

4.2. REGULARITY OF THE FOLIATION WITH RESPECT TO (t, x)

81

0 -tangent tensor which admits a decomposition of
Proposition 4.27. Assume F is an Ps,u0
the form, ∇/ F = A∇
/ L0 P + E. Then,kF kL∞0 L2s . N10 (F ) + N10 (P )(kAkL∞ + k∇
/ 0 AkL2 0 L∞
+ k∇
/ L0 AkL2 0 L∞
) + kEkP 0 0 .
s
s
x

x

(4.87)

x4.2.4.2. Estimates in the time foliation. In this section, we obtain the L∞ bound for trχ,and the trace bounds for χb and ζ by relying on the corresponding estimates in the geodesic
foliation (4.81). We start by establishing the relation between the Ricci coefficients in the timeand in the geodesic foliation. Recall from (4.4) that L = bL0 . Since (e1 , e2 ) and (e01 , e02 ) are bothorthonormal vectors in the tangent space of Hu which are both orthogonal to L, we may chose
0 satisfying:these vectors such that there is a tensor F 0 on Ps,ueA = e0A + FA0 L0 , A = 1, 2.
We then easily express L in the frame (L0 , L0 , e0A ). Finally, we have the following relations:L = bL0 ,eA = e0A + FA0 L0 , A = 1, 2,
L = b−1 L0 + 2b−1 FA0 e0A + b−1 |F 0 |2 L0 .

(4.88)

Next, using the definition (4.8) and (4.75) of the Ricci coefficients respectively in the timeand geodesic foliation, and the identities (4.88), we easily obtain0χ = bχ0 , trχ = btrχ0 , χ
b = bbχ0 , ζ A = ζ A+ χ0AC FC0 .

(4.89)(4.89) together with the bootstrap assumption (4.58) and the estimate (4.81) yields:
ktrχkL∞ (Hu ) ≤ kbkL∞ (Hu ) ktrχ0 kL∞ (Hu ) . ε,kb
χkL∞0 L2t ≤ kbkL∞ (Hu ) kb
χ0 kL2 0 L∞
. ε,
x
x s
0
0
kζkL∞0 L2t . kζ kL∞0 L2s + kχ kL∞0 L2s kF 0 kL∞ . ε + εkF 0 kL∞ ,xx(4.90)x2 ∞where we have used the fact that the trace norms L2x0 L∞
t and Lx0 Ls are equivalent by Remark
4.26.
In view of the trace estimate for ζ given by (4.90), we need to estimate kF 0 kL∞ . To thisend, we estimate ∇/ 0 F 0 . Using the definition (4.8) of χ and (4.75) of χ0 , and the identities (4.88),
we obtain:
1
g(De0A F 0 , e0B ) = − χ0AB + · · · ,2where we only kept the main term. Together with the estimate for χ0 (4.85), this yieldsk∇/ 0 F 0 kB0 0 . Dε
which together with (4.86) implies:
kF 0 kL∞ . Dε.

(4.91)kζkL∞0 L2t . ε.

(4.92)

In particular, (4.90) and (4.91) imply:x

Note that (4.90) and (4.92) are improvements of the corresponding estimates in the bootstrap
assumptions (4.61)-(4.63).

82

4. CONTROL OF THE SPACE-TIME FOLIATION

4.2.5. Trace norm bounds for δ and . The goal of this section is to improve the estimate
for kδkL∞0 L2t and kkL∞0 L2t given by the bootstrap assumption (4.60), where δ and  are definedxxin (4.16). Let us first define kLL and kLA :kLL = −g(DL T, L), kLA = −g(DL T, eA ), A = 1, 2.

(4.93)

Then, using in particular the definition (4.16), we have:δ = kLL and A = kLA .

(4.94)

kL0 L0 = −g(DL0 T, L0 ), kL0 A = −g(DL0 T, e0A ), A = 1, 2.

(4.95)We also define kL0 L0 and kL0 A :Then, the relations (4.88) between L, e1 , e2 and L0 , e01 , e02 together with the definitions (4.93) and(4.95) yield:
kLL = b2 kL0 L0 and kLA = bkL0 A + bFA0 kL0 L0 .
(4.96)
Thus, (4.94) and (4.96) imply:kδkL∞0 L2t. kbkL0 L0 kL∞0 L2s . kkL0 L0 kL∞0 L2skkL∞0 L2t. kbkL0 A kL∞0 L2s + kbFA0 kL0 L0 kL∞0 L2s . kkL0 L0 kL∞0 L2s + kkL0 A kL∞0 L2sxxx(4.97)

x

x

x

x

x

where we used the bootstrap assumption (4.58), the L∞ bound for F 0 (4.91) and Remark 4.26.
In view of (4.97), it is enough to bound the trace norms kkL0 L0 kL∞0 L2s and kkL0 A kL∞0 L2s . Toxx
this end, we would like to apply the trace estimate (4.87), which requires to show that ∇/ 0 kL0 L00and ∇/ kL0 A admit a decomposition of the form, A∇/ L0 P + E. We only discuss the estimate for000kL L , and we refer the reader to [75] for kL A . We have:∇/ 0e0 kL0 L0 = −De0A g(DL0 T, L0 ) = −g(De0A DL0 T, L0 ) − g(DL0 T, De0A L0 ).
A

Introducing the commutator term [De0A , DL0 ], and decomposing the corresponding componentof R, we obtain0∇/ 0e0 kL0 L0 = −b−1 FB0 αAB+ ··· ,
(4.98)
A

where we only kept a typical term for simplicity. Relying on the Bianchi identities, the following
decomposition for α0 was obtained in [39]:α0 = ∇/ L0 (P ) + E,

(4.99)

0where P = D0 −1
2 β , andN10 (P ) + kEkP 0 0 . ε.
Together with (4.98), we obtain a decomposition of the following form

(4.100)∇/ 0 kL0 L0 = A1 ∇/ L0 P1 + E1 ,(4.101)kA1 kL∞ + k∇
/ 0 A1 kL2 0 L∞
+ k∇
/ L0 A1 kL2 0 L∞
+ N10 (P1 ) + kE1 kP 0 0 . ε.
ss

(4.102)

where
x

x

Using (4.101), (4.102) and the trace estimate (4.87), we deducekkL0 L0 kL∞0 L2s . ε,xwhich together with (4.97) allows us to improve the estimate for kδkL∞0 L2t and kkL∞0 L2t given byxx
the bootstrap assumption (4.60).

4.3. AN ESTIMATE FOR LLTRχ

83

4.2.6. Remaining estimates for trχ, χb, ζ and b. In order to improve the remaining
estimates in the bootstrap assumptions (4.58)-(4.63), we use the null structure equation ofsection 4.1.2, which consists of transport equations along L and Hodge systems on Pt,u . We
refer the reader to section 4.8 in [75], where using the L∞ bound of trχ, the trace estimates forχ
b, δ and , and the estimates for the lapse n, we easily obtain the remaining estimates. Thus,
there exists a universal constant D > 0 such that (4.58)-(4.63) hold. This yields (4.40)-(4.45)
which concludes the proof of Theorem 4.15.
4.3. An estimate for LLtrχ
In this section, we outline the main ideas of the proof of Theorem 4.16. Let µ1 = b L(µ).
Then, we first derive a transport equation for µ1 , and a Hodge system for ∇/ L ζ. For simplicity,
we only discuss the transport equation for µ1 . We differentiate the transport equation (4.27)
satisfied by µ with respect to L and multiply it by nb. We also use commutator formulas ofsection 4.1.3, the Bianchi identity (4.34) for ρ, the curvature bound (4.39) and the estimates(4.40)-(4.45) obtained in Theorem 4.15. We obtain−1b / L (ζ) + b ∇bnL(µ1 ) + ntrχµ1 = −2bn∇/ L (ζ) · ∇/ trχ − 2bnb
χ· ∇
/ ⊗∇
/ b∇
/ L (ζ) + 2∇
/ L ζ ⊗ζ+2ntrχbn−1 ∇/n · ∇/ L (ζ) + div/ (F1 ) + f2 ,where the tensor F1 and the scalar f2 satisfy
kF1 kL2 (Hu ) + kf2 kL1 (Hu ) . ε.
This yields:jkPj (µ1 )kL2 (Hu ) . 2 2 λ(u)ε + 2j ε + Pj

Z t
0


b / L (ζ))dτ(bnb
χ · (∇/ ⊗∇+ ··· .

(4.103)

L2 (Hu )Here, the term 2j ε comes from the estimate for F1 and f2 together with Bernstein and the finite
jband property for Pj , and the term 2 2 λ(u)ε comes from the initial data term for the transport
equation - i.e. µ1 at t = 0 which is estimated in [73] - together with Bernstein for Pj . We have
only kept one typical term in the right-hand side of (4.103) for the sake of simplicity.
In view of the desired estimate (4.47), we are bootstrapping an estimate of the typejkPj ∇/ L (ζ)kL2 (Hu ) . Dε + D2− 2 λ(u)for some large enough bootstrap constant D, and where λ is a function in L2 (R). Thus, estimating directly the term in the right-hand side of (4.103) would yield an upper bound of the
type
X
|q−l|
(2)
(2)
2j 2− 2 γq(1) γl , where γq(1) ∈ `2 (N) and γl ∈ `∞ (N)(4.104)l,qwhich is not summable. Instead, we rely on the following decomposition for bnb
χ:∇/ (bnbχ) = ∇/ nL P + E
where P , E are Pt,u -tangent tensors, and P , E satisfy:
N1 (P ) + kEkP 0 . ε.

(4.105)

84

4. CONTROL OF THE SPACE-TIME FOLIATION

Remark 4.28. A similar decomposition has been proved in the geodesic foliation in [39], andadapted to the time foliation in the spirit of section 4.2.4. In order to obtain (4.105), we use thefact that the proof in the geodesic foliation relies on a specific structure of certain commutatorsand of the Bianchi identities, which can be recovered in the time foliation. We refer to [75] forthe details.
Using the decomposition(4.105), we decompose the term in the right-hand side of (4.103) ina sum of two terms which are estimated as follows (see [75] for the details):• For the term involving ∇/ nL P , we integrate by parts in ∇/ nL , and consider the term where
the L derivative falls on ∇/ L ζ. Differentiating the transport equation (4.21) satisfied byζ with respect to ∇/ L , commutators formula, and the Bianchi identity (4.32), we obtain/ nL D0 (β) + · · ·∇/ nL ∇/ Lζ = ∇for some elliptic operator of order 0 D0 on Pt,u . We then integrate by parts the Lderivatives, and obtain for this term an upper bound of the type
2j N1 (P )kβkL2 (Hu ) + · · · .
Then, using the estimate for P and the the curvature bound (4.39) for β, this is enoughto bound the term involving ∇/ nL P in the right-hand side of (4.103).
• For the term involving E, we we rely on the Besov improvement for E, and we derive
an upper bound of the form
X
|q−l|
(2)
(2)
2j 2− 2 γq(1) γl , where γq(1) ∈ `1 (N) and γl ∈ `∞ (N),l,qwhich is summable unlike (4.104). This is enough to bound the term involving E in
the right-hand side of (4.103).
Remark 4.29. The reader may wonder why the estimate for LLtrχ is not better than L2with respect to the variable t - instead of L∞ as one should expect since we rely on a transport
equation for the corresponding quantity µ1 . The reason is that boundary terms arise from several
integration by parts in ∇/ L in the course of the proof. The point is that we do not have better
estimates than L2 with respect to the variable t for these terms.
4.4. Regularity of the foliation with respect to ω
4.4.1. First order derivatives with respect to ω. In this section, we outline the mainideas of the proof of Theorem 4.17. Let us first explain how to control ∂ω trχ. Differentiating
the Raychaudhuri equation (4.19) with respect to ω, we obtainL(∂ω trχ) = [L, ∂ω ]trχ + · · · .
Now, we have[L, ∂ω ]trχ = −∂ω N (trχ) = −∇/ ∂ω N trχ
where we used the fact that g(N, N ) = 1, which differentiated with respect to ω implies that∂ω N is a vectorfield tangent to Pt,u . Thus, we obtainL(∂ω trχ) = −∇/ ∂ω N trχ + · · ·4.4. REGULARITY OF THE FOLIATION WITH RESPECT TO ω

85which together with the estimate (4.43) for ∇/ trχ immediately yieldsk∂ω trχkL2 0 L∞
. ε.
t
x

Remark 4.30. In view of the commutator[L, ∂ω ] = −∇/ ∂ω N ,a derivative with respect to ω has essentially the same regularity as a ∇/ -derivative.
The estimates in (4.48) and (4.49) are obtained in the same way, i.e. by differentiating the
ricci equations (4.15) and the transport equations (4.18) (4.19) (4.20) and (4.21) with respectto ω, computing the commutators [∂ω , DL ] and [∂ω , ∇/ L ], and estimating the correspondingtransport equations (see [75] for the details).
Next, let us explain how to derive the decomposition (4.50) for χb:χb = χ1 + χ2 ,where χ1 and χ2 are two symmetric traceless Pt,u -tangent 2-tensors satisfying the estimates(4.51) and (4.52). Recall the Codazzi type equation (4.22) satisfied by χ
b:
1
div
/χ
b= ∇
/ trχ − β + · · · .
2This is an elliptic system on Pt,u , and we may write formally1χb = D−1 ∇/ trχ − D−1 β + · · · ,2where D−1 is a pseudodifferential operator of order -1 on Pt,u . This allows us to define χ1 andχ2 as
1χ1 = D−1 ∇/ trχ + · · · and χ2 = −D−1 β.
2The estimate (4.51) corresponds to the estimate (4.44) for χb and is prove similarly, so we focus
on the estimate (4.52). For the sake of clarity, we only explain why, compared to χb, χ1 hasbetter regularity with respect to (t, x) while χ2 has better regularity with respect to ω, which isthe point of the decomposition (4.50) (see Remark 4.18). Indeed, since the estimate (4.43) fortrχ is better than the estimate (4.44) for χb, and since ∇/ D−1 is a pseudodifferential operator oforder 0 on Pt,u , we are able to obtain better regularity in (t, x) for χ1 compared to χb. Next,
we focus on χ2 . Now, note that the curvature tensor R does not depend on ω. Thus, whendifferentiating β with respect to ω, the ω derivative falls on the frame (L, L, eA ), and we obtain
schematically∂ω β = (α + ρ + σ)∂ω N.
In particular, we have+ kρkL∞+ kσkL∞) . ε,k∂ω βkL∞. k∂ω N kL∞ (kαkL∞2
2
22
u L (Hu )u L (Hu )u L (Hu )u L (Hu )where we used the curvature bound (4.39) for α, ρ and σ, and the estimate (4.48) for ∂ω N .
Thus, ∂ω β has the same regularity with respect to (t, x) than β. In view of the definition ofχ2 , we obtain that ∂ω χ2 has essentially the same regularity as χ2 , while the estimate (4.49) for∂ω χb looses one ∇/ -derivative with respect to the estimate (4.44) for χb. Thus, the regularity ofχ2 with respect to ω is better than the corresponding regularity for χb.

86

4. CONTROL OF THE SPACE-TIME FOLIATION

4.4.2. Second order derivatives with respect to ω. In this section, we outline the mainideas of the proof of Theorem 4.19. We focus on the estimate (4.55) for ∂ω2 ζ which is typical.
Differentiating twice with respect to ω the transport equation (4.21) for ζ, and computing thecommutator [∇/ L , ∂ω2 ], we obtain∇/ L (Π(∂ω2 ζ)) = −χ · Π(∂ω2 ζ) + ∇/ (F1 ) + F2 + · · · ,(4.106)

where the Pt,u -tangent tensors F1 and F2 satisfy
kF1 kL2 (Hu ) + kF2 kL1 0 L2t . ε.
x

We first get rid of the first term in the right-hand side of (4.106) which is troublesome. To thisend, we use the following lemma.
Lemma 4.31. Let γ denotes the metric induced by g on Pt,u . Let M the Pt,u -tangent 2-tensor
defined as the solution of the following transport equation:∇/ L MAB = MAC χCB , MAB = γAB on P0,u ,(4.107)

Then, MAB satisfies the following estimate:kM − γkL∞ + k∇
/ M kB0 . ε.

(4.108)Using the transport equation (4.106) for Π(∂ω2 ζ) and the transport equation (4.107), for M
allows us to get rid of the troublesome term χ · Π(∂ω2 ζ):∇/ L (M · Π(∂ω2 ζ)) = ∇/ (M · F1 ) − ∇/ (M ) · F1 + M · F2 + · · · .
Together with the finite band property and the Bernstein inequality for Pj , the estimates for F1
and F2 , and the estimate (4.108) for M , we obtain for M · Π(∂ω2 ζ) the estimate corresponding
to (4.55). Then, we obtain the wanted estimate (4.55) for Π(∂ω2 ζ) by proving that the estimate(4.108) for M is enough to ensure that the multiplication by M −1 preserves the estimate (4.55).
4.5. Additional decompositions
In this section, we outline the main ideas of the proof of Theorem 4.20. We need to compare
N and trχ at two different angles ω and ν. The basic tool is the following lemma.
Lemma 4.32. Let ω and ω 0 in S2 . Let u = u(t, x, ω) and u0 = u(t, x, ω 0 ). Then, for any
tensor F , we have:!! 1
Z u+|ω−ω0 |
4
1
2
0 41
2
kF
k
sup
kDF
k
dτ
.
kF kL∞0 L2 (Hu0 ) . kF kL∞
+
|ω
−
ω
|
2
L2 (Hτ )L∞ L2 (Hu )u L (Hu )
u

u

u

u

2
∞ 2
In order to compare the norms L∞
u0 L (Hu0 ) and Lu L (Hu ), we need coordinate systems. We
3define Φt,ω : Σt → R defined by:

Φt,ω (t, x) := u(t, x, ω)ω + ∂ω u(t, x, ω).
is a global C 1 diffeomorphism from Σ(4.109)to R3 and therefore provides a

Then we claim that Φt,ωt
global coordinate system on Σt (see Proposition 6.6 for a related result on Σ0 ). Next, we prove
that
Z 1Z
0
2 0
kF k2L2 (Hu ) '|F (Φ−1(4.110)t,u,ω (y ))| dy dt.
0R2

4.5. ADDITIONAL DECOMPOSITIONS

87

2
∞ 2
This formula allows us to compare the norms L∞
u0 L (Hu0 ) and Lu L (Hu ). In turn, one needs
to evaluate
−1
0
2
0
2
|F (Φ−1t,u,ω (y ))| − |F (Φt,u0 ,ω 0 (y ))|In particular, we need to estimate0∂ω Φ−1t,u,ω (y ) L∞ .

We refer the reader to [75] for details on the proof of Lemma 4.32.
Using Lemma 4.32 as well as commutator estimates for [D, Pl ] among others, we may prove
the following corollary.
Corollary 4.33. Let f a scalar function and ω, ω 0 in S2 . Then, for any l ≥ 0, we have:1l),+ kDf kL∞
kPl f kL∞0 L2 (Hu0 ) . (2−l + |ω − ω 0 | 2 2− 2 )(kf kL∞2
2
u L (Hu )u L (Hu )uandkP≤l f kL∞0 L2 (Hu0 )
u

1

l

1

1

. (1 + |ω − ω 0 | 2 2 2 )kf kL∞+ |ω − ω 0 | 4 kf kL2 ∞ L2 (Hu )2u L (Hu )
u
1

4
Z
X u+|ω−ω0 |22(kPq (nL(f ))kL2 (Hτ ) + kPq (bN (f ))kL2 (Hτ ) )dτ  .
× supuq≤lu

We also need the following non sharp commutator lemma.
Lemma 4.34. Let f a scalar function and ω, ω 0 in S2 . Then, for any l ≥ 0, we have:k[∂ω , P≤l ]f kL∞0 L2 (Hu0 ) . kDf kL∞.
2
u L (Hu )u

Using Corollary 4.33 and Lemma 4.34 together with the estimates (4.43) and (4.49) for trχ
jand the fact that |ω − ν| . 2− 2 , we are able to prove the decomposition (4.57) for trχ.
Next, we consider N − Nν . We haveZjj2∂ω N (., ω 00 )dω 00 (2 2 (ω − ν)),2 (N − Nν ) =[ω,ν]jwhere [ω, ν] denotes the arc of S2 joining ω and ν. Since |ω − ν| . 2− 2 , we want to proceed as
for the decomposition of trχ. More precisely, we want to use Corollary 4.33 and Lemma 4.34
together with the estimates (4.48), (4.49), (4.53), (4.54) and (4.55) for ∂ω N , in order to provethe decomposition (4.56) for N − Nν . Now, unlike trχ which is a scalar, ∂ω N is a tensor. Since
Corollary 4.33 and Lemma 4.34 only apply to scalars, we need one last ingredient to prove thedecomposition (4.56) for N − Nν and conclude the proof of Theorem 4.20. Namely we need toscalarize ∂ω N using a basis of the tangent space of Σt which does not depend on ω. We refer to[75] for the details.

CHAPTER 5

Construction and control of the parametrix at initial time
In this chapter, and the next one, we will only consider the leave Σ0 of the foliation Σt ofM, and we denote it by Σ for simplicity. Recall the plane wave type parametrix given by (1.22)1Z Z ∞ωeiλ u(t,x) f (λω)λ2 dλdωS20where u(., ., ω) is a solution to the eikonal equation gαβ ∂α u∂β u = 0 on M such that u(0, x, ω) ∼x · ω when |x| → +∞ on Σ. The goal of this chapter is to outline the main ideas allowing us toobtain the control for that parametrix restricted to Σ in [76].
5.1. Geometric set-up and main results
5.1.1. Presentation of the parametrix. In this section, we construct a parametrix forthe following homogeneous wave equation:
g φ = 0 on M,(5.1)
φ|Σ = φ0 , T (φ)|Σ = φ1 ,where φ0 and φ1 are two given functions on Σ and T is the future oriented unit normal to Σ inthe space-time M.
We recall the plane wave representation of the solution of the flat wave equation. This
corresponds to the case where g is the Minkowski metric. (5.1) becomes:φ = 0 on R1+3 ,(5.2)
φ(0, .) = φ0 , ∂t φ(0, .) = φ1 on R3 .
The plane wave representation of the solution φ of (5.2) is given by:
Z Z +∞
Fφ1 (λω)i(−t+x·ω)λ 1
Fφ0 (λω) + i
dλdω
e
2 
λ
2 0
SZ

Z +∞
1
Fφ1 (λω)+ei(t+x·ω)λFφ0 (λω) − i
dλdω,
2
λ
S2 0

(5.3)

where F denotes the Fourier transform on R3 .
We would like to construct a parametrix in the curved case similar to (5.3). We introduce
two solutions u± of the eikonal equationgαβ ∂α u± ∂β u± = 0 on M,(5.4)

T (u± ) = ∓|∇u± | = ∓a−1
± on Σ,(5.5)

such that:
1This is actually a half wave parametrix. See (5.6) below for the full parametrix
8990

5. CONSTRUCTION AND CONTROL OF THE PARAMETRIX AT INITIAL TIME

where ∇ is the covariant derivative on Σ associated to the metric g induced by g on Σ, | · | isthe length associated to g for vectorfields on Σ, and a± is the lapse of u± on Σ. We look for aparametrix for (5.1) of the form:
S(t, x) = S+ f+ (t, x) + S− f− (t, x),
where

Z Z +∞
S± f± (t, x) =S2

eiλu± (t,x,ω) f± (λω)λ2 dλdω.

(5.6)

(5.7)

0

In the next two sections, we specify the parametrix (5.6) by prescribing u± on Σ and by making
our choice for f± explicit.
5.1.1.1. Prescription of u+ and u− on Σ. (5.4) and (5.5) are not enough to define u± in aunique manner. Indeed, we still need to prescribe u± on Σ. To motivate our choice, we need to
introduce some geometric objects connected to u± . Let N± the vectorfield on Σ defined by:
N± =

∇u±
= a± ∇u± ,|∇u± |(5.8)

and L± the vectorfield on M which is given on Σ by:
L± = a± gαβ ∂α u± ∂β = a± (−T (u± )T + ∇u± ) = ±T + N± .

(5.9)

Let Pu± = {x ∈ Σ/ u± (x) = u± } denote the level surfaces of u± in Σ. Since N± is the unit
normal to Pu± , the second fundamental form of Pu± in Σ is given by:±±
θ± (e±A , eB ) = g(De± N± , eB ), A, B = 1, 2,(5.10)

A±
where (e±1 , e2 ) is an arbitrary orthonormal frame of T Pu± . Let

Hu± = {(t, x) ∈ M/ u± (t, x) = u± }denote the null level hypersurfaces of u± in M. Since L± is null and orthogonal to Pu± in Hu± ,the null second fundamental form χ± is given on Pu± by:±±
χ± (e±A , eB ) = g(De± L± , eB ), A, B = 1, 2.

(5.11)A

Taking the trace in (5.10) and (5.11), and using (5.9) and the fact that k is the second fundamental form of Σ, we obtain:trχ± = ±trk + trθ± .
(5.12)
Note that trg k = trk + kN N , where trg denotes the trace for 2-tensors on Σ. In view of the
maximal foliation assumption (1.6), we have trg k = 0. Together with (5.12), this yields:trχ± = ∓kN± N± + trθ± .

(5.13)

Now, in [75] (see also Theorem 4.15), we prove that trχ± belongs to L∞ (M) using a transport
equation (the Raychaudhuri equation, see (4.19)) provided that it belongs to L∞ (Σ) at t = 0.
Thus, one needs the following estimatetrχ± ∈ L∞ (Σ),(5.14)∓kN± N± + trθ± ∈ L∞ (Σ).

(5.15)

which in view of (5.13) is equivalent to:5.1. GEOMETRIC SET-UP AND MAIN RESULTS

91

We construct in [73] (see also Chapter 6) a function u(x, ω) on Σ × S2 such that−kN N + trθ ∈ L∞ (Σ).

(5.16)

kN N + trθ ∈ L∞ (Σ).

(5.17)

Note that −u(x, −ω) satisfies:
Thus, in view of (5.15), (5.16) and (5.17), we initialize u± on Σ by:
u+ (0, x, ω) = u(x, ω) and u− (0, x, ω) = −u(x, −ω) for (x, ω) ∈ Σ × S2 .

(5.18)

Remark 5.1. Note that in the particular case where k ≡ 0 - the so-called time symmetriccase-, we may takeu+ (0, x, ω) = u− (0, x, ω) = u(x, ω) for (x, ω) ∈ Σ × S2 .
In particular, we have u+ (0, x, ω) = u− (0, x, ω) = x · ω in the flat case.
5.1.1.2. The choice of f+ and f− . Having defined u± , we still need to define f± in theparametrix (5.6). According to (5.1), the half wave parametrix S+ and S− should satisfy on Σ:S+ f+ (0, x) + S− f− (0, x) = φ0 (x),(5.19)T (S+ f+ )(0, x) + T (S− f− )(0, x) = φ1 (x).
Let us introduce the following operators acting on functions of R3 :Z Z +∞
M± f (x) =e±iλu(x,±ω) f (λω)λ2 dλdωS2(5.20)0and
Z Z +∞
Q± f (x) =S2e±iλu(x,±ω) a(x, ±ω)−1 f (λω)λ2 dλdω,(5.21)

0where a(x, ω) = |∇u(x, ω)|−1 is the lapse of u. Using (5.5), the definition of S± in (5.6), (5.18),
the definition (5.20) of M± and the definition (5.21) of Q± , we may rewrite (5.19) as:
M+ f+ + M− f− = φ0 ,(5.22)Q+ (λf+ ) − Q− (λf− ) = iφ1 .
The goal of this chapter will be to show that there exist a unique (f+ , f− ) satisfying (5.22), andthat (f+ , f− ) satisfies the following estimate:kλf+ kL2 (R3 ) + kλf− kL2 (R3 ) . k∇φ0 kL2 (Σ) + kφ1 kL2 (Σ) .

(5.23)

Remark 5.2. In the case of the flat wave equation (5.2), we have (Σ, g, k) = (R3 , δ, 0),u± (t, x, ω) = ∓t + x · ω, u(x, ω) = x · ω and a(x, ω) = 1. In particular, the operators M± and Q±
defined respectively by (5.20) and (5.21) all coincide with the inverse Fourier transform. Then,the system (5.22) admits the following solutions:1Fφ1 (λω)f± (λω) =Fφ0 (λω) ± i,2λwhich clearly satisfy the estimate (5.23).

92

5. CONSTRUCTION AND CONTROL OF THE PARAMETRIX AT INITIAL TIME

5.1.2. Geometric set-up. We define the lapse a(x, ω) = |∇u(x, ω)|−1 , and the unit vector
N such that ∇u(x, ω) = a(x, ω)−1 N (x, ω). We also define the level surfaces Pu = {x / u(x, ω) =u} so that N is the normal to Pu .
For 1 ≤ p, q ≤ +∞, we define the spaces Lpu Lq (Pu ) using the norm
Z
1/p
p
kF kLq (Pu ) du.
kF kLpu Lq (Pu ) =uWe assume that 1/2 ≤ a(x) ≤ 2 for all x ∈ Σ (see (5.24) below) so that Lpu Lp (Pu ) coincides
with Lp (Σ) for all 1 ≤ p ≤ +∞. We denote by γ the metric induced by g on Pu , and by ∇/ theinduced covariant derivative.
Before stating precisely the main results of this chapter, we first record the regularity obtained for the phase u(x, ω) constructed in [73] (see also Chapter 6).
5.1.3. Regularity assumptions on the phase u(x, ω). In this section, we collect theestimates for the phase u(x, ω) of our Fourier integral operators that are needed to follow thediscussion of the control of the parametrix at initial time contained in this chapter. An outlineof the proof of these estimates will be given in Chapter 6 (see [73] for the complete proof).
We start with the regularity in x of the lapse a. We need:k∇akL∞+ ka − 1kL∞ (Σ) + k∇
/ ∇akL2 (Σ) . ε.
2u L (Pu )(5.24)We also need a decomposition for ∇N a. For all j ≥ 0, there are scalar functions aj1 and aj2 suchthat2:∇N a = aj1 + aj2 where kaj1 kL2 (Σ) . 2−j/2 ε, kaj2 kL∞.ε2u L (Pu )(5.25)and k∇N aj2 kL2 (Σ) + kaj2 kL2u L∞ (Pu ) . 2j/2 ε.
Next, we consider the regularity with respect to ω. We have:k∂ω akL2 (Σ) + k∇∂ω akL2 (Σ) . ε,

(5.26)k∂ωα akL∞ (Σ) . 1 for some 0 < α < 1,

(5.27)

where (5.27) should be understood in the Hölder sense,k∂ω N kL∞ (Σ) . 1,

(5.28)k∂ω3 ukL∞. 1.
loc (Σ)(5.29)

andWe will need the following global change of variable on Σ. Let ω ∈ S2 . Let φω : Σ → R3 definedby:φω (x) := u(x, ω)ω + ∂ω u(x, ω).
(5.30)
Then φω is a bijection, and the determinant of its Jacobian satisfies the following estimate:k| det(Jacφω )| − 1kL∞ (Σ) . ε.

(5.31)

j2we choose aj = P>j/2 (∇N a) and a2 = P≤j/2 (∇N a), and then obtain (5.25) using (5.24) and an estimate for1
2∇N a (see [73] for the details)

5.1. GEOMETRIC SET-UP AND MAIN RESULTS

93

Finally, we can compare u(x, ω) with a phase linear in ω. Let ν ∈ S2 and φν the map defined in(5.30). Then, we have:u(x, ω) − φν (x) · ω = O(ε|ω − ν|2 ),∂ω u(x, ω) − ∂ω (φν (x) · ω) = O(ε|ω − ν|),∂ω2 u(x, ω) − ∂ω2 (φν (x) · ω) = O(ε).

(5.32)

Remark 5.3. In (5.24)-(5.32), all inequalities hold for any ω ∈ S2 with the constant in theright-hand side being independent of ω. Thus, one may take the supremum in ω everywhere. To
ease the notations, we do not explicitly write down this supremum.
Remark 5.4. In the case of the flat wave equation (5.2), we have (Σ, g) = (R3 , δ), u(x, ω) =x · ω, a = 1, N = ω and φω = IdR3 . Thus, (5.24)-(5.32) are clearly satisfied with ε = 0.
Remark 5.5. Recall that the lapse a is at the level of one derivative of u with respect to x.
Thus, we obtain from (5.24) that some components of ∇3 u are in L2 (Σ). Note that this is nottrue for all components since (5.25) does not allow us to control ∇2N a in L2 (Σ). In fact, (5.25)
is consistent with 3/2 derivatives of a with respect to N in L2 .
5.1.4. Main results. We first state a result of boundedness on L2 for Fourier integral
operators with phase u(x, ω).
Theorem 5.6. Let u be a function on Σ × S2 satisfying suitable assumptions (we refer to[74] for the complete set of assumptions, and to section 5.1.3 for some typical assumptions).
Let U the Fourier integral operator with phase u(x, ω) and symbol b(x, ω):Z Z +∞U f (x) =eiλu(x,ω) b(x, ω)f (λω)λ2 dλdω.
(5.33)S2

0

Let D > 0. We assume furthermore that b(x, ω) satisfies:
kbkL∞ (Σ) + k∇bkL∞+ k∇/ ∇bkL2 (Σ) . D,
2u L (Pu )(5.34)k∂ω bkL2 (Σ) + k∇∂ω bkL2 (Σ) . D,

(5.35)

andj∇N b = bj1 + bj2 where kbj1 kL2 (Σ) . 2− 2 D, kbj2 kL∞. D,
2u L (Pu )jand k∇N bj2 kL2 (Σ) + kbj2 kL2u L∞ (Pu ) . 2 2 D.

(5.36)

Then, U is bounded on L2 and satisfies the estimate:kU f kL2 (Σ) . Dkf kL2 (R3 ) .

(5.37)

Remark 5.7. We intend to apply Theorem 5.6 to the Fourier integral operators M± and Q±
introduced in section 5.1.1.2 whose symbol are respectively 1 and a−1 . Thus, our assumptions
on the regularity of the symbol b(x, ω) are consistent with the assumptions on the regularity ofa(x, ω) given by (5.24)-(5.29).
Remark 5.8. Under the additional assumption (5.32) on u, and under some restrictions on
the constant D appearing in (5.34), (5.35), and (5.36), we may prove the opposite of (5.37):
kf kL2 (R3 ) . kU f kL2 (Σ)(see Proposition 5.19). This will be a major ingredient in the proof of Theorem 5.9 below, andin particular of (5.41).

94

5. CONSTRUCTION AND CONTROL OF THE PARAMETRIX AT INITIAL TIME

Recall the definition of the Fourier integral operators M± and Q± introduced in section
5.1.1.2:
Z Z+∞

e±iλu(x,±ω) f (λω)λ2 dλdω,(5.38)e±iλu(x,±ω) a(x, ±ω)−1 f (λω)λ2 dλdω.

(5.39)

M± f (x) =
S2

0

and
Z Z +∞
Q± f (x) =S2

0

The following theorem is the main result of this chapter.
Theorem 5.9. Let u be a function on Σ × S2 satisfying suitable assumptions (we refer to[74] for the complete set of assumptions, and to section 5.1.3 for some typical assumptions).
Then, there exist a unique (f+ , f− ) satisfying:M+ f+ + M− f− = φ0 ,(5.40)Q+ (λf+ ) − Q− (λf− ) = iφ1 .
Furthermore, (f+ , f− ) satisfies the following estimate:kλf+ kL2 (R3 ) + kλf− kL2 (R3 ) . k∇φ0 kL2 (Σ) + kφ1 kL2 (Σ) .

(5.41)

Proving the estimates (5.37) and (5.41) for the Fourier integral operators U , M± and Q±
will require taking several integrations by parts. The main difficulty in proving Theorem 5.6
and Theorem 5.9 will be to perform these integrations by parts within the very low level ofregularity for the phase u(x, ω) given by (5.24)-(5.32) and for the symbol b(x, ω) given by (5.34)(5.35) (5.36). The proof will rely both on harmonic analysis decompositions and the geometryof the foliation of Σ by u. Theorem 5.6 will be reviewed in section 5.2 and Theorem 5.9 will be
reviewed in section 5.3.
5.2. Control of Fourier integral operators
5.2.1. Structure of the proof of Theorem 5.6. The proof of Theorem 5.6 proceeds inthree steps. We first localize in frequencies of size λ ∼ 2j . We then localize the angle ω inpatches on the sphere S2 of diameter 2−j/2 . Finally, we estimate the diagonal terms.
Remark 5.10. Note that the structure of the proof is analogous to the one on the control ofthe error term in Chapter 3 (see section 3.2.2). However, the proof each step (almost orthogonality in frequency, almost orthogonality in angle, and control of the diagonal term) is different,
more particularly the last two steps.
5.2.1.1. Step 1: decomposition in frequency. For the first step, we introduce ϕ and ψ twosmooth compactly supported functions on R such that:X
ϕ(λ) +
ψ(2−j λ) = 1 for all λ ∈ R.
(5.42)
j≥0

We use (5.42) to decompose U f as follows:U f (x) =Xj≥−1

Uj f (x),(5.43)

5.2. CONTROL OF FOURIER INTEGRAL OPERATORS

95

where for j ≥ 0:
Z Z +∞
Uj f (x) =S2andeiλu b(x, ω)ψ(2−j λ)f (λω)λ2 dλdω,(5.44)0Z Z +∞
U−1 f (x) =S2eiλu b(x, ω)ϕ(λ)f (λω)λ2 dλdω.

(5.45)

0

The goal of this first step is to prove the following proposition:
Proposition 5.11. The decomposition (5.43) satisfies an almost orthogonality property:X(5.46)
kUj f k2L2 (Σ) + D2 kf k2L2 (R3 ) .
kU f k2L2 (Σ) .
j≥−1

The proof of Proposition 5.11 is postponed to section 5.2.2.
5.2.1.2. Step 2: decomposition in angle. Proposition 5.11 allows us to estimate kUj f kL2 (Σ)
instead of kU f kL2 (Σ) . We perform a second dyadic decomposition. We introduce a smooth
partition of unity on the sphere S2 :X
ηjν (ω) = 1 for all ω ∈ S2 ,(5.47)
ν∈Γjwhere Γ is a lattice on S2 of size 2− 2 , where the support of ηjν is a patch on S2 of diameter∼ 2−j/2 . We use (5.47) to decompose Uj f as follows:X
Uj f (x) =Ujν f (x),(5.48)
ν∈Γ

where:
Ujν f (x) =We also define:Z Z +∞
S2

eiλu b(x, ω)ψ(2−j λ)ηjν (ω)f (λω)λ2 dλdω.

(5.49)

0γ−1 = kϕ(λ)f kL2 (R3 ) , γj = kψ(2−j λ)f kL2 (R3 ) , j ≥ 0,γjν = kψ(2−j λ)ηjν (ω)f kL2 (R3 ) , j ≥ 0, ν ∈ Γ,

(5.50)

which satisfy:
kf k2L2 (R3 ) =

X
j≥−1

γj2 =

X X

(γjν )2 .

(5.51)

j≥−1 ν∈ΓThe goal of this second step is to prove the following proposition:
Proposition 5.12. The decomposition (5.48) satisfies an almost orthogonality property:X
kUj f k2L2 (Σ) .
kUjν f k2L2 (Σ) + D2 γj2 .
(5.52)
ν∈Γ

The proof of Proposition 5.12 is postponed to section 5.2.3.
5.2.1.3. Step 3: control of the diagonal term. Proposition 5.12 allows us to estimate kUjν f kL2 (Σ)
instead of kUj f kL2 (Σ) . The diagonal term is estimated as follows.
Proposition 5.13. The diagonal term Ujν f satisfies the following estimate:kUjν f kL2 (Σ) . Dγjν .
The proof of Proposition 5.13 is postponed to section 5.2.4.

(5.53)

96

5. CONSTRUCTION AND CONTROL OF THE PARAMETRIX AT INITIAL TIME

5.2.1.4. Proof of Theorem 5.6. Proposition 5.11, 5.12 and 5.13 immediately yield the proofof Theorem 5.6. Indeed, (5.46), (5.51), (5.52) and (5.53) imply:X
kUj f k2L2 (Σ) + D2 kf k2L2 (R3 )kU f k2L2 (Σ) .
.

j≥−1
X X

kUjν f k2L2 (Σ) + D2

j≥−1X
ν∈ΓX
2

.D

(γjν )2 + D2

j≥−1 ν∈ΓX

γj2 + D2 kf k2L2 (R3 )Xj≥−1
γj2 + D2 kf k2L2 (R3 )(5.54)j≥−1. D2 kf k2L2 (R3 ) ,which is the conclusion of Theorem 5.6.
The remainder of section 5.2 is dedicated to the proof of Proposition 5.11, 5.12 and 5.13.
5.2.2. Proof of Proposition 5.11 (almost orthogonality in frequency). We have toprove (5.46):
X
kU f k2L2 (Σ) .
kUj f k2L2 (Σ) + D2 kf k2L2 (R3 ) .
(5.55)
j≥−1

This will result from the following inequality using Shur’s Lemma:Z|j−k|
Uj f (x)Uk f (x)dΣ . D2 2− 2 γj γk for |j − k| > 2.

(5.56)

Σ

We consider a coordinate system (u, x0 ) on Σ where x0 denotes a coordinate system on Pu ,and we would like to integrate by parts with respect to ∂u . Since ∇u = a−1 N and ∇u0 = a0 −1 N 0 ,we have:i0 00 0
eiλu−iλ u = −∂u (eiλu−iλ u ),(5.57)λ − λ0 aa0 g(N, N 0 )where we use the notation u for u(x, ω), a for a(x, ω), N for N (x, ω), u0 for u(x, ω 0 ), a0 fora(x, ω 0 ) and N 0 for N (x, ω 0 ). Then, the proof of (5.56) is analogous to the proof of (3.45), so
we skip it for the sake of simplicity. Let us just say that, as for the proof of (3.45), most terms
require one integration by parts using (5.57) and the estimate (5.34) for b (as in section 3.3.1),
while one term requires a second integration by parts using (5.57) and the decomposition (5.36)
for ∇N b (as in sections 3.3.2 and 3.3.3).
5.2.3. Proof of Proposition 5.12 (almost orthogonality in angle). We have to prove(5.52):X
kUj f k2L2 (Σ) .
kUjν f k2L2 (Σ) + D2 γj2 .
(5.58)
ν∈Γ

R
05.2.3.1. Presence of a log-loss. Integrating by parts twice in Σ Ujν f (x)Ujν f (x)dΣ would
ultimately imply:
0Z
D2 γjν γjν0ννUj f (x)Uj f (x)dΣ . j/2, |ν − ν 0 | =6 0.
(5.59)(2 |ν − ν 0 |)2
ΣThis yields to a log-loss since we have:
X
1
sup
∼ j.
(5.60)(2j/2 |ν − ν 0 |)2ν
0ν5.2. CONTROL OF FOURIER INTEGRAL OPERATORS

97

Remark 5.14. Recall that there is an analogous log-loss in the almost orthogonality argument
in angle for the error term (see section 3.5.2). In section 3.5, we removed the log-loss inparticular by using integration by parts with respect to the null vectorfield L. On the other
hand, we work here on Σ which is Riemannian, so there is no equivalent of the null vectorfield
L. Instead, we will use a second decomposition in λ (see section 5.2.3.2). Note that such a
strategy can not be used to control the error term (see Remark 5.16).
To avoid the log-loss present in (5.59), we will instead derive the following inequality:
00Z
D2 γjν γjν
D2 γjν γjν0ννUj f (x)U f (x)dΣ . jα/2 j/2+ j/2, |ν − ν 0 | =6 0,(5.61)
0 |)2−α
0 |)3
2
(2
|ν
−
ν
(2
|ν
−
ν
Σ
where α > 0. Indeed, since S2 is 2 dimensional and 1 ≤ 2j/2 |ν − ν 0 | ≤ 2j/2 for ν, ν 0 ∈ Γ andν 6= ν 0 , we have:X
1≤ C < +∞,
(5.62)
supj/2(2 |ν − ν 0 |)3
ν
0
ν

and
sup

X

ν

ν0

1
2jα/2 (2j/2 |ν − ν 0 |)2−α≤ Cα < +∞ ∀α > 0.

(5.63)

Thus, (5.61), (5.62) and (5.63) together with Shur’s Lemma imply (5.58).
5.2.3.2. A second decomposition in frequency. To avoid the log-loss present in (5.59), we doa second decomposition in frequency. λ belongs to the interval [2j−1 , 2j+1 ] which we decompose
in intervals Ik :[[2j−1 , 2j+1 ] =Ik where diam(Ik ) ∼ 2j |ν − ν 0 |α .
(5.64)
1≤k≤|ν−ν 0 |−αLet φk a partition of unity of the interval [2j−1 , 2j+1 ] associated to the Ik ’s. We decompose Ujν f
as follows:
X
Ujν f (x) =Ujν,k f (x),(5.65)
1≤k≤|ν−ν 0 |−αwhere:
Ujν,k f (x) =Z Z +∞
S2

eiλu b(x, ω)ψ(2−j λ)ηjν (ω)φk (λ)f (λω)λ2 dλdω.

(5.66)

0Remark 5.15. The point of this additional decomposition is to exploit the volume in λ.
Indeed, after performing Cauchy-Schwarz in λ, we obtainpjα|Ik | ∼ 2 2 |ν − ν 0 | 2αwhich displays the crucial gain |ν − ν 0 | 2 .
We also define:γjν,k = kψ(2−j λ)ηjν (ω)φk (λ)f kL2 (R3 ) , j ≥ 0, ν ∈ Γ, 1 ≤ k ≤ |ν − ν 0 |−α ,(5.67)

which satisfy:(γjν )2 =X
1≤k≤|ν−ν 0 |−α(γjν,k )2 .

(5.68)

98

5. CONSTRUCTION AND CONTROL OF THE PARAMETRIX AT INITIAL TIME

5.2.3.3. The two key estimates. We will prove the following two estimates:
00Z
D2 γjν,k γjν ,kD2 γjν,k γjν ,kν,kν 0 ,kUj f (x)Uj f (x)dΣ . jα/2 j/2+
2(2 |ν − ν 0 |)2−α (2j/2 |ν − ν 0 |)3Σ
0for |ν − ν | =6 0, 1 ≤ k ≤ |ν − ν 0 |−α ,(5.69)

and
0

Z
Σ

0 0
Ujν,k f (x)Ujν ,k f (x)dΣD2 γjν,k γjν ,k,|k − k 0 |2j/2(1−4α) (2j/2 |ν − ν 0 |)1+4α
for |ν − ν 0 | =6 0, 1 ≤ k, k 0 ≤ |ν − ν 0 |−α , k 6= k 0 .
.

(5.69) and (5.70) imply:
Z
0
Ujν f (x)Ujν f (x)dΣ
Σ
Z
X
0
≤
Ujν,k f (x)Ujν ,k f (x)dΣ +Σ

1≤k≤|ν−ν 0 |−α2jα/2 (2j/2 |ν − ν 0 |)2−α

.

0

0

+

1≤k≤|ν−ν 0 |−αD2 γjν,k γjν ,k

X

Σ

0

Ujν,k f (x)Ujν ,k f (x)dΣD2 γjν,k γjν ,k

X
0

+

Z

X

0

1≤k≤|ν−ν 0 |−α(5.70)(5.71)

1≤k6=k0 ≤|ν−ν 0 |−αD2 γjν,k γjν ,k

X

.

0

(2j/2 |ν − ν 0 |)30j

0 2 (1−4α) (2j/2 |ν − ν 0 |)1+4α
1≤k6=k0 ≤|ν−ν 0 |−α |k − k |2D2 γjν γjν0+2jα/2 (2j/2 |ν − ν 0 |)2−α

D2 γjν γjν

0(2j/2 |ν − ν 0 |)3,

where we have used (5.68) and the fact that we may choose 0 < α < 1/5, together with the fact
that:
X
1
sup
. α| log(|ν − ν 0 |)|.
(5.72)
0|
|k
−
k
0
−α
1≤k≤|ν−ν |00 −α 01≤k ≤|ν−ν |, k 6=kSince (5.71) yields the wanted estimate (5.61), we are left with proving (5.69) and (5.70). The
discussion in the following section will be very informal for the sake of simplicity. We refer to[74] for the details.
5.2.3.4. Proof of (5.69). The estimate (5.69) will result of two integrations by parts withrespect to tangential derivatives (in the spirit of section 3.5.1.1). Let us consider for instancethe case where the two tangential derivatives fall on the symbol b of Ujν,k f defined in (5.66).
This yields a term of the form
Z +∞

Z
∇
/ 2b
eiλu ψ(2−j λ)φk (λ)f (λω)λ2 dλ ηjν (ω)dω.
(5.73)
S2

0

Then, in view of the estimate (5.34) for b, we have in particular ∇/ 2 b ∈ L2 (Σ) which will force
us to estimate the λ integral in (5.73) in L∞
u . To this end, we do Cauchy-Schwarz and obtainin particular the square root of the diameter of Ik . Due to (5.64), we thus gain an additional
factor of |ν − ν 0 |α with respect to (5.59), which yields (5.69).

5.2. CONTROL OF FOURIER INTEGRAL OPERATORS

99

Remark 5.16. Here, the log-loss is removed by exploiting the size of the diameter of Ik .
This is possible since we estimate the λ integral of (5.73) in L∞
u using Cauchy Schwartz. In
turn, this is a consequence of our estimate for ∇/ 2 b in L2 (Σ). This explains why this method
cannot be used to remove the log-loss of the error term in Chapter 3. Indeed, our estimates for2
the space-time foliation in Chapter 4 are typically of the type L∞
u L (Hu ), so that the integral
2in λ is estimated in Lu using Plancherel. In turn, this does not allow us to see the size of thelocalization in λ, so that a second decomposition in frequency of the type (5.65) would be uselessin that case.
5.2.3.5. Proof of (5.70). Note that we not only need to gain summability in (ν, ν 0 ) for thisterm, but also in (k, k 0 ). This is achieved through the presence of the additional gain k − k 0 in
the right-hand side of (5.70). The estimate (5.70) will result of two integrations by parts, onewith respect to the normal derivative N , and one with respect to tangential derivatives. We
obtain a term analogous to (5.73)
Z +∞

Z
iλu
−j
2
∇
/ ∇N be ψ(2 λ)φk (λ)f (λω)λ dλ ηjν (ω)dω.
(5.74)
S2

0

In view of the estimate (5.34) for b, we have in particular ∇/ ∇N b ∈ L2 (Σ). Thus, the log-loss ofthe summation in (ν, ν 0 ) is removed as in the previous section, in particular using the size of thediameter of Ik . Note also that the gain k − k 0 in the right-hand side of (5.70) comes from theintegration by parts in N . Indeed, we use
ia

0 0

eiλu−iλ u = −0 0

λ − λ0 aa0 g(N, N 0 )∇N (eiλu−iλ u ).

(5.75)

Now, since λ ∈ Ik , λ ∈ Ik0 , we have in view of (5.64), the assumption (5.27) for ∂ωα a, and theassumption (5.28) for ∂ω Na
λ − λ0 0 g(N, N 0 ) ∼ |k − k 0 |2j |ν − ν 0 |α ,a0which yields the gain k − k in the right-hand side of (5.70).
5.2.4. Proof of Proposition 5.13 (control of the diagonal term). We have to prove(5.53):
kUjν f kL2 (Σ) . Dγjν .
(5.76)
Recall that Ujν is given by:
Ujν f (x) =ZbFj (u)ηjν (ω)dω,(5.77)eiλu ψ(2−j λ)f (λω)λ2 dλ.

(5.78)

S2

where Fj (u) is defined by:Z +∞Fj (u) =0

We decompose Ujν in the sum of two terms:
ZZ(b(x, ω) − b(x, ν))Fj (u)ηjν (ω)dω.
Ujν f (x) = b(x, ν)Fj (u)ηjν (ω)dω +
S2

S2

100

5. CONSTRUCTION AND CONTROL OF THE PARAMETRIX AT INITIAL TIME

Then, using in particular the assumption (5.34) for b and the assumption (5.35) for ∂ω b, we
obtainZ
kUjν f kL2 (Σ) . D

S2

Fj (u)ηjν (ω)dω+ Dγjν .

(5.79)

L2 (Σ)In order to estimate the right-hand side of (5.79), we use the following proposition.
Proposition 5.17. We have the following bound:Z
Fj (u)ηjν (ω)dωS2. γjν .

(5.80)

L2 (Σ)The proof of Proposition 5.17 is postponed to the next section. Finally, (5.79) and (5.80)
yield the wanted estimate (5.76) which concludes the
R proof ofν Proposition 5.13
5.2.4.1. Proof of Proposition 5.17. Recall that S2 Fj (u)ηj (ω)dω is given by:Z
Z Z +∞νFj (u)ηj (ω)dω =eiλu ψ(2−j λ)ηjν (ω)f (λω)λ2 dλdω.
(5.81)S2

S2

0

Relying on the classical T T ∗ argument, (5.80) is equivalent to proving the boundedness on L2 (Σ)
of the operator whose kernel K is given by:
Z Z +∞
K(x, y) =eiλu(x,ω)−iλu(y,ω) ψ(2−j λ)ηjν (ω)λ2 dλdω, x, y ∈ Σ.
(5.82)S2

0

The decay satisfied by this kernel is stated in the following proposition.
Proposition 5.18. The kernel K defined in (5.82) satisfies the following decay estimate forall x, y in Σ:|K(x, y)| .

2j(1 + |2j |u(x, ν) − u(y, ν)| − 2j/2 |∂ω u(x, ν) − ∂ω u(y, ν)||)22j.
×(1 + 2j/2 |∂ω u(x, ν) − ∂ω u(y, ν)|)3(5.83)The proof of Proposition 5.18 is postponed to section 5.2.4.2. In the rest of this section,
we show how (5.83) implies Proposition 5.17. According to Schur’s Lemma, the operator whose
kernel is K is bounded on L2 (Σ) provided we can prove the following bound:ZZ(5.84)
sup |K(x, y)|dy < +∞, sup |K(x, y)|dx < +∞.
x∈Σ Σ

y∈Σ

Σ

Due to the symmetry of K in x, y, the two bounds in (5.84) are obtained in the same way.
We focus on establishing the first bound. Using in particular (5.83) and the global change ofvariable on Σ given by (5.30)3, we are able to obtain:
Z
Z
2j
2j
|K(x, y)|dy .
dy,(5.85)
j/2 |y 0 ||)2 (1 + 2j/2 |y 0 |)3jΣ
R3 (1 + |2 |y · ν| − 2
3using also the bound on the Jacobian (5.31)

5.3. CONTROL OF THE PARAMETRIX AT INITIAL TIME

101

where y = y · ν + y 0 and y 0 · ν = 0. Making the change of variable y → z where z is defined byz · ν = 2j y · ν and z 0 = 2j/2 y 0 in the right-hand side of (5.85), and remarking that z · ν is one
dimensional, and z 0 is two dimensional, we obtain:
Z
Z
dz
. 1.
(5.86)|K(x, y)|dy .
0 2
0 3
R3 (1 + ||z · ν| − |z ||) (1 + |z |)Σ
(5.86) implies the first bound in (5.84). K being symmetric with respect to x, y, the second
bound in (5.84) is also true. Thus, the operator whose kernel is K is bounded on L2 (Σ) whichconcludes the proof of Proposition 5.17.
5.2.4.2. Proof of Proposition 5.18. Recall the definition of K:
Z Z +∞
K(x, y) =eiλu(x,ω)−iλu(y,ω) ψ(2−j λ)ηjν (ω)λ2 dλdω, x, y ∈ Σ.
(5.87)
S2

0

We need to prove that K satisfies the following decay estimate for all x, y in Σ:|K(x, y)| .

2j(1 + |2j |u(x, ν) − u(y, ν)| − 2j/2 |∂ω u(x, ν) − ∂ω u(y, ν)||)22j×.
(1 + 2j/2 |∂ω u(x, ν) − ∂ω u(y, ν)|)3(5.88)For the sake of simplicity, let us just describe the general strategy of the proof of Proposition
5.18. In view of the regularity for u(x, ω) with respect to ω provided by (5.24) and (5.26)-(5.29),
we have|u(x, ω)| + |∂ω u(x, ω)| + |∂ω2 u(x, ω)| + |∂ω3 u(x, ω)| . 1 + |x|, ∀x ∈ Σ, ∀ω ∈ S2 .

(5.89)

This regularity allows us to integrate by part three times with respect to ω, while we may
integrate as much as we want with respect to λ. The estimate (5.88) is then obtained after
performing in (5.87) three integrations by parts with respect to ω and two integrations by parts
with respect to λ.
5.3. Control of the parametrix at initial time
In this section, we discuss the proof of Theorem 5.9. To this end, we first show that theFourier integral operator U of Theorem 5.6 almost preserves the L2 norm provided we makeadditional assumptions on its symbol. We then use this observation to prove the estimate (5.41).
Finally, we conclude the proof of Theorem 5.9 by establishing the existence and uniqueness of(f+ , f− ) solution of the system (5.40).
5.3.1. A refinement of Theorem 5.6. In Theorem 5.6, we have proved that the Fourier
integral operator U with phase u and symbol b is bounded on L2 (Σ) provided u satisfies (5.24),(5.26)-(5.29) and (5.30)-(5.31), and the symbol b satisfies (5.34) (5.35). We now would like toprove that U satisfies the following bound from below:
kf kL2 (R3 ) . kU f kL2 (Σ) ,(5.90)

provided u also satisfies (5.32) and under additional assumptions on the symbol b. This is theaim of the following proposition.

102

5. CONSTRUCTION AND CONTROL OF THE PARAMETRIX AT INITIAL TIME

Proposition 5.19. Let u be a function on Σ × S2 satisfying suitable assumptions (we refer
to [74] for the complete set of assumptions, and to section 5.1.3 for some typical assumptions).
Let U the Fourier integral operator with phase u(x, ω) and symbol b(x, ω):Z Z +∞eiλu(x,ω) b(x, ω)f (λω)λ2 dλdω.
(5.91)U f (x) =S20

We assume furthermore that b(x, ω) satisfies:k∂ω bkL2 (Σ) + k∇∂ω bkL2 (Σ) . 1,

(5.92)+ k∇/ ∇bkL2 (Σ) . ε,kb − 1kL∞ (M) + k∇bkL∞
2
u L (Pu )(5.93)

andj∇N b = bj1 + bj2 where kbj1 kL2 (Σ) . 2− 2 ε, kbj2 kL∞.ε2u L (Pu )jand k∇N bj2 kL2 (Σ) + kbj2 kL2u L∞ (Pu ) . 2 2 ε.

(5.94)

Then, U is bounded on L2 and satisfies the estimate:
kf kL2 (R3 ) . kU f kL2 (Σ) .

(5.95)

Remark 5.20. Notice that the only difference in the assumptions with respect to Theorem
5.6 lies in the fact that u also satisfies (5.32) and in the constant D which has been replaced by
1 in (5.92) and by ε in (5.93) (5.94).
The proof of Proposition 5.19 uses the decomposition in frequency and angle of the operator
U introduced in section 5.2. In order to control the diagonal term in a third step (see next
section), we have to modify slightly the size of the support of our partition of unity ηjν on S2
introduced in (5.47). Let δ > 0 such that:√0 < ε << δ << 1.
(5.96)
We now require that the support of ηjν is a patch on S2 of diameter ∼ δ2−j/2 . With thismodification, the assumptions for b in Proposition 5.19, and by carefully tracking the size of thevarious terms in the almost orthogonality argument in frequency and angle, and the control ofthe diagonal term, we obtain
Z
ε

X
X
0
2
(5.97)
kU f kL2 (Σ) =Sjν f (x)Slν f (x)dΣ + O 2 + δ kf k2L2 (R3 ) ,δ−j/2 Σ
0|j−l|≤2 |ν−ν |≤2δ2

where the operator Sjν is defined on Σ by:
Z Z +∞
Sjν f (x) =eiλu(x,ω) ψ(2−j λ)ηjν (ω)f (λω)λ2 dλdω.
Σ(5.98)

0

Recall (5.30)-(5.31) which states that the map φν : Σ → R3 defined by:φν (x) := u(x, ν)ν + ∂ω u(x, ν),(5.99)

is a bijection, such that the determinant of its Jacobian satisfies the following estimate:k| det(Jacφν )| − 1kL∞ (M) . ε.

(5.100)5.3. CONTROL OF THE PARAMETRIX AT INITIAL TIME

103

Let us note F −1 the inverse Fourier transform on R3 . We introduce the operator Sejν on Σ defined
by:
Z
ν
−1
−j
ν
e
eiλφν (x)·ω ψ(2−j λ)ηjν (ω)f (λω)λ2 dλdω.
(5.101)
Sj f (x) = F (ψ(2 ·)ηj f )(φν (x)) =
R3

R
R
0
0
The following proposition shows that the term Σ Sjν f (x)Slν f (x)dΣ is close to the term Σ Sejν f (x)Selν f (x)dΣ.
Proposition 5.21. We have the following bound:1

kSjν f − Sejν f kL2 (Σ) . δ 2 γjν .

(5.102)The proof of Proposition 5.21 relies on the classical T T ∗ argument, and the comparison
between u(x, ω) and φν (x) · ω provided by (5.32) (see [74] for the details). Now, (5.97) and(5.102) yield:
Z
ε

X
X
1
0
2
(5.103)
Sejν f (x)Selν f (x)dΣ + O 2 + δ 2 kf k2L2 (R3 ) .
kU f kL2 (Σ) =δ−j/2 Σ
0|j−l|≤2 |ν−ν |≤2δ2

R
0
Making the change of variable y = φν (x) in Σ Sejν f (x)Selν f (x)dΣ and using (5.100) and (5.101)
implies:Z
X
X
0
Sejν f (x)Selν f (x)dΣ|j−l|≤2 |ν−ν 0 |≤2δ2−j/2 ZΣ

X

=

X

3
|j−l|≤2 |ν−ν 0 |≤2δ2−j/2 R
2+O(ε)kf kL2 (R3 )
Z

X

=

X

|j−l|≤2 |ν−ν 0 |≤2δ2−j/2+O(ε)kf k2L2 (R3 ) ,R30

F −1 (ψ(2−j ·)ηjν f )(y)F −1 (ψ(2−l ·)ηjν f )(y)dy(5.104)0

ψ(2−j λ)ηjν (ω)f (λω)ψ(2−l λ)ηjν (ω)f (λω)dywhere we have used the fact that F −1 is an isomorphism on L2 (R3 ) in the last equality of(5.104). Now, we have:ZX
X
0ψ(2−j λ)ηjν (ω)f (λω)ψ(2−l λ)ηjν (ω)f (λω)dy = kf k2L2 (R3 ) ,(5.105)|j−l|≤2 |ν−ν 0 |≤2δ2−j/2R3

which together with (5.103) and (5.104) yields:
kU f k2L2 (Σ) = kf k2L2 (R3 ) + O

ε
δ


1
2
+
δ
kf k2L2 (R3 ) .
2(5.106)

1

Choosing δ 2 and εδ −2 small enough, we deduce from (5.106):
kf kL2 (R3 ) . kU f kL2 (Σ) ,which is the wanted estimate. This conclude the proof of Proposition 5.19.

(5.107)

104

5. CONSTRUCTION AND CONTROL OF THE PARAMETRIX AT INITIAL TIME

5.3.2. Proof of Theorem 5.9. The symbol of the Fourier integral operators M± andQ± are respectively given by 1 and a(x, ±ω)−1 . Thus, they clearly satisfy the assumptions of
Proposition 5.19. Relying on Proposition 5.19, we are then able to prove the estimate (5.41).
We refer to [74] for the details.
The uniqueness of (f+ , f− ) solution of the system (5.40) is an immediate consequence ofthe estimate (5.41), so there only remains to prove the existence of (f+ , f− ) to conclude theproof of Theorem 5.9. Recall that the phase u(x, ω) of our Fourier integral operators has been
constructed in [73] (see also Chapter 6) on Σ × S2 under the assumption that (Σ, g, k) satisfies
the following bounds consistent with the bounded L2 curvature conjecture:kRkL2 (Σ) ≤ ε, k∇kkL2 (Σ) ≤ ε,

(5.108)where the fact that we may take ε > 0 small comes from a reduction to the small data case.
(Σ, g, k) also satisfies the constraint equations and the maximal foliation assumption
 j ∇ kij = 0,
R = |k|2 ,(5.109)
Trk = 0.
We introduce two sets V and W :V = {(Σ, g, k) such that (5.108) and (5.109) are satisfied},(5.110)and
W = { (Σ, g, k) ∈ V such that (f+ , f− ) solution of (5.40) exist for all (φ0 , φ1 )such that ∇φ0 ∈ L2 (Σ) and φ1 ∈ L2 (Σ)}.

(5.111)

Not first that W is not empty since (Σ, g, k) = (R3 , δ, 0) belongs to W in view of Remark 5.2.
We then show that V is connected and W is both open and closed in V for a suitable topology(see [74] for the details). We infer W = V . This proves the existence of (f+ , f− ) solution of(5.40) and concludes the proof of Theorem 5.9.

CHAPTER 6

Control of the foliation at initial time
In this chapter, we will only consider u(t, x, ω) and Σt at t = 0. Thus, for simplicity, we
denote in the rest of this chapter u(0, x, ω) by u(x, ω) and Σ0 by Σ. The goal of this chapter isto prove the estimates on the control of the foliation of Σ by u(x, ω) which are needed for the
proof of Theorem 5.9 (see section 5.1.3), i.e. for the control of the parametrix at initial time.
The estimates obtained for u(x, ω) in this chapter must also be consistent with the control on
M for u(t, x, ω) obtained in Chapter 4 (see section 4.1.5). Here, we outline the main ideas and
we refer to [73] for the details.
6.1. Geometric set-up and main results
6.1.1. Reduction to small data. Recall from section 1.2.3 that we have reduced ourselves
to an asymptotically flat initial data set (Σ, g, k) solution to the constraint equations which
satisfies the bounds
kRkL2 (Σ) ≤ ε, k∇kkL2 (Σ) ≤ ε,
(6.1)
and is smooth outside of a small neighborhood U . In order to construct u(x, ω) satisfying theasymptotic behavior u(x, ω) ∼ x · ω when |x| → +∞ on Σ, we need to modify (Σ, g, k) outsideof U . We can glue it in a trivial way to (R3 , δ, 0) so that the new initial data set is still smoothoutside of U , satisfies (6.1), and coincides with (R3 , δ, 0) outside of a slightly larger neighborhood.
We still denote this initial data set (Σ, g, k). Of course, (Σ, g, k) does not satisfies the constraint
equations in the annulus where the gluing takes place. However, for the construction of u(x, ω),
we only require (Σ, g, k) to satisfy the constraint equations in U . Outside of U , (Σ, g, k) issmooth, so things are much easier.
6.1.2. Geometry of the foliation of Σ by a scalar function u. We define the lapse
a = |∇u|−1 , and the unit vector N such that ∇u = a−1 N . We also define the level surfaces
Pu0 = {x / u = u0 } so that N is the normal to Pu . The second fundamental form θ of Pu is
defined by
θ(X, Y ) = g(∇X N, Y )(6.2)
with X, Y arbitrary vectorfields tangent to the u-foliation Pu of Σ and where ∇ denotes thecovariant differentiation with respect to g. We denote by trθ the trace of θ, i.e. trθ = δ AB θAB
where θAB are the components of θ relative to an orthonormal frame (eA )A=1,2 on Pu .
6.1.3. Structure equations of the foliation of Σ by a scalar function u. We recall
some of the structure equations of the foliation of Σ by a scalar function u which will be needed
for the discussion of the main result of this chapter (see [17] for a proof).
105

106

6. CONTROL OF THE FOLIATION AT INITIAL TIME

Proposition 6.1. The orthonormal frame frame N, eA , A = 1, 2 of Σ satisfies the following
system:∇A N = θAB eB ,(6.3)
∇N N = −a−1 ∇/ a.
Also, the lapse a satisfies the following equationa−1 ∆/ (a) = −∇N trθ − |θ|2 + RN N ,(6.4)

where ∆/ is the Laplace-Beltrami for the metric γ on Pu induced by g. Finally, the second
fundamental form θ satisfies the following Codazzi equation∇/ B θAB = ∇/ A trθ + RN A .

(6.5)

6.1.4. Choice of u(x, ω). We look for u(x, ω) satisfying the three following conditions:(a) u(x, ω) ∼ x.ω when |x| → +∞ on Σ(b) The regularity of u(x, ω) with respect to x and ω is consistent with the regularity ofu(t, x, ω) with respect to (t, x) and ω obtained in Chapter 4 (see section 4.1.5). In
particular, we have trθ − kN N ∈ L∞ (Σ) (see the discussion in section 5.1.1.1)(c) u(x, ω) has as enough regularity in x and ω to control the parametrix at initial time,i.e. to obtain the conclusion of Theorem 5.9
where the initial data set (Σ, g, k) satisfies
 j ∇ kij = 0,
R = |k|2 ,(6.6)

trg k = 0,
in U (see section 6.1.1), and where R and ∇k are in L2 (Σ) and satisfy the smallness assumption(6.1).
In order to motivate our choice of u(x, ω), we investigate the regularity of the lapse a, whichby (6.4) satisfies the following equation:a−1 ∆/ (a) = −∇N trθ − |θ|2 − RN N .

(6.7)

Since R is in L2 (Σ), (6.7) implies that a has at most two derivatives in L2 (Σ). Thus, u(x, ω)
has at most three derivatives with respect to x in L2 (Σ). This is not enough to satisfy (c). In
fact, the classical T ∗ T argument (see for example [68]) relies on integrations by parts in x andwould require at least one more derivative since Σ has dimension 3.
Alternatively, we could try to use the T T ∗ argument which relies on integrations by partsin ω. Indeed, R being independent of ω, one would expect the regularity of u(x, ω) with respectto ω to be better. Differentiating (6.7) with respect to ω, we obtain:a−1 ∆/ (∂ω a) = 2∇
/ ∇N a + · · · ,(6.8)

where the term on the right-hand side comes from the commutator [∂ω , ∆/ ] (see section 6.4).
Thus, obtaining an estimate for ∂ω a from (6.8) requires to control ∇N a. Unfortunately, (6.7)
seems to give control of tangential derivatives of a only. This is where the specific choice ofu(x, ω) comes into play.
Having in mind the equation of minimal surfaces (i.e. trθ = 0), condition (b) suggest the
choice trθ − kN N = 0. Unfortunately, this equation together with (6.7) does not provide any
control of ∇N a. We might propose as a second natural guess to take instead trθ − kN N = ∇N a.

6.1. GEOMETRIC SET-UP AND MAIN RESULTS

107

Plugging in (6.7) yields an elliptic equation for a: ∇2N a + a−1 ∆/ (a) = −|θ|2 − ∇N kN N − RN N .
2
2This allows us to control ∇N a in L (Σ). However, trθ − kN N = ∇N a, and ∇N a is at most in
H 1 (Σ) which does not embed in L∞ (Σ) - since Σ has dimension 3 - so that condition (b) is notsatisfied. To sum up, the first guess trθ − kN N = 0 satisfies (b), but not (c), whereas the second
guess trθ − kN N = ∇N a might satisfy (c), but does not satisfy (b).
The correct choice is the intermediate onetrθ − kN N = 1 − a.

(6.9)

We will see in section 6.2 that a − 1 belongs to L∞ (Σ) so that (b) is satisfied.

Also, plugging(6.9) in (6.7) yields the parabolic equation:∇N a − a−1 ∆/ (a) = |θ|2 + ∇N kN N + RN N .

(6.10)

This will allow us to control normal derivatives of a. In turn, we will control derivatives of awith respect to ω using (6.8). Ultimately, we will prove enough regularity with respect to bothx and ω, such that (c) is satisfied.
6.1.5. Main results. For 1 ≤ p, q ≤ +∞, we define the spaces Lpu Lq (Pu ) for tensors F on
Σ using the norm:
Z
1/p
p
kF kLpu Lq (Pu ) =kF kLq (Pu ) du.
u

Remark 6.2. In the rest of the paper, all inequalities hold for any ω ∈ S2 with the constant in
the right-hand side being independent of ω. Thus, one may take the supremum in ω everywhere.
To ease the notations, we do not explicitly write down this supremum.
We first state a result of existence and regularity with respect to x for u.
Theorem 6.3. Let (Σ, g, k) chosen as in section 6.1.1. There exists a scalar function u on
Σ × S2 satisfying assumption (a) and such that:ka − 1kL∞+ k∇akL∞+ ka − 1kL∞ (Σ) + k∇
/ ∇akL2 (Σ) . ε,22
u L (Pu )u L (Pu )ktrθ − kN N kL∞ (Σ) + k∇θkL2 (Σ) . ε,

(6.11)

where Pu , a, N and θ are associated to u as in section 6.1.2.
Notice that condition (b) is included in (6.11). In order to state our second result, we
introduce fractional Sobolev spaces H b (Pu ) on the surfaces Pu for any b ∈ R (see [73] for theirprecise definition). We have the following estimate for ∇2N a, and improved estimate for ∇N a.
Theorem 6.4. Let (Σ, g, k) chosen as in section 6.1.1. Let u the scalar function on Σ × S2
constructed in theorem 6.3, and let Pu , a and N be associated to u as in section 6.1.2. We have:k∇N akL∞+ k∇2N ak 2
4
u L (Pu )1

Lu H − 2 (Pu ). ε.

(6.12)The third theorem investigates the regularity of u with respect to ω:
Theorem 6.5. Let (Σ, g, k) chosen as in section 6.1.1. Let u the scalar function on Σ × S2
constructed in theorem 6.3, and let Pu , a, N and θ be associated to u as in section 6.1.2. Wehave:k∂ω akL∞ (Σ) + k∇
/ ∂ω akL∞+ k∇/ 2 ∂ω akL2 (Σ) + k∇N ∂ω ak 2 12
2
u L (Pu )Lu H (Pu )(6.13)+k∇2N ∂ω ak 2 − 23
+ k∇∂ω θkL2 (Σ) . ε, k∂ω N kL∞ (Σ) . 1,
Lu H

(Pu )

108

6. CONTROL OF THE FOLIATION AT INITIAL TIMEk∂ω2 ak 2

3

Lu H 2 (Pu )+ k∂ω2 ak ∞1

Lu H 2 (Pu )+ k∇N ∂ω2 ak 2

1

Lu H − 2 (Pu )+ k∇∂ω2 θkL2 (Σ) . ε,k∂ω2 N kL∞ (Σ) . 1,

(6.14)

andk∂ω3 ukL∞. 1.
loc (Σ)(6.15)

6.1.6. Additional results. The following proposition establishes the existence of a global
coordinate system on Σ.
Proposition 6.6. Let ω ∈ S2 . Let φω : Σ → R3 defined by:φω (x) := u(x, ω)ω + ∂ω u(x, ω).

(6.16)

Then φω is a bijection, and the determinant of its Jacobian satisfies the following estimate:k| det(Jacφω )| − 1kL∞ (Σ) . ε.

(6.17)

Below, we state several additional estimates. We start with a first proposition.
Proposition 6.7. Let (Σ, g, k) chosen as in section 6.1.1. Let u the scalar function on
Σ × S2 constructed in theorem 6.3. Let ν ∈ S2 and φν the map defined in (6.16). Then, we havefor all x ∈ Σ and ω ∈ S2u(x, ω) − φν (x) · ω = O(ε|ω − ν|2 ),∂ω u(x, ω) − ∂ω (φν (x) · ω) = O(ε|ω − ν|),∂ω2 u(x, ω) − ∂ω2 (φν (x) · ω) = O(ε).

(6.18)

Using the geometric Littlewood Paley projections Pj on Pu constructed in [41] (see section
3.1.5) together with the estimates for ∇N a in (6.11), and the estimate for ∇2N a in (6.12), we
obtain the following proposition:
Proposition 6.8. Let (Σ, g, k) chosen as in section 6.1.1. Let u the scalar function on
Σ × S2 constructed in theorem 6.3, and let a and N be associated to u as in section 6.1.2. For
all j ≥ 0, there are scalar functions aj1 and aj2 such that:jj∇N a = aj1 + aj2 where kaj1 kL2 (Σ) . 2− 2 ε and k∇N aj2 kL2 (Σ) . 2 4 ε.

(6.19)

Remark 6.9. Recall from section 6.1.4 that we do not have enough regularity in x to apply
the T ∗ T method. Alternatively, we could try the T T ∗ method which relies on integration by partsin ω. But ∂ω3 u ∈ L∞ (Σ) is also not enough and we would need at least one more derivative inω. Nevertheless, using the regularity in x and ω obtained for u in the present and the previous
section, we are able to control the parametrix at initial time (see Chapter 5).
Let us conclude this section by mentioning several ingredients of [73] that have been omitted here for the sake of simplicity, and that have to be proved by relying on low regularity
assumptions for u which are consistent with the results stated in this section and the previousone:• estimates for the parabolic operator (∇N − a−1 ∆/)• estimates for θ and N• product estimates in the Sobolev spaces H b (Pu )• embeddings on Σ and Pu• a control of the Gauss curvature of Pu

6.2. CONSTRUCTION OF THE FOLIATION AND REGULARITY WITH RESPECT TO x

109

• Bochner inequalities on Pu• estimates for various commutator terms of the type: [∇N , ∇/ ], [∇N , Pj ], ...
The rest of this chapter is as follows. In section 6.2, we discuss the proof of Theorem 6.3.
In section 6.3, we discuss the proof of Theorem 6.4. In section 6.4, we discuss the proof of
Theorem 6.5. In section 6.5, we discuss the proof of Proposition 6.13 and Proposition 6.6.
Finally, Proposition 6.7 and Proposition 6.8 are discussed in section 6.6.
6.2. Construction of the foliation and regularity with respect to x
In this section, we discuss the proof of Theorem 6.3. By section 6.1.1, we may assume that(Σ, g, k) coincides with (R3 , δ, 0) outside of a compact, say |x| ≥ 1. Notice that in |x| ≥ 1 andfor all ω ∈ S2 , the scalar function x.ω satisfies the equation (6.9) and the estimate (6.11), sincea ≡ 1, θ ≡ 0 and N ≡ ω in this region. Thus, we would like to construct a function u solution of(6.9) satisfying (6.11) in a region containing |x| ≤ 2 and to glue it to x.ω in 1 ≤ |x| ≤ 2. Now,
(6.9) is of parabolic type - see (6.10) - where u plays the role of time. For each fixed ω ∈ S2 ,we start with u = −2 on x · ω = −2. Then, we propagate with the parabolic equation (6.10),
coupled with the equation for θ (6.5), to the strip S = {x ∈ Σ such that − 2 < u(x, ω) < 2}.
This strip covers the entire region |x| ≤ 1, and we then glue u to x · ω outside of |x| ≤ 1 (seesection 6.2.2). In the next section, we prove a priori estimates consistent with the estimate(6.11) and valid on −2 < u < 2 for the solution u of:trθ − kN N = 1 − a, on − 2 < u < 2,

(6.20)

where u is initialized on x.ω = −2 by:u(x, ω) = −2 on x.ω = −2.

(6.21)

Note that the first equation of (6.25), (6.21) and the fact that (g, k, Σ) coincides with (δ, 0, R3 )for |x| ≥ 2 yields:∇p (a − 1) = 0, ∇p θ = 0, ∇p (N − ω) = 0 for all p ∈ N on u = −2.

(6.22)

6.2.1. A priori estimates for lower order derivatives. Let (Σ, g, k) chosen as in section
6.1.1. In particular, we assume:k∇kkL2 (Σ) + kRkL2 (Σ) ≤ ε.

(6.23)Let u a scalar function on Σ × S2 , and let Pu , a, N and θ be associated to u as in section 6.1.2.
Assume that u satisfies the additional equation (6.9), which we recall below together with (6.3)
and (6.4):
∇A N = θAB eB ,(6.24)∇N N = −∇/ log(a),andtrθ − kN N = 1 − a,(6.25)∇N a − a−1 ∆/ (a) = |θ|2 + ∇N kN N + RN N .
In this section, we establish a priori estimates for a, N and θ corresponding to (6.11) in theregion S of Σ between P−2 and P2 (i.e. S = {x / − 2 < u(x, ω) < 2}) where u is initialized onx.ω = −2 by (6.21). In particular, we have (6.22), so that the subsequent integrations by parts
will not create boundary terms at u = −2.

110

6. CONTROL OF THE FOLIATION AT INITIAL TIME

For the sake of simplicity, let us just discuss the estimate (6.11) for the lapse a. We rewrite
the second equation of (6.25) as:(∇N − a−1 ∆/ )(a − 1) = h,

(6.26)

h = ∇N kN N + RN N + · · · .

(6.27)

where h is given by:
Using in particular (6.23), we obtain:
khkL2 (Σ) . ε.
Together with (6.26) and an L2 parabolic estimate for the operator (∇N − a−1 ∆/ ), we obtainka − 1kL∞+ k∇/ akL∞+ k∇N akL2 (Σ) + k∇
/ 2 akL2 (Σ) . ε.
22
u L (Pu )u L (Pu )(6.28)In order to obtain estimates for ∇/ ∇N a and ∇2N a, we differentiate the second equation of(6.25) by ∇N :(∇N − a−1 ∆/ )∇N a = ∇2N kN N + ∇N RN N + · · · ,(6.29)
where we only kept two typical terms. Note that ∇2N kN N and ∇N RN N are dangerous terms
which cannot be estimated directly. We first need to trade a ∇N derivative with a ∇/ derivative
using Bianchi identities and the constraints equations. We use the twice-contracted Bianchi
identity on Σ1
∇j Rij = ∇i R.
(6.30)
2
In particular, using also the constraint equations (6.6), we have
∇N RN N = −∇A RAN + ∇N |k|2 .
Also, the constraint equations (6.6) yield∇2N kN N = −∇N ∇A kAN + · · · = −∇A ∇N kAN + · · · .
Together with (6.29), we obtain(∇N − a−1 ∆/ )∇N a = div/ (H) + h1 ,(6.31)

H = −∇N k.N − R.N ,(6.32)

where
and h1 satisfies
kh1 k 2

4Lu L 3 (Pu ). ε.

(6.33)

Using the smallness assumption (6.23) and the definition of H (6.32), we have
kHkL2 (Σ) . ε(6.34)

which together with (6.33), (6.31), and an L2 parabolic estimate for the operator (∇N − a−1 ∆/ ),yieldsk∇N akL∞+ k∇/ ∇N akL2 (Σ) . ε.
(6.35)2u L (Pu )Finally, (6.28) and (6.35) yield the wanted estimate (6.11).

6.3. ESTIMATES FOR ∇N a AND ∇2N a

111

6.2.2. End of the proof of Theorem 6.3. We briefly sketch the rest of the proof of
Theorem 6.3, and we refer to [73] for the details. In (6.28) and (6.35), we have obtained apriori estimates consistent with the estimate (6.11) and valid on −2 < u < 2 for the solution uof (6.20). Then, we also prove on −2 < u < 2 a priori estimates for higher derivatives of thesolution u of (6.20). We then use the existence of u solution to1:trθ − kN N = 1 − a, on α < u < α + T,
(6.36)
u = α on u = α,where −2 ≤ α ≤ 2, u is smooth, and T > 0 is small enough. Together with the a priori estimates,
this allows us to control the solution of (6.36) on −2 + kT < u < −2 + (k + 1)T uniformly withrespect to k = 0, . . . , [4/T ] in order to obtain a solution u of (6.20) on −2 < u < 2. Finally, we
conclude the proof of Theorem 6.3 by showing how to glue the solution u of (6.20) to x.ω in
1 ≤ |x| ≤ 2 in order to obtain a solution on Σ satisfying (6.11).
6.3. Estimates for ∇N a and ∇2N a
In this section, we discuss the proof of Theorem 6.4. Recall the decomposition (6.31), (6.32),
and the estimate (6.33). We introduce the scalar functions on S a1 and a2 solutions of:(∇N − a−1 ∆/ )a1 = h1 on S, a1 (−2, .) = 0,

(6.37)(∇N − a−1 ∆/ )a2 = div/ (H) on S, a2 (−2, .) = 0,

(6.38)

and:which yields, in view of (6.31) and (6.22), the decomposition:∇N a = a1 + a2 .

(6.39)The idea behind the decomposition (6.39) is to take advantage of the better regularity of
h1 for a1 (see (6.33) compared to (6.34)), and to use the structure of div/ (H) to obtain a usefulequation for ∇N a2 . Indeed, in view of the equation (6.38) satisfied by a2 , ∇N a2 satisfies:(∇N − a−1 ∆/ )(∇N a2 ) = ∇N (div/ (H)) + · · · ,(6.40)

and using in particular the twice-contracted Bianchi identity on Σ, the constraint equations in
the maximal foliation (6.6), and (6.40), we obtain(∇N − a−1 ∆/ )∇N a2 = div/ div/ (H1 ) + div/ (H2 ) + h2 + · · · ,(6.41)

where the tensors H1 , H2 and the scalar h2 satisfy
kH1 kL2 (S) + kH2 k 2

4

Lu L 3 (Pu )+ kh2 kL1 (S) . ε.

These ideas allow us to derive the following two propositions (see [73] for the detailed proof ofthese propositions).
Proposition 6.10. Let a1 be the solution of (6.37), where h1 satisfies (6.33). Then, wehave:
ka1 kL∞. ε,(6.42)4u L (Pu )1this local existence result could be proved either using a Nash Moser procedure or a combination of Cauchy-

Kowalewska and enhanced a priori estimates for all derivatives

112

6. CONTROL OF THE FOLIATION AT INITIAL TIME

and:X

2−j kPj (∇N a1 )k2L2 (Σ) . ε2 ,(6.43)

j≥0

Proposition 6.11. Let a2 be the solution of (6.38), where H is defined in (6.32). Then,
we have:ka2 kL∞. ε,(6.44)4u L (Pu )and:supkPj (∇N a2 )kL2 (Σ) . ε.

(6.45)

j≥0

In view of the decomposition (6.39) for ∇N a, the estimates (6.42) (6.43) for a1 , and theestimates (6.44) (6.45) for a2 , we immediately obtain the estimate (6.12) for ∇N a and ∇2N a.
This concludes the proof of Theorem 6.4.
6.4. Regularity of the foliation with respect to ω
Let u(x, ω) the function constructed in section 6.2. In this section, we discuss the proof of
Theorem 6.5 which deals with the control of the derivatives with respect to ω of the foliationof Σ provided by u(x, ω). Recall that (Σ, g, k) coincides with (R3 , δ, 0) in |x| ≥ 2. Also, u(x, ω)
coincides with x.ω in |x| ≥ 2, and so a ≡ 1, N ≡ ω and θ ≡ 0 in this region. Thus, u clearlysatisfies the estimates of Theorem 6.5 in |x| ≥ 2 and it is enough to control the derivatives withrespect to ω of the function u(x, ω) solution to:trθ − kN N = 1 − a, on − 2 < u < 2,
(6.46)
u(., ω) = −2 on x.ω = −2,
in the strip S = {x/ − 2 < u < 2}.
6.4.1. First order derivatives with respect to ω. The goal of this section is to prove(6.13). For the sake of simplicity, we only outline of the proof of the estimate for ∂ω a. Differentiating the second equation of (6.25) with respect to ω, we obtain:(∇N − a−1 ∆/ )∂ω a = 2∇
/ ∇N a + 2RN ∂ω N + · · ·(6.47)

where the first term on the right-hand side comes from the commutator [∂ω , ∆/ ] (see [73]). Since∇/ ∇N a and R are in L2 (Σ) respectively by (6.11) and (6.23), we obtain using in particular an
L2 parabolic estimate for the operator (∇N − a−1 ∆/)k∇N ∂ω akL2 (Σ) + k∇
/ ∂ω akL∞+ k∇/ 2 ∂ω akL2 (Σ) . ε.
2u L (Pu )(6.48)

Next, we differentiate (6.47) with respect to ∇N . We obtain:(∇N − a−1 ∆/ )∇N ∂ω a = 2∇
/ ∇2N a + 2∇N RN ∂ω N + · · · .

(6.49)The term ∇N RN ∂ω N may be treated using the contracted Bianchi identity for R - as we did for∇N RN N in section 6.2.1 - and turns out to be in L2u H −1 (Pu ). On the other hand, in view of3
the estimate (6.12) for ∇2N a, ∇/ ∇2N a belongs to L2u H − 2 (Pu ). We obtain using in particular arefined parabolic estimate for the operator (∇N − a−1 ∆/)k∇N ∂ω ak 2

1

Lu H 2 (Pu )+ k∇2N ∂ω ak 2

3

Lu H − 2 (Pu ). ε.

(6.50)

6.5. A GLOBAL COORDINATE SYSTEM ON Pu AND Σ

113
5

Finally, by interpolation between (6.48) and (6.50), we obtain ∂ω a in L∞
u H 4 (Pu ) which embeds∞
in L (Σ) since Pu has dimension 2. Together with (6.48) and (6.50), we obtain the estimate
corresponding to ∂ω a in (6.13).
6.4.2. Second order derivatives with respect to ω. The goal of this section is toprove (6.14). For the sake of simplicity, we only outline of the proof of the estimate for ∂ω2 a.
Differentiating the equation (6.47) for ∂ω a with respect to ω, we obtain:(∇N − a−1 ∆/ )∂ω2 a = 2∇2N a + ∇/ ∇N ∂ω a + 2R∂ω N ∂ω N + · · ·(6.51)

where the first two terms on the right-hand side come respectively from the commutators [∂ω , ∇/]1
and [∂ω , ∆/ ]. Since R is in L2 (Σ) by (6.23), ∇2N a is in L2u H − 2 (Pu ) by (6.12), and ∇N ∂ω a is in
1
1
L2u H 2 (Pu ) by (6.13), the right-hand side of (6.51) belongs to L2u H − 2 (Pu ). Using in particularestimates for the parabolic operator (∇N − a−1 ∆/ ), we deducek∂ω2 ak 2

3

Lu H 2 (Pu )+ k∂ω2 ak ∞1

Lu H 2 (Pu )+ k∇N ∂ω2 ak 2

. ε,

1

Lu H − 2 (Pu )(6.52)

which is the estimate corresponding to ∂ω2 a in (6.14).
Remark 6.12. Note that we may not differentiate the equation (6.51) for ∂ω2 a with respect
to ∇N . Indeed, the term ∇N R∂ω N ∂ω N has no structure: unlike RN N and RN ∂ω N which wereinvolved in the equation for a and ∂ω a, R∂ω N ∂ω N does not contain any contraction with N since∂ω N is tangent to Pu . Thus, unlike ∇N RN N and ∇N RN ∂ω N , we can not write ∇N R∂ω N ∂ω N asa tangential derivative using the contracted Bianchi identities for R. Consequently, we can not
obtain any estimate for ∇2N ∂ω2 a.
6.4.3. Third order derivatives with respect to ω. The goal of this section is to prove(6.15). Recall that div(N ) = trθ, N = ∇u/|∇u|, a = 1/|∇u| and trθ = 1 − a + kN N , so that:


∇u
1
div
=1−
+ kN N .
(6.53)|∇u||∇u|
Differentiating (6.53) three times with respect to ω yields:(∇N − a−1 ∆/ )∂ω3 u = ∇/ ∂ω2 log(a) + · · · .

(6.54)

Using in particular (6.54), the estimate (6.14) for ∂ω2 a and refined parabolic estimates for theoperator (∇N − a−1 ∆/ ) we obtaink∂ω3 uk 2

5

Lu H 2 (Pu )+ k∂ω3 uk ∞3

Lu H 2 (Pu )+ k∇N ∂ω3 uk 2

1

Lu H 2 (Pu )

. 1.

3

3
∞
Now, since ∂ω3 u ∈ L∞
u H 2 (Pu ) and Pu is 2-dimensional, we obtain that ∂ω u belongs to Lloc (Σ),
which is the desired estimate (6.15).

6.5. A global coordinate system on Pu and Σ
The goal of this section is to discuss the proof of Proposition 6.6. We start by constructing
a global coordinate system on Pu .

114

6. CONTROL OF THE FOLIATION AT INITIAL TIME

6.5.1. A global coordinate system on Pu . We have the following proposition
Proposition 6.13. Let ω ∈ S2 . Let Φu : Pu → Tω S2 defined by:Φu (x) := ∂ω u(x, ω),(6.55)

where Tω S2 is the tangent space to S2 at ω. Then Φu is a global C 1 diffeomorphism from Pu toTω S2 .
For the sake of simplicity, we only briefly sketch the proof. We start by showing that Φu is
a local C 1 diffeomorphism. We haveg(∂ϕ N, ∂ϕ N ) g(∂ψ N, ∂ϕ N )T−2(JacΦu ) JacΦu = a,g(∂ψ N, ∂ϕ N ) g(∂ψ N, ∂ψ N )where (ϕ, ψ) denotes the usual spherical coordinates on S2 . Using the estimates (6.11) and(6.13), we are able to derive the following estimatek(JacΦu )T JacΦu − IkL∞ (Σ) . ε,

(6.56)

so that Φu is a C 1 local diffeomorphism. In turn, this yields:
k| det(JacΦu )| − 1kL∞ (Σ) . ε.

(6.57)

It remains to show that Φu is onto and one-to-one. The proof relies on the estimate (6.56)
for the Jacobian of Φu , the fact that u coincides with x.ω in the region |x| ≥ 2 and geometricconsiderations on the level sets of ∂ϕ u and ∂ψ u. We refer to [73] for the details.
6.5.2. Proof of Proposition 6.6. Let ω ∈ S2 . Recall the definition (6.16) of φω : Σ → R3 :φω (x) := u(x, ω)ω + ∂ω u(x, ω) = u(x, ω)ω + Φu (x),where Φu has been defined in Proposition 6.13. The fact that φω is a bijection is an easy
consequence of the fact that Φu is a bijection for all u. Then, it remains to prove (6.17). We
are able to obtain
T
−2
(Jacφ
 ω ) Jacφω = a
1−∂ϕ log(a)−∂ψ log(a)(∂ϕ log(a))2 + g(∂ϕ N, ∂ϕ N )∂ϕ log(a)∂ψ log(a) + g(∂ψ N, ∂ϕ N )  .
×  −∂ϕ log(a)−∂ψ log(a) ∂ϕ log(a)∂ψ log(a) + g(∂ψ N, ∂ϕ N )(∂ψ log(a))2 + g(∂ψ N, ∂ψ N )Taking the determinant yields:det((Jacφω )T Jacφω ) = a−2 det((JacΦu )T JacΦu ),(6.58)

which together with (6.56) and the estimate (6.11) for a implies:
kdet((Jacφω )T Jacφω ) − 1kL∞ (Σ) . ε.

(6.59)(6.59) yields (6.17). This concludes the proof of Proposition 6.6.
6.6. Additional estimates
The proof of Proposition 6.7 and Proposition 6.8 follow from the estimates of Theorem 6.3,
Theorem 6.4, and Theorem 6.5 using also for some estimates the fact that u coincides with x.ω
in the region |x| ≥ 2 or the properties of the Littlewood-Paley projections Pj . For the sake ofsimplicity, we skip these proofs and refer the reader to [73] for the details.

Part 2Other works

Formation of singularities for the nonlinear Schrödinger equation
The focusing nonlinear Schrödinger equation
i∂t u + ∆u + |u|p−1 u = 0, (t, x) ∈ [0, T ) × Rd ,u(0, x) = u0 (x),(6.60)

is one of the canonical nonlinear equations in physics arising in various fields such as nonlinear
optics, plasma physics, Bose Einstein condensates and surface waves (see for example [71] fora complete introduction to these problems). It is well-known that the mass (i.e. the L2 normof u(t, .)) is conserved by the flow of (6.60). With respect to this quantity, (6.60) is said to be
subcritical if p < 1 + 4/d, critical if p = 1 + 4/d, and supercritical if p > 1 + 4/d.
We are interested in the possible breakdown in finite time of solutions to (6.60), i.e. theexistence of 0 < T < +∞ such thatlim ||∇u(t, .)||L2 (Rd ) = +∞.

t→T−In the subcritical case, all solutions exists globally in time since the mass is conserved andcontrols the flow. On the other hand, it is known since the 60’s by a simple algebraic argument
relying on the virial identity that some solutions do break down in finite time in the criticaland supercritical cases. However, this argument does not provide any qualitative informationon the blow-up dynamic. In particular, it does not yield any information on the behavior of thesolution near the blow-up time such as the rate of the blow-up, or the profile of the solution.
The goal of the four results presented below is to provide a precise description of the mechanismof formation of singularities in the critical and in the supercritical case.
Solutions blowing up on a sphere for quintic NLS with d ≥ 3. In this section, we
present the results obtained in the following paper
Raphaël, P., Szeftel, J., Standing ring blow up solutions to the N-dimensional quintic nonlinear
Schrödinger equation, Comm. Math. Phys., 290 (3), 973-996, 2009.
We prove the existence of solutions blowing up on a sphere under the assumption of radialsymmetry for quintic NLS in dimension d ≥ 34+iut = −∂r2 u − d−1r ∂r u − |u| u, (t, r) ∈ [0, T ) × R ,(6.61)+u(0, r) = u0 (r), u0 : R → C.
This equation is critical in dimension d = 1, and supercritical for dimensions d ≥ 2. One of the
motivations for considering this problem is the fact that the supercritical case (i.e. p > 1 + 4/d)
is poorly understood. Indeed, the only available result in the supercritical setting at the time
was the analog result in dimension d = 2 (i.e. the existence of solutions blowing up on a circleof strictly positive radius) which was obtained by P. Raphaël in [62]. Our goal was to extend
this result to higher dimensions. The basic heuristics is that in radial variables, the problem
becomes:N −1
i∂t u + ∂r2 u +∂r u + u|u|4 = 0r

FORMATION OF SINGULARITIES FOR THE NONLINEAR SCHRÖDINGER EQUATION

117

and hence one can expect that if the singularity formation happens along the circle r = 1, then∂r u
∼ |∂r u| << ∂r2 u
r
and the leading order blow up dynamics should be given by the one dimensional quintic NLS
for which a stable log-log dynamics is known from the breakthrough work of F. Merle and P.
Raphaël [48] [49] [50] [51] [52]. This heuristic can be justified in any dimension.
The dimension plays an important role near the origin, where we need to control the critical
norm. More precisely, in order to control the flow, we need to prove the following bound on thesolution u of (6.61)≤ δ, ∀t ∈ [0, T ),(6.62)
ku(t)k d − 1
1
H2

2 (r≤

2

)

where δ > 0 is small enough. Now, using in particular the smoothing effect of the linear
Schrödinger flow, we haveku(t)k1

H 2 ( 14 ≤r≤ 12 )<< 1, ∀t ∈ [0, T ),(6.63)

which is sufficient to conclude the proof in the case d = 2, as noticed in [62].
In dimensions d ≥ 3, (6.63) is not enough to obtain (6.62). We proceed in two steps. The
first step consists in constructing approximate energies EN which provide enough cancellation
to ensure a structure of the form:d2−αEN (u) . kukH
kuk2H N . EN (u) andfor some 0 < α < 2.
Ndt
The gain 2 − α < 2 is the key to bootstrap the global rough boundCku(t)kH N . N +δ , ∀t ∈ [0, T ),λ(t)

(6.64)The second step is to use space localization outside the singular circle and the smoothing
effect of the linear flow. We show that the initial gain of 1/2 a derivative (6.63) together withthe global bound (6.64) are in fact enough to propagate (6.62). In both steps of the proof, we
use in a crucial way the fact that the log-log singularity is an almost self similar blow up regime
which allows us to gain from various time integrations.
Stable self similar blow up dynamics for slightly L2 supercritical NLS. In thissection, we present the results obtained in the following paper
Merle, F., Raphaël, P., Szeftel, J., Stable self similar blow up dynamics for slightly L2 supercritical NLS equations. Geom. Funct. Anal., 20 (4), 1028-1071, 2010.
We consider the focusing nonlinear Schrödinger equations (6.60), with p > 1+ d4 , and p sufficiently
close to 1+ d4 , i.e. the slightly supercritical case. As in the previous section, one of the motivations
for considering this problem is the fact that the supercritical case (i.e. p > 1 + 4/d) is poorly
understood. Furthermore, the self similar regime, i.e. the regime corresponding to a behaviorof the following type1
k∇u(t)kL2 ∼when t → T,
(6.65)
1+1−N(T − t) p−1 2 4

118

is the dynamic conjectured to be the stable blow-up regime in view of the numerical computations(see for example [71]).
In this work, we prove this conjecture in the slightly supercritical case. Indeed, we prove theexistence and stability in the energy space H 1 of a self similar finite time blow up dynamics andprovide a qualitative description of the singularity formation near the blow up time. We startwith the construction of an approximate solution Qb depending on a parameter b > 0. Then,
we chose initial data such that on a short time, the solution admits a decomposition
1x − x(t) iγ(t)− πu(t, x) =(Qb(t) + ε) t,ewith |∇ε(t)|2L2 << e b(t)2
λ(t)λ p−1 (t)and aim at deriving a dynamical trapping for the parameter b(t) and the deformation ε(t). More1
precisely, introducing the global rescaled time ds
dt = λ2 , the dynamical system driving λ is
−λλt = −λs∼ b(t)
λand finite time blow up in the self similar regime will follow fromb(t) ∼ b0 > 0, |∇ε(t)|2L2 << eπ
− b(t)(1−c)

for some small constant c > 0 in the maximum time interval of existence. In order to derive such
dynamical controls, we run the log-log analysis developed in [51] by keeping track of the leading
order O(p − 1 − d4 ) deformation. The outcome is the derivation of two structural monotonicity
formulae for the parameter b. The first one is inherited from the local virial control first derived
in [48] and roughly leads to
p−1−

π
4
+ |∇ε|2L2 − e− b . bs .
d

The main observation is that the new term with respect to the log-log analysis, i.e. p − 1 − d4 in
the above left-hand side, has the good sign. The second monotonicity formula is a consequenceof the L2 conservation law and the control of the mass ejection phenomenon:
bs . p − 1 −π4− e− b .
dUsing these two monotonicity formulae, we derive the dynamical system for b:πbs ∼ σc − e− bwhich traps b around the valueπb ∼ b∗ with σc ∼ e− b∗ .
The nontrivial pointwise control on ε|∇ε(t)|2L2 << eπ−(1−c) b(t)also follows from the obtained Lyapunov controls. The existence of a self similar finite time blow
up solution is now a simple consequence of this dynamical trapping of the solution.

FORMATION OF SINGULARITIES FOR THE NONLINEAR SCHRÖDINGER EQUATION

119

Existence and uniqueness of minimal blow up solutions to an inhomogeneousmass critical NLS. Let us consider the cubic NLS in dimension 2:
iut = −∆u − |u|2 u, (t, x) ∈ [−1, T ) × R2 ,(6.66)u(−1, x) = u0 (x), u0 : R2 → C.
This equation is critical. It admits a groundstate Q which is the unique H 1 nonzero positive
radial solution to∆Q − Q + Q3 = 0.
Then the variational characterization of Q ensures that initial data u0 ∈ H 1 with ku0 kL2 <kQkL2 yield global and bounded solutions T = +∞, [83]. On the other hand, finite time blow
up may occur for data ku0 kL2 ≥ kQkL2 (R2 ) . At the critical mass threshold, the pseudo conformal
symmetry
1 x i |x|2
1
v(t, x) = N u,(6.67)
e 4t ,t t|t| 2
which preserves the flow of (6.66), applied to the periodic solitary wave solution u(t, x) = Q(x)eit
yields a minimal mass blow-up solution:
1  x  i |x|2 − i(6.68)
S(t, x) = Q
e 4t t , kS(t)kL2 = kQkL2ttwhich blows up at t = 0. In [47], Merle proves the uniqueness of the critical mass blow up
solution: a solution u ∈ H 1 with ku0 kL2 = kQkL2 and blowing up at t = 0 is equal to S(t) up
to the symmetries of the flow.
The existence of the minimal solution S(t), as well as the uniqueness proof of F. Merle rely
in a crucial way on the pseudo conformal symmetry (6.67). Now, this symmetry is unstable
under perturbation, which is not necessary the case of the property of existence and uniquenessof a minimal blow-up solution. In
Raphaël, P., Szeftel, J., Existence and uniqueness of minimal blow up solutions to an inhomogeneous mass critical NLS. J. Amer. Math. Soc., 24, 471-546, 2011,
our goal was therefore to obtain a proof of the result of F. Merle allowing us to bypass thepseudo conformal symmetry. To this end, we consider the cubic NLS in dimension 2 with an
inhomogeneity k(x):iut = −∆u − k(x)|u|2 u, (t, x) ∈ [−1, T ) × R2 ,(6.69)u(−1, x) = u0 (x), u0 : R2 → C.
The case k = 1 corresponds to the homogeneous case (6.66). The canonical effect of the inhomogenity is to destroy the large group of symmetry of the homogeneous case, and in this sense(6.69) is a toy model to analyze the properties of NLS systems in the absence, in particular, ofthe pseudo conformal symmetry.
The proof starts with the construction of an approximate solution at a very high order QP
depending on four parameters P = (b, λ, β, α). Then, we decompose the solution u of (6.69) as1
x − α(t) iγ(t)u(t, x) = N QP(t) t,e+ ũ.
(6.70)λ(t)λ 2 (t)120

We then derive a first monotonicity formula which consists of a mixed energy/Morawetz functional
Z
kũk2L2d2k∇ũkL2 ++=∂r ũũ≥ 0 + O(l.o.t.).
(6.71)dtλ2
r≤λThis functional provides us with a backward control on the solution which together with compactness arguments is enough to obtain the existence of a minimal solution, which we denoteby uc .
We then turn to the uniqueness part of the proof. We can adapt the monotonicity formula(6.71) to estimate the difference between a minimal solution u, and the minimal solution uc wehave constructed. However, we need to know a priori that u is sufficiently close to uc as weapproach the blow up time in order to use this monotonicity formula. This requires a second
monotonicity formula of Morawetz type. In turn, we obtain a strong a priori bound on u − uc ,which together with the backward control provided by the monotonicity formula (6.71) allows
us to obtain uniqueness (i.e. u = uc ).
The instability of Bourgain-Wang solutions. We consider again the cubic NLS indimension 2 (6.66) which is critical. We consider the finite time blow up dynamic exhibited by
J. Bourgain and W. Wang in [10]. Let z ∗ an asymptotic profile which is sufficiently smooth andvanishes at high order at the origin∂ α z ∗ (0) = 0 for |α| ≤ m − 1,
where m is a large enough integer. Let z the solution of equation (6.66) with initial data z ∗ :i∂t z + ∆z + z|z|2 = 0, z|t=0 = z ∗ .
Then, there exists a solution uBW which blows up at time t = 0 such thatuBW (t) = S(t) + z(t) + O(|t|m ).

(6.72)

Here, the singular part of the dynamics is given by S(t) which is defined in (6.68), and theasymptotic profile is given by z ∗ . Note that there is some decoupling between the dynamics ofS(t) and z(t) due to the fact that z ∗ vanishes at a high order at the origin.
These solutions are never observed numerically (unlike the log-log regime), and are thus
conjectured to be unstable. We prove that this is indeed the case in the following work
Merle, F., Raphaël, P., Szeftel, J., On the instability of the pseudoconformal blow up for the L2
critical NLS. Accepted for publication in Amer. J. Math, 41 pp.
More precisely, we show that a Bourgain Wang solution lies on the boundary of both H 1 open
sets of solutions that scatter forward in time, and solutions that blow up in finite time on theright in the log-log regime. The proof consists in four steps:• We first construct an explicit family of profiles Sη (t) suchp that Sη=0 = S, Sη is globalfor η > 0, and Sη formally blows-up for η < 0 at t = − |η|. We then decompose thesolution uη of (6.66) asuη = Sη (t) + z(t) + ũη (t).

(6.73)

This decomposition is the analog of the decomposition (6.72) for the Bougain Wang
solution. We then need to prove that the map η → uη is continuous, and that uη isFORMATION OF SINGULARITIES FOR THE NONLINEAR SCHRÖDINGER EQUATION

121

global for η > 0, coincides with uBW for η = 0, and blows up in the log-log regime forη < 0.
• In the case η > 0, we choose at t = 0 the initial datauη (t = 0) = Sη (0) + z ∗ .
We the obtain a backwards control of ũη all the way down to some time t = t1 , witht1 < 0 independent of η, by adapting to this context the mixed energy/Morawetz
functional (6.71) discussed in the previous section. The same argument also yields asimilar control on [0, |t1 |]. From there, forward scattering for uη is easily obtained.
√• In the case η < 0, we choose at t = tη , with tη ∼ − η, the initial datauη (tη ) = Sη (tη ) + z(tη ).
The backward control all the way down to some time t = t1 , with t1 < 0 independentof η, is obtained as in the previous step. We then show that uη for η > 0 blows up
forward in the log-log regime. Here we use the sharp open characterization of the log
log set exhibited in [61] which allows one to predict loglog blow up directly from theinitial data uη (tη ).
• We now pass to the limit as η → 0 and obtain in both cases a Bourgain Wang typesolution with regular part z ∗ as t → 0. It remains to show some uniqueness statement
about this type of dynamics to ensure the continuity of the map at η = 0. This last
step relies again on a variant of the mixed energy/Morawetz functional (6.71).
Perspective. Many open problems remain concerning the description of the various blowup dynamics for the focusing nonlinear Schrödinger equation (6.60). We suggest in particular
two problems which are related to the two results we obtained in the supercritical case in [63][53].
Ring blow up solutions. Recently, Fibich, Gavish and Wang [20] observed numerically forthe supercritical NLS radial solutions blowing up in finite time by focusing on spheres withradius collapsing to zero. The rigorous understanding of this blow up scenario and its role ina more general theory of radial super critical blow up in the continuation of [62] [63] is an
important problem.
Selfsimilar blow up for the cubic NLS in dimension 3. The cubic NLS in dimension 3, i.e.
i∂t u + ∆u + |u|2 u = 0, (t, x) ∈ [0, T ) × R3 ,u(0, x) = u0 (x),is supercritical. It is of particular interest since it is the physical relevant case. For this equation,
the self similar regime, i.e. the regime corresponding to a behavior of the typek∇u(t)kL2 ∼1
1(T − t) 4

when t → T,

is the dynamic conjectured to be the stable blow-up regime in view of the numerical computations(see for example [71]). Now, recall that we obtained in [53] the corresponding result in theslightly supercritical case, where the proof bifurcated from the critical case (i.e. p = 34 indimension 3). In order to prove the statement for the cubic NLS in dimension 3, one will needto rely on a new approach.

122

Domain decomposition methods
When discretizing a partial differential equation, one obtains for some applications (forexample oceanography, meteorology,. . . ) an enormous number of unknowns, often too many
to be managed by a single computer. Domain decomposition methods propose solutions bypartitioning the initial domain into several subdomains. Each subdomain is then solved by one
processor and an iterative exchange of information at the common interface leads to convergence
toward the global solution.
Domain decomposition methods have been mainly developed and analyzed for elliptic coercive problems and their convergence theory is well understood (see [85] [65] [60] and references
therein). When treating evolution problems, the classical approach consists of discretizing thetime dimension uniformly on the whole domain by an implicit scheme and then treating theobtained problems at each time step by a classical domain decomposition method for steadyproblems (see [13] [54] for parabolic problems and [4] [84] for hyperbolic problems).
This approach has two disadvantages. First one needs to impose a uniform time discretization
for all subdomains and thus looses one of the main features of domain decomposition algorithms,
namely to adapt the solution process to the physical properties of the subdomain. It is still
possible to refine in space, but for evolution systems this is not sufficient, since the space and time
discretization are linked in general by stability constraints and conditions on the dispersion of thenumerical scheme. Second the algorithm needs to communicate small amounts of information
at each time step. Each communication involves in addition to the cost for the data transmitted
a start up cost independent of the amount of data transmitted. It can thus be of interest tocommunicate larger packages of data at once over several time steps instead of many small
packages to save communication time.
To avoid the above disadvantages, one can decompose the original domain into subdomains
like in the classical case, but without discretizing the time dimension. Instead, one solves timedependent subproblems on each subdomain. This approach is related to waveform relaxation
algorithms for ordinary differential equations and has first been considered for partial differential
equations by Bjorhus in [8] where first order hyperbolic problems where analyzed. An overlapping Schwarz algorithm of this type has been analyzed for the heat equation in [22] [26] andfor more general parabolic problems in [27] [23]. For overlapping and non-overlapping Schwarz
waveform relaxation methods for the wave equation and convection-diffusion equations see [24][25].
Nonlinear Schwarz waveform relaxation (SWR) for semilinear wave propagation. There are very few results on the convergence of domain decomposition methods (DDM)
for nonlinear partial differential equations. In
Halpern, L., Szeftel, J. (2009): Nonlinear Nonoverlapping Schwarz Waveform Relaxation for
Semilinear Wave Propagation. Math. Comp., 78, 865-889,
we consider the model problem of the semilinear wave equation in dimension 1(∂t2 − ∂x2 )u = f (u, ∂t u, ∂x u).

(6.74)

Our goal is to show that one may optimize the convergence of SWR methods by choosing theabsorbing boundary conditions we derived for (6.74) in [72]. A special feature of these boundary

DOMAIN DECOMPOSITION METHODS

123

conditions is that they are nonlinear and thus provide us with nonlinear transmission condition,
which is not standard but not more costly to implement numerically. We prove the convergenceof our algorithm at the continuous and discrete level for the nonlinear problem. The numericalresults confirm that our method converges without overlap (this allows to avoid the delicate
issue of the gluing of different grids in the case of nonconforming grids, i.e. different grids in
different subdomains) as opposed to the classical Schwarz method, that it converges much fasterthan the classical Schwarz method, and that it provides optimal results within a large class ofalgorithms.
SWR for semilinear reaction-diffusion equations. In
Caetano, F. , Gander, M., Halpern, L., Szeftel, J. (2010): Schwarz waveform relaxation algorithms for semilinear reaction-diffusion equations. Netw. Heterog. Media, 5 (3), 487-505,
we consider the model problem of the semilinear reaction-diffusion equation in dimension 2.
The application we have in mind is the C02 geological storage, which is one of possible solutions
for reducing the emissions of greenhouse gases. We introduce two algorithms with transmission
conditions of order 0 (of Robin type), with one involving linear transmission conditions, and theother nonlinear transmission conditions. We also introduce two algorithms with transmission
conditions of order 2 (of Ventcell type), with one involving linear transmission conditions, andthe other nonlinear transmission conditions. We prove the convergence of the two algorithms
involving linear transmission conditions at the continuous level. We provide numerical tests, inparticular for an equation modeling the CO2 geological storage, which illustrate the efficiencyof the four algorithms, with a significant improvement between the order 0 and the order 2.
Optimized and Quasi-Optimal SWR for the linear Schrödinger equation. In
Halpern, L., Szeftel, J. (2010): Optimized and Quasi-Optimal Schwarz Waveform Relaxation
for the One Dimensional Schrödinger equation, Math. Models Methods Appl. Sci, 20 (12),2167-2199,
we consider the linear Schrödinger equation in dimension 1 with a variable potential V . We have
two motivations for studying this problem. On the one hand, there were no results concerning
DDM for the Schrödinger equation in the literature. On the other hand, the optimal transmission
conditions√ in the case of the free Schrödinger equation (i.e. V = 0) involve the nonlocal operator
in time ∂t . Nonlocal operators are usually not implemented in DDM due to the belief thatthe implementation cost may be too high. Now, these nonlocal operators are discretized in
the context of absorbing boundary conditions and we wanted to test the efficiency of suchdiscretizations as transmission conditions for a model problem. We first show that the classicalSchwarz algorithm converges extremely slowly. Then, we introduce two algorithms - one ofRobin type, with the optimization of the corresponding Robin√ parameter, and a ”quasi-optimal”
algorithm using the discretization of the nonlocal operator ∂t . We obtain a convergence proof
for both algorithms at the continuous level, and at the discrete level in the case of the algorithmof Robin type. Finally, numerical tests show that both proposed algorithms converge very fast,
and that the gain obtained in the case of the ”quasi-optimal” algorithm counterbalances its
higher implementation cost.

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Optimized SWR and discontinuous Galerkin time stepping. In
Halpern, L., Japhet, C., Szeftel, J., Optimized Schwarz waveform relaxation and discontinuous
Galerkin time stepping for heterogeneous problems. Submitted, 33 pp,
we consider a linear advection-diffusion equation in arbitrary dimension with non constant coefficients with discontinuities along interfaces. This problem is motivated by the fact that DDM
algorithms are not only introduced to run computations on very large domains. Indeed, one of
their main feature is also to take into account different physical parameters in each subdomain,e.g. in the case of the interface between ocean and atmosphere in meteorology. It is therefore natural to consider a model problem with discontinuities at the interfaces of subdomains.
Furthermore, in order to take into account different physical phenomenon in each subdomain,
it is relevant to consider DDM algorithms allowing for nonconforming grids both in space andtime, i.e time steps and space steps which are different in each subdomain. Thus, we consider
here discontinuous Galerkin methods in space and time, which are well-suited to this context.
It also allows for possible adaptive mesh refinement relying on an a posteriori error analysis.
In this paper, we introduce a DDM algorithm of order 0, and another of order 2, using botha discontinuous Galerkin discretization. We prove the convergence of these algorithms at thecontinuous and semi-discrete level for subdomains which can have curved interfaces. We provide
error estimates between the solution of the DDM after convergence and the original solution.
Finally, we present numerical results showing a fast convergence of both proposed algorithms
and confirming our error estimates.
Perspectives. Recall that in [12], we considered the model problem of the semilinear
reaction-diffusion equation. Next, we would like to consider parabolic problems with a nonlinear advective part (i.e. with a derivative in the nonlinearity as opposed to [12]). The obvious
model problem is the design of DDM methods for the viscous Burgers equation in dimension 1.
This would be the first step towards nonlinear parabolic systems like Navier Stokes. There arevery few results in the literature concerning DDM algorithms for these types of equations. We
are more particularly interested in applications concerning the coupling between the ocean andthe atmosphere in meteorology, where one may assume in a first approximation that both theocean and the atmosphere satisfy Navier Stokes.
Long-time existence for the Klein-Gordon equation on compact manifolds
Let u be the solution of the nonlinear Klein-Gordon equations of type(∂t2 − ∆g + m2 )u = f (u, ∂t u, ∇g u)(6.75)on (M, g) with smooth Cauchy data of size ε → 0 where m > 0. Local existence theory implies
that if the nonlinearity vanishes at order p ≥ 2 at 0, there is a solution defined on an intervalof length at least cε−p+1 . A natural question is to decide if the solution can be constructed ona longer interval, and if its Sobolev norms can be bounded uniformly on that interval.
The first results have been obtained in the case of the Euclidean space. It has been proved
independently by Klainerman [30] and Shatah [64] that the quasi-linear Klein-Gordon equation
with small, smooth, rapidly decaying Cauchy data on Rd has global solutions when d ≥ 3. The
proof relies on the use of dispersive properties of the linear Klein-Gordon equation. In theseLONG-TIME EXISTENCE FOR THE KLEIN-GORDON EQUATION ON COMPACT MANIFOLDS

125

dimensions, the nonlinearity of the equation can be considered as a short range perturbationof the linear problem, since linear solutions decay when t → ±∞ at an integrable rate. On
the other hand, in d = 2 (resp. d = 1) space dimensions, and for quadratic (resp. quadratic
or cubic) nonlinearities, the nonlinear equation is a long range perturbation of the linear one.
Nevertheless, global existence in two space dimensions for quadratic nonlinearities has been
proved by Ozawa, Tsutaya and Tsutsumi [55] combining the use of dispersive properties of theequation together with a method of normal forms. The latter, introduced initially by Shatah inhigher space dimensions, consists in modifying the solution by a quadratic perturbation chosen
in such a way that it cancels out the quadratic part of the nonlinearity. One reduces thus the
problem to an equation with a cubic nonlinearity, which is a short range perturbation of thelinear problem.
In the framework of compact manifolds, the first results have been obtained by Bourgain [9]and Bambusi [5]. They proved that for the Dirichlet problem on an interval, when the nonlinearity f is Hamiltonian – i.e. depends only on u – one gets, for almost all values of the mass
m > 0, a solution uniformly bounded in H s (s >> 1) on intervals of length cε−N for any N oreven on intervals of exponential length. A natural question is then to try to extend these resultson compact manifolds of dimension d ≥ 2.
Long time existence on compact manifolds. In the case of the equation on a compact
manifold, the linear equation no longer displays any dispersive effect in contrast to the Euclideancase. We can anyway ask if the use of a normal form method alone can improve the estimatecε−p+1 given by the local existence theory. The strategy is to use the normal form method to
get rid of the terms of lower order in the right hand side of (6.75). In
Delort J.-M. and Szeftel, J. (2004): Long-time existence for small data nonlinear Klein-Gordon
equations on tori and spheres, Int. Math. Res. Not., 37, 1897-1966,
we obtain a result of long-time existence on spheres of arbitrary dimensions for non Hamiltonian
nonlinearities. We prove in particular that if, for instance, f is an homogeneous polynomial ofeven degree p, for almost all values of m, one gets solutions which are bounded in H s (s >> 1)
on intervals of time of length cε−2p+2 (instead of ε−p+1 , as given by local existence theory).
The same type of conclusion holds true for p odd if the nonlinearity depends only on u and ∂t u
and is even in ∂t u. In order to implement the normal form method,
we introduce multilinearp
operators involving the spectral projections of the operator −∆g , and we study in details
their boundedness on Sobolev spaces as well as their composition properties. In the course of
the proof, we encounter a small divisors problem, involving quantities of the type
q
p
`,p
Fm
(λ1 , . . . , λp ) = m2 + λ21 + · · · + m2 + λ2`
qp− m2 + λ2`+1 − · · · − m2 + λp+1 ,where m is the mass in (6.75),
p and ` are integers with 0 ≤ ` ≤ p, and λ1 , . . . , λp+1 arepeigenvalues of the operator −∆g , where ∆g is the Laplacien-Beltrami on the sphere. We thenmust show that these quantities do not vanish. This does not hold in general, but we prove it
for almost every mass m > 0 relying on tools coming from sub-analytic geometry, in particular
the Lojaciewiecz inequalities (see for example [7]).

126

In
Delort J.-M. and Szeftel, J. (2006): Almost orthogonality properties of products of eigenfunctions and applications to long-time existence for semi-linear Klein-Gordon equations on Zoll
Manifolds, Amer. J. Math., 128, 1187-1218,
we extend the previous results to the case of Zoll manifolds. These manifolds are defined by thefact that all geodesic are periodic with samep period. For these manifold, the spectrum as well as
the spectral projections of the operator −∆g are in general not explicit unlike the case of thesphere. In order to extend the constructions
on the sphere, we exploit the well-known propertiespof localization of the eigenvalues of −∆g , and we prove an almost orthogonality property ofthe eigenvectors.
Almost global existence on compact manifolds. A natural question in view of theprevious result is the one of almost global existence. In other words, is it possible to gain any
negative power of ε with respect to the energy method? To this end, we have to reiterate thenormal form method as much as needed (and not just one time as in the previous section), until
we reach any gain in ε. Now, when we use the normal form method, certain terms - called
resonant - can not be eliminated. The key is that they vanish in the energy estimate due to a
special structure. The difficulty when reiterating the normal form method is then to make surethat the special structure of these resonant terms is preserved.
We obtained a first result in this direction on compact hypersurfaces of revolution in
Delort J.-M. and Szeftel, J. (2006): Bounded almost global solutions for non hamiltonian semilinear Klein-Gordon equations with radial data on compact revolution hypersurfaces, Ann. Inst.
Fourier (Grenoble), 56, 1419-1456.
To this end, we prove that the Laplacien-Beltrami restricted to the space invariant under thesymmetries of the hypersurface of revolution has simple eigenvalues, which allows for a simple
verification of the fact that reiterating the normal form method preserves the structure of the resonant terms. However, the result is not entirely satisfactory since it is morally one-dimensional,
due to the fact that we restrict to the case of simple eigenvalues.
To get a true multidimensional result, we restrict ourselves to the Hamiltonian case (i.e.
f (u, ∂t u, ∇g u) = f (u) in (6.75)). We obtain almost global existence on Zoll manifolds (e.g.
spheres of arbitrary dimensions) in
Bambusi, D., Delort, J.-M., Grébert, B., Szeftel, J. (2007): Almost global existence for Hamiltonian semi-linear Klein-Gordon equations with small Cauchy data on Zoll manifolds, Comm.
Pure. Appl. Math., 60, 1665-1690.
This is to our knowledge the first result of almost global existence for a nonlinear partial differential equation on a compact manifold of dimension d ≥ 3 (i.e. dimensions for which theconservation of the Hamiltonian is not enough to control the flow unlike the case of dimensions
1 and 2). Here, we rely on the normal form method at the level of the Hamiltonian (Birkhoff
normal form) instead of working at the level of the equation (Poincarré normal form). It turns

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127

out that the Hamiltonian structure simplifies a lot the verification of the fact that the reiterationof the normal form method preserves the structure of the resonant terms.

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