Concentration Compactness for Critical Wave Maps

Contents 1

Introduction and overview . . . . . . . . . . . . . . . . . . . 1.1 The main result and its history . . . . . . . . . . . . . . 1.2 Wave maps to H2 . . . . . . . . . . . . . . . . . . . . . 1.3 The small data theory . . . . . . . . . . . . . . . . . . . 1.4 The Bahouri–Gérard concentration compactness method 1.5 The Kenig–Merle agument . . . . . . . . . . . . . . . . 1.6 An overview of the book . . . . . . . . . . . . . . . . .

2

The spaces S Œk and N Œk . . . . . . . . . . . . . . . . . . . . . . . 2.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 The null-frame spaces . . . . . . . . . . . . . . . . . . . . . . . 2.3 The energy estimate . . . . . . . . . . . . . . . . . . . . . . . . 2.4 A stronger S Œk-norm, and time localizations . . . . . . . . . . 2.5 Solving the inhomogeneous wave equation in the Coulomb gauge

3

Hodge decomposition and null-structures . . . . . . . . . . . . . . . 65

4

Bilinear estimates involving S and N spaces . . . . . . . . . . 4.1 Basic L2 -bounds . . . . . . . . . . . . . . . . . . . . . 4.2 An algebra estimate for S Œk . . . . . . . . . . . . . . . 4.3 Bilinear estimates involving both S Œk1  and N Œk2  waves 4.4 Null-form bounds in the high-high case . . . . . . . . . 4.5 Null-form bounds in the low-high and high-low cases . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

73 73 90 93 101 111

5

Trilinear estimates . . . . . . . . . . . . . . . . . . . . . 5.1 Reduction to the hyperbolic case . . . . . . . . . . 5.2 Trilinear estimates for hyperbolic S-waves . . . . . 5.3 Improved trilinear estimates with angular alignment

. . . .

. . . .

. . . .

. . . .

115 116 148 180

6

Quintilinear and higher nonlinearities . . . . . . . . . . . . . . . . . 197 6.1 Error terms of order higher than five . . . . . . . . . . . . . . . 217

7

Some basic perturbative results . . . . . . . . . . . . . . . . . . . . . 223 7.1 A blow-up criterion . . . . . . . . . . . . . . . . . . . . . . . . 223 7.2 Control of wave maps via a fixed L2 -profile . . . . . . . . . . . 244

8

BMO, Ap , and weighted commutator estimates . . . . . . . . . . . . 277

. . . .

. . . .

. . . .

. . . . . . .

. . . . . . .

. . . . . . .

. 1 . 1 . 4 . 8 . 11 . 16 . 18 25 25 28 42 57 63

vi

Contents

9

The Bahouri–Gérard concentration compactness method . . 9.1 The precise setup for the Bahouri–Gérard method . . . 9.2 Step 1: Frequency decomposition of initial data . . . . 9.3 Step 2: Frequency localized approximations to the data 9.4 Step 3: Evolving the lowest-frequency nonatomic part . 9.5 Completion of some proofs . . . . . . . . . . . . . . . 9.6 Step 4: Adding the first large component . . . . . . . . 9.7 Step 5: Invoking the induction hypothesis . . . . . . . 9.8 Completion of proofs . . . . . . . . . . . . . . . . . . 9.9 Step 6 of the Bahouri–Gérard process; adding all atoms

. . . . . . . . . .

293 293 294 302 309 358 358 397 421 422

10

The proof of the main theorem . . . . . . . . . . . . . . . . . . . . . 10.1 Some preliminary properties of the limiting profiles . . . . . . . 10.2 Rigidity I: Harmonic maps and reduction to the self-similar case 10.3 Rigidity II: The self-similar case . . . . . . . . . . . . . . . . .

427 429 444 450

11

Appendix . . . . . . . . . 11.1 Completing a proof . 11.2 Completion of proofs 11.3 Completion of a proof 11.4 Completion of a proof 11.5 Competion of proofs

459 459 462 463 467 469

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . . . . . .

. . . . . .

. . . . . . . . . .

. . . . . .

. . . . . . . . . .

. . . . . .

. . . . . . . . . .

. . . . . .

. . . . . .

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 477 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 481

1

Introduction and overview

1.1

The main result and its history

Formally speaking, wave maps are the analogue of harmonic maps where the Minkowski metric is imposed on the independent variables. More precisely, for a smooth u W RnC1 ! M with .M; g/ Riemannian, define the Lagrangian Z
L.u/ WD j@ t uj2g jruj2g dt dx: RnC1

Then the critical points are defined as L0 .u/ D 0 which means that u ? Tu M in case M is embedded in some Euclidean space. This is called the extrinsic formulation, which can also be written as u C A.u/.@˛ u; @˛ u/ D 0 where A.u/ is the second fundamental form. In view of this, it is clear that ı  is a wave map for any geodesic in M and any free scalar wave . Moreover, any harmonic map is a stationary wave map. The intrinsic formulation is D ˛ @˛ u D 0, where j D˛ X j WD @˛ X j C i k ı u X i @˛ uk is the covariant derivative induced by u on the pull-back bundle of T M under u (with the summation convention in force). Thus, in local coordinates u D .u1 ; : : : ; ud / one has uj C

j ik

ı u @˛ ui @˛ uk D 0:

(1.1)

The central problem for wave maps is to answer the following question: For which M does the Cauchy problem for the wave map u W RnC1 ! M with P t D0 D .u0 ; u1 / have global smooth solutions? smooth data .u; u/j In view of finite propagation speed, one may assume that the data .u0 ; u1 / are trivial outside of some compact set (i.e., u0 is constant outside of some compact set, whereas u1 vanishes outside of that set). Let us briefly describe what is known about this problem.

2

1 Introduction and overview

First, recall that the wave map equation is invariant under the scaling u 7! n u.
/ which is critical relative to HP 2 .Rn /, whereas the conserved energy n Z 1X E.u/ D j@˛ u.t; x/j2 dx n 2 R ˛D0

is critical relative to HP 1 .Rn /. In the supercritical case n  3 it was observed by Shatah [41] that there are self-similar blow-up solutions of finite energy. In the critical case n D 2, it is known that there can be no self-similar blow-up, see [42]. Moreover, Struwe [50] observed that in the equivariant setting, blow-up in this dimension has to result from a strictly slower than self-similar rescaling of a harmonic sphere of finite energy. His arguments were based on the very detailed well-posedness of equivariant wave maps by Christodoulou, Tavildar-Zadeh [4], [5], and Shatah, Tahvildar-Zadeh [44], [45] in the energy class for equivariant wave maps into manifolds that are invariant under the action of SO.2; R/. Finally, Rodnianski, Sterbenz [38], as well as the authors together with Daniel Tataru [26] exhibited finite energy wave maps from R2C1 ! S 2 that blow up in finite time by suitable rescaling of harmonic maps. Let us now briefly recall some well-posedness results. The nonlinearity in (1.1) displays a null-form structure, which was the essential feature in the subcritical theory of Klainerman–Machedon [16]–[18], and Klainerman–Selberg [20], [21]. These authors proved strong local well-posedness for data in H s .Rn / when s > n2 . The important critical theory s D n2 was begun by Tataru [64], [63]. These seminal papers proved global well-posedness for smooth data satisfying a n n 2 2 1 smallness condition in BP 2;1 .Rn /  BP 2;1 .Rn /. In a breakthrough work, Tao [59], n n [58] was able to prove well-posedness for data with small HP 2  HP 2 norm and the sphere as target. For this purpose, he introduced the important microlocal gauge in order to remove some “bad” interaction terms from the nonlinearity. Later results by Klainerman, Rodnianski [19], Nahmod, Stephanov, Uhlenbeck [36], Tataru [61], [60], and Krieger [23], [24], [25] considered other cases of targets by using similar methods as in Tao’s work. Recently, Sterbenz and Tataru [47], [48] have given the following very satisfactory answer1 to the above question: If the energy of the initial data is smaller than the energy of any nontrivial harmonic map Rn ! M, then one has global existence and regularity.

1

The conclusions of our work were reached before the appearance of [47], [53].

1.1 The main result and its history

3

Notice in particular that if there are no harmonic maps other than constants, then one has global existence for all energies. A particular case of this are the hyperbolic spaces Hn for which Tao [57]– [53] has achieved the same result (with some a priori global norm control). The purpose of this book is to apply the method of concentration compactness as in Bahouri, Gérard [1] and Kenig, Merle [14], [15] to the large data wave map problem with the hyperbolic plane H2 as target. We emphasize that this gives more than global existence and regularity as already in the semilinear case considered by the aforementioned authors. The fact that in the critical case the large data problem should be decided by the geometry of the target is a conjecture going back to Sergiu Klainerman. Let us now describe our result in more detail. Let H2 be the upper half-plane 2 y2 model of the hyperbolic plane equipped with the metric ds 2 D d x yCd . Let u W 2 R2 ! H2 be a smooth map. Expanding the derivatives f@˛ ug˛D0;1;2 (with @0 WD @ t ) in the orthonormal frame fe1 ; e2 g D fy@x ; y@y g gives rise to smooth coordinate P j 2 1 functions ˛1 ; ˛2 . In what follows, k@˛ ukX will mean . j2D1 k˛ kX / 2 for any norm k
kX on scalar functions. For example, the energy of u is

E.u/ WD

2 X

k@˛ uk22 :

˛D0

Next, suppose  W H2 ! M is a covering map with M some hyperbolic Riemann surface with the metric that renders  a local isometry. In other words, M D H2 = for some discrete subgroup  PSL.2; R/ which operates totally discontinuously on H2 . Now suppose u W R2 ! M is a smooth map which is constant outside of some compact set, say. It lifts to a smooth map uQ W R2 ! H2 uniquely, up to composition with an element of . We now define Q X . In particular, the energy E.u/ WD E.u/. Q Note that due to k@˛ ukX WD k@˛ uk 2 the fact that is a group of isometries of H , these definitions are unambiguous. Our main result is as follows. Theorem 1.1. There exists a function K W .0; 1/ ! .0; 1/ with the following property: Let M be a hyperbolic Riemann surface. Suppose .u0 ; u1 / W R2 ! M  TM are smooth and u0 D const, u1 D 0 outside of some compact set. Then the wave map evolution u of these data as a map R1C2 ! M exists globally 1 as a smooth function and, moreover, for any p1 C 2q  41 with 2  q < 1,

4

D1

1 Introduction and overview 1 p

2 q, 2 X

2

k.
/

@˛ ukLpt Lqx  Cq K.E/:

(1.2)

˛D0

Moreover, in the case when M ,! RN is a compact Riemann surface, one has scattering:

max @˛ u.t / @˛ S.t /.f; g/ L2 ! 0 as t ! ˙1 x

˛D0;1;2

jrj/ where S.t/.f; g/ D cos.t jrj/f C sin.t g and suitable .f; g/ 2 .HP 1  jrj L2 /.R2 I RN /. Alternatively, if M is non-compact, then lifting u to a map j R1C2 ! H2 with derivative components ˛ as defined above, one has

max ˛j .t / @˛ S.t /.f j ; g j / L2 ! 0 as t ! ˙1 x

˛D0;1;2

where .f j ; g j / 2 .HP 1  L2 /.R2 I R/. We emphasize that (1.2) can be strengthened considerably in terms of the type of norm applied to the Coulomb gauged derivative components of the wave map: 2 X

k

2 ˛ kS

 C K.E/2

(1.3)

˛D0

The meaning ˛ as well as of the S norm will be explained below. We now turn to describing this result and our methods in more detail. For more background on wave maps see [13], [61], and [42].

1.2

Wave maps to H2

The manifold H2 is the upper half-plane equipped with the metric ds 2 D d x2 Cd y2 . Expanding the derivatives f@˛ ug˛D0;1;2 (with @0 WD @ t ) of a smooth y2 map u W R1C2 ! H2 in the orthonormal frame fe1 ; e2 g D fy@x ; y@y g yields @˛ u D .@˛ x; @˛ y/ D

2 X j D1

˛j ej ;

1.2 Wave maps to H2

whence yDe

P

j D1;2

1@ 2 j j




;

xD

5

X

1




@j .j1 y/;

(1.4)

j D1;2

provided we assume the normalization limjxj!1 j ln yj D limjxj!1 jxj D 0. Energy conservation takes the form Z

Z 2 X 2 X ˇ j ˇ ˇ .t; x/ˇ2 dx D ˛

2 X 2 X ˇ j ˇ ˇ .0; x/ˇ2 dx ˛

(1.5)

R2 ˛D0 j D1

R2 ˛D0 j D1

where x D .x1 ; x2 / and @0 D @ t . If u.t; x/ is a smooth wave map, then the j functions f˛ g for 0  ˛  2 and j D 1; 2 satisfy the div-curl system @ˇ ˛1

@˛ ˇ1 D ˛1 ˇ2

@ˇ ˛2

@˛ ˇ2 @˛  1˛ @˛  2˛

ˇ1 ˛2

(1.6)

D0 D

(1.7) ˛1  2˛

(1.8)

D ˛1  1˛

(1.9)

for all ˛; ˇ D 0; 1; 2. As usual, repeated indices are being summed over, and lowering or raising is done via the Minkowski metric. Clearly, (1.6) and (1.7) are integrability conditions which are an expression of the curvature of H2 . On the other hand, (1.8) and (1.9) are the actual wave map system. Since the choice of frame was arbitrary, one still has gauge freedom for the system (1.6)–(1.9). We shall exclusively rely on the Coulomb gauge which is given in terms of complex notation by the functions 1

P2

1

j D1 @j j : D .˛1 C i ˛2 /e i
P If j1 are Schwartz functions , then j2D1 @j j1 has mean zero whence

˛




1

2 X j D1

WD

1 ˛

Ci

2 ˛


1 @j j1 .z/ D 2

Z R2

log jz

j

2 X

@j j1 ./ d  ^ d N

(1.10)

(1.11)

j D1

is well-defined and moreover decays like jzj 1 (but in general no faster). The gauged components f ˛ g˛D0;1;2 satisfy the new div-curl system X 2 1 @˛ ˇ @ˇ ˛ D i ˇ
1 @j . ˛1 j2 ˛ j/ j D1;2

i

˛


1

@j .

1 2 ˇ j

2 1 ˇ j/

(1.12)

6

1 Introduction and overview

@





Di




2 X

1

@j .

1 2  j

2 1  j /:

(1.13)

j D1

In particular, one obtains the following system of wave equations for the 

˛

D i @ˇ



˛


1

X

@j .

1 2 ˇ j

˛:

2 1 ˇ j/



j D1;2

i @ˇ



ˇ


1

@j .

C i @˛



ˇ

1 2 ˛ j




2 1 ˛ j/



X

1

@j .

1 2 ˇ j

2 1 ˇ j/



(1.14)

j D1;2

Throughout this book we shall only consider admissible wave maps u. These are characterized as smooth wave maps u W I  R2 ! H2 on some time interval I so j that the derivative components ˛ are Schwartz functions on fixed time slices. By the method of Hodge decompositions from2 [23]– [25] one exhibits the null-structure present in (1.12)–(1.14). Hodge decomposition here refers to writing 2 X Rk k C ˇ (1.15) ˇ D Rˇ kD1

where Rˇ WD @ˇ jrj 1 are the usual Riesz transform. Inserting the hyperbolic P terms Rˇ 2kD1 Rk k into the right-hand sides of (1.12)–(1.14) leads to trilinear nonlinearities with a null structure. As is well-known, such null structures are amenable to better estimates since they annihilate “self-interactions”, or more precisely, interactions of waves which propagate along the same characteristics, cf. [18]– [17], as well as [20], [21], [11]. Furthermore, inserting at least one “elliptic term” ˇ from (1.15) leads to a higher order nonlinearity, in fact quintic or higher which are easier to estimate (essentially by means of Strichartz norms). To see this, note that 2 X

@j j D 0

j D1

@j ˇ

2

@ˇ j D @j

ˇ

@ˇ

j;

In these papers this decomposition is also referred to as “dynamic decomposition”.

1.2 Wave maps to H2

7

whence ˇ D i

2 X

@j


1



ˇ


1

@k .

1 j

2 k

1 2 k j/

j


1

@k .

1 ˇ

2 k

1 k

2 ˇ/

 :

j;kD1

(1.16) j Since we are only going to obtain a priori bounds on ˛ , it will suffice to assume j throughout that the ˛ are Schwartz functions, whence the same holds for ˛ . In what follows, we shall never actually solve the system (1.12)–(1.14). To go further, the wave-equation (1.14) by itself is meaningless without assuming the ˛ to satisfy the compatibility relations (1.12) and (1.13). In fact, it is not even clear that (1.12) and (1.13) will hold for all t 2 . T; T / if they hold at time t D 0 and (1.14) holds for all t 2 . T; T /. Nonetheless, assuming that the ˛ are defined in terms of the derivative components ˛ of a ‘sufficiently nice’ wave map, it is clear that all three of (1.12)–(1.14) will be satisfied. This being said, we will only use the system (1.14) to derive a priori estimates for ˛ , which will j then be shown to lead to suitable bounds on the components ˛ of derivatives of a wave map u. This is done by means of Tao’s device of frequency envelope, see [59] or [23]. This refers to a sequence fck gk2Z of positive reals such that ck 2

 jk `j

 c`  ck 2 jk

`j

(1.17)

where  > 0 is a small number. The most relevant example is given by ck WD

X

2

 jk `j

kP` .0/k22

 21

`2Z

which controls the initial data. While it is of course clear that (1.6)–(1.9) imply the system (1.12)–(1.14), the reverse implication is not such a simple matter since it involves solving an elliptic system with large solutions. On the other hand, transferring estimates on the ˛ in H s .R2 / spaces to similar bounds on j the derivative components ˛ does not require this full implication. Indeed, assume the bound k kL1 .. T0 ;T1 /IH ı1 .R2 // < 1 for some small ı1 > 0 (we will t obtain such bounds via frequency envelopes with 0 < ı1 <  ). For any fixed time t 2 . T0 ; T1 / one now has with Pk being the usual Littlewood–Paley projections to frequency 2k , P2

1 1 kP` ˛ kH ı2 D P` Œe i j D1
@j j ˛  H ı2 P2

1 1  P` ŒP 0 is sufficiently small. The latter claim is of course the entire objective of this book. We should also remark that we bring (1.14) into play only because it fits into the framework of the spaces from [59] and [63]. This will allow us to obtain the crucial energy estimate for solutions of (1.14), whereas it is not clear how to do this directly for the system (1.12), (1.13). As already noted in [23], the price one pays for passing to (1.14) lies with the initial conditions, or more precisely, the time derivative @ t ˛ .0;
/. While ˛ .0;
/ only involves one derivative of the wave map u, this time derivative involves two. This will force us to essentially “randomize” the initial time.

1.3

The small data theory

In this section we give a very brief introduction to the spaces which are needed to control the system (1.12), (1.13), and (1.14). A systematic development will

9

1.3 The small data theory

be carried out in Chapter 2 below, largely following [58] (we do need to go beyond both [58] and [23] in some instances such as by adding the sharp Strichartz spaces with the Klainerman–Tataru gain for small scales, and by eventually modifying k
kS Œk to the stronger jjj
jjjS Œk which allows for a high-high gain in the S  S ! L2tx estimate). First note that it is not possible to bound the trilinear nonlinearities in this system in Strichartz spaces due to slow dispersion in dimension two. Moreover, it is not possible to adapt the X s;b -space of the subcritical theory to the scaling invariant case as this runs into logarithmic divergences. For this reason, Tataru [63] devised a class of spaces which resolve these logarithmic divergences. His idea was to allow characteristic frames of reference. More precisely, fix ! 2 S 1 and define p !˙ WD .1; ˙!/= 2; t! WD .t; x/
!C ; x! WD .t; x/ t! !C ; which are the coordinates defined by a generator on the light-cone. Now suppose that i are free waves such that 1 is Fourier supported on 1  jj  2, and both 2 and 3 are Fourier supported on jj  2k where k is large and negative. Finally, we also assume that the three waves are in “generic position”, i.e., that their Fourier supports make an angle of about size one. Clearly, 2 k 1 2 3 is then a representative model for the nonlinearities arising in (1.14). With Z e i ŒtjjCx
 f ./ d  3 .t; x/ D R2

p R we perform the plane-wave decomposition 3 .t; x/ D ! . 2t! / d! where Z ! .s/ WD e i rs f .r!/ rdr: By inspection, Z k! kL2

k

1 t! L x !

d! . 2 2 k

3 kL1 L2x :

(1.19)

t

Hence, 2

k

Z k!

1

2 kL1

2 t! Lx!

d! . 2 .k

k

Z k! kL2

1 t! Lx!

3 kL1 L2x t

k

d!k

1

2 kL2 L2x t! !

1 kL1 L2x k 2 kL1 L2x t

t

example3

which is an of a trilinear estimate which will be studied systematically in Chapter 5. Here we used both (1.19) and the standard bilinear L2tx bilinear 3

Note that one does not obtain a gain in this case. This fact will be of utmost importance in this book, forcing us to use a “twisted” wave equation resulting from these high-low-low interactions in the linearized trilinear expressions.

10

1 Introduction and overview

L2 -bound for waves with angular separation: k

1

2 kL2 L2x t! !

Dk

1

2 kL2 L2x t

k

. 22 k

2 kL1 L2x k 1 kL1 L2x t

This suggests introducing an atomic space with atoms 1 and satisfying k ! kL1 L2x  1 t!

!

t

of Fourier support jj 

!

as part of the space N Œ0 which holds the nonlinearity (the zero here refers to the Littlewood–Paley projection P0 . Below, we refer to this space as NF). In addition, the space defined by (1.19) is also an atomic space and should be incorporated in the space SŒk holding the solution at frequency 2k (we refer to this below as the 2 PW space). By duality to L1t! L2x! in N Œ0, we then expect to see L1 t! Lx! as part of SŒ0. The simple observation here (originating in [63]) is that one can indeed bound the energy along a characteristic frame .t! ; x! / of a free wave as long as its Fourier support makes a positive angle with the direction !. Indeed, recall the local energy conservation identity @ t e div.@ t r / D 0 for a free wave where 1 e D .j@ t j2 C jr j2 / 2 is the energy density, over a region of the form f T  t  T g \ ft! > ag. From the divergence theorem one obtains that Z Œ T t T  j! ? r j2 d L2 . k k2L1 L2 t

t! Da

x

where L2 is the planar Lebesgue measure on ft! D ag. Sending T ! 1 and letting  denote the distance between ! and the direction of the Fourier support of j tD0 , one concludes that k kL1 L2x .  t!

!

1

k kL1 L2x : t

Hence, we should include a piece sup d.!; /k kL1 2 ! Lx

!622

!

in the norm S Œ0 holding P0 provided is a wave packet oriented along the cone of dimensions 1  2k  22k , projecting onto an angular sector in the -plane associated with the cap   S 1 , where  is of size 2k (this is called NF below). Recall that we have made a genericity assumption which guaranteed that the Fourier supports were well separated in the angle. In order to relax this condition,

1.4 The Bahouri–Gérard concentration compactness method

11

it is essential to invoke the usual device of null-forms which cancel out parallel interactions. One of the discoveries of [23] is a genuinely trilinear null-form expansion, see (5.44) and (5.45), which exploit the relative position of all three waves simultaneously. It seems impossible to reduce the trilinear nonlinearities of (1.14) exclusively to the easier bilinear ones. It is shown in [63] (and then also in [58] which develops much of the functional framework that we use, as well as [23]) that in low dimensions (especially n D 2 but these spaces are also needed for n D 3), these null-frame spaces are strong enough – in conjunction with more traditional scaling invariant X s;b spaces – to bound the trilinear nonlinearities, as well as weak enough to allow for an energy estimate to hold. This then leads modulo passing to an appropriate gauge to the small energy theory.
12 P 2 The norm k
kS in (1.3) is of the form k kS WD where k2Z kPk kS Œk 2 s;b , L4 L1 Strichartz norms, as well as the S Œk is built from L1 t Lx , critical X t x null-frame spaces which we just described.

1.4

The Bahouri–Gérard concentration compactness method

We now come to the core of the argument, namely the Bahouri–Gérard type decomposition and the associated perturbative argument. We remark that independently of and simultaneously with Bahouri, Gérard, Merle and Vega [30] introduced a similar concentration compactness method into the study of nonlinear evolution equations (they considered the L2 -critical nonlinear Schrödinger equation). In [12] P. Gérard considered defocusing semilinear wave equations in R3C1 of the form u C f .u/ D 0 with data given by a sequence .n ; n / of energy data going weakly to zero. Denote the resulting solutions to the nonlinear problem by un , and the free waves with the same data by vn . Gérard proved that provided f .u/ is subcritical relative to energy then kun

vn kL1 .I IE/ ! 0;

as n ! 1

where E is the energy space. In contrast, for this to hold for the energy critical problem he found via the concentrated compactness method of P. L. Lions that it is necessary and sufficient that kvn kL1 .I IL6 .R3 // ! 0. In other words, the critical problem experiences a loss of compactness. The origin of this loss of compactness, as well as the meaning of the L6 condition were later made completely explicit by Bahouri–Gérard [1]. Their result

12

1 Introduction and overview

2 3 P1 reads as follows: Let f.n ; n /g1 nD1  H  L .R / be a bounded sequence, and define vn to be a free wave with these initial data. Then there exists a subsequence fvn0 g of fvn g, a finite energy free wave v, as well as free waves V .j / and C .".j / ; x .j / / 2 .RC ; R3 /Z for every j  1 with the property that for all `  1, ! ` .j / .j / X 1 t t x x n n vn0 .t; x/ D v.t; x/ C V .j / ; C wn.`/ .t; x/ (1.20) q .j / .j / .j / "n "n j D1 "n

where lim sup kwn.`/ kL5 .R;L10 !0 3 x .R // t

n!1

as ` ! 1;

and for any j ¤ k, .j /

"n

.k/

"n

.k/

C

"n

.j /

"n

C

ˇ .j / ˇxn

ˇ .j / .k/ ˇ xn ˇ C ˇtn .j /

"n

.k/ ˇ tn ˇ

! 1;

as n ! 1:

Furthermore, the free energy E0 satisfies the following orthogonality property: E0 .vn0 /

D E0 .v/ C

` X

E0 .V .j / / C E0 .wn.`/ / C o.1/;

as n ! 1:

j D1

Note that this result characterized the loss of compactness in terms of the appearance of concentration profiles V .j / . Moreover, [1] contains an analogue of this result for so-called Shatah–Struwe solutions of the semi-linear problem u C juj4 u D 0 which then leads to another proof of the main result in [12]. One of the main applications of their work was to show the existence of a function A W Œ0; 1/ ! Œ0; 1/ so that every Shatah–Struwe solution satisfies the bound
kukL5 RIL10 .R3 /
 A E.u/ (1.21) t

x

where E.u/ is the energy associated with the semi-linear equation. This is proved by contradiction; indeed, assuming (1.21) fails, one then obtains sequences of bounded energy solutions with uncontrollable Strichartz norm which is then shown to contradict the fact the nonlinear solutions themselves converge weakly to another solution. The decomposition (1.20) compensates for the aforementioned loss of compactness by reducing it precisely to the effect of the symmetries, i.e., dilation and scaling. This is completely analogous to the elliptic (in fact, variational) origins of the method of concentration compactness, see Lions [28] and Struwe [49]. See [1] for more details and other applications.

1.4 The Bahouri–Gérard concentration compactness method

13

The importance of [1] in the context of wave maps is made clear by the argument of Kenig, Merle [14], [15]. This method, which will be described in more detail later in this section, represents a general method for attacking global well-posedness problems for energy critical equations such as the wave map problem. Returning to the Bahouri–Gérard decomposition, we note that any attempt at implementing this technique for wave maps encounters numerous serious difficulties. These are of course all rooted in the difficult nonlinear nature of the system (1.6)–(1.9). Perhaps the most salient feature of our decomposition, performed in detail in Section 9.2, as compared to [1] is that the free wave equation no longer captures the correct asymptotic behavior for large times; rather, the atomic components V .j / are defined as solutions of a covariant (or “twisted”) wave equation of the form  C 2iA˛ @˛ (1.22) where the magnetic potential A˛ arises from linearizing the wave map equation in the Coulomb gauge. More precisely, the magnetic term here captures the highlow-low interactions in the trilinear nonlinearities of the wave map system where there is no a priori smallness gain. We shall then obtain the concentration profiles via an inductive procedure over increasing frequency scales; in particular, in (1.22) the Coulomb potential A˛ (this is a slight misnomer, but the “Coulomb” here and in all other instances where we use this phrase is a reference to the gauge) is defined in terms of lower-frequency approximations which are already controlled, see the next subsection for more details. In keeping with the Kenig–Merle method, the Bahouri–Gérard decomposition is used to show the following: assume that a uniform bound of the form k kS  C.E/ for some function C.E/ fails for some finite energy levels E. In particular, the set ˚ A WD E 2 RC j sup k kS D 1 ¤ ; k kL2 E x

P 1 where we loosely denote the energy by k kL2x D . 2˛D0 k ˛ k2 2 / 2 , and we can Lx then define a number, denoted throughout the rest of this book by Ecrit , as follows: Ecrit D inf E E 2A

(1.23)

Then there must exist a weak wave map ucritical W . T0 ; T1 / ! S to a compact Riemann surface uniformized by H2 , which enjoys certain compactness properties. In the final part of the argument we then need to rule out the existence of such an object, arriving at an eventual contradiction at the end of the book.

14

1 Introduction and overview

Starting this grand contradiction argument here, we now assume as above that A ¤ ;; this implies that there is a sequence of Schwartz class (on fixed time slices) wave maps un W . T0n ; T1n /  R2 ! H2 with the properties that ı k n kL2x ! Ecrit , ı limn!1 k n kS.. T0n ;T1n /R2 / D 1. Thus all these wave maps have t D 0 in their domain of definition. We shall call such a sequence of wave maps essentially singular. Roughly speaking, we shall proceed along the following steps. First, recall that the Bahouri–Gérard theorem is a genuine phase-space result in the sense that it identifies the main asymptotic .`/ carriers of energy which are not pure radiation, which would then sit in wn . This refers to the free waves V .j / above, which are “localized” in frequency (namely at .j / scale ."n / 1 ) as well as in physical spaces (namely around the space-time points .j / .j / .j / .tn ; xn / ). The procedure of filtering out the scales "n is due to Metivier– Schochet, see [33]. (1) Bahouri–Gérard I: Filtering out frequency blocks. If we apply the frequency localization procedure of Metivier–Schochet to the n n derivative components ˛n D . @˛ynx ; @˛yny / of an essentially singular sequence at time t D 0, we run into the problem that the resulting frequency components are not necessarily related to an actual map from R2 ! H2 . We introduce a procedure to obtain a frequency decomposition which is “geometric”, i.e., the frequency localized pieces are themselves derivative components of maps from R2 ! H2 . More specifically, in Section 9.2, we start with decompositions ˛n D

A X

Q ˛na C w˛nA ; ˛ D 0; 1; 2

aD1

where the Q ˛na are “frequency atoms” obtained from the first stage of the standard Bahouri–Gérard process, see [1]. Here it may be assumed that the frequency scales in the cases ˛ D 0; 1; 2 are identical. Since the Q ˛na do not necessarily form the derivative components of admissible maps into H2 , one replaces them by components ˛na which are derivative components of admissible maps, subject to the same frequency scales. (2) Refining the considerations on frequency localization; frequency localized approximative maps. In order to deal with the non-atomic (in the frequency sense) derivative components, which may still have large energy, we need to be able to truncate the derivative components arbitrarily in frequency while still retaining the geometric interpretation. Here we shall use arguments just as in the first step to allow us to “build up” the components ˛n from low fre-

15

1.4 The Bahouri–Gérard concentration compactness method

quency ones. In the end, we of course need to show that for some subsequence of the ˛n , the frequency support is essentially atomic. If this were to fail, we deduce an a priori bound on k ˛n kS.. T0n ;T1n /R2 / . Specifically, we show in Section 9.3 that judicious choice of an interval J , depending on the position of the Fourier support of the frequency atoms ˛na allows us to truncate the components ˛n to PJ ˛n while retaining their “geometric significance”, i.e., the components PJ ˛na , ˛ D 0; 1; 2 are also derivative components of a map up to arbitrarily small errors. (3) Assuming the presence of a lowest energy non-atomic type component, establish an a priori estimate for its nonlinear evolution. More precisely, in nA

.0/

Section 9.4, we replace ˛n by components ˚˛ 0 , which arise by truncating the frequency support of ˛n to sufficiently low frequencies such that all frequency atoms with energy above a certain threshold are eliminated. In order to obtain a priori bounds on the evolution of the associated Coulomb .0/

components

nA ˛ 0

.0/

.0/

P nA0 1 nA D ˚˛ 0 e i kD1;2 4 @k ˚k , .0/ nA0 the ˚˛ by frequency truncated

we use the previous .0/

nA

step to approximate PJj ˚˛ 0 for judiciously chosen increasing intervals Jj , whose number only depends on the energy Ecrit . A finite induction procedure then leads to a priori bounds on nA

.0/

the ˛ 0 , provided n is chosen large enough (only depending on Ecrit ). Here we already encounter the difficulty that the low frequency components appear to interact strongly with the high-frequency components in the nonlinearity, a stark contrast to the defocussing nonlinear critical wave equation. In particular, in order to “bootstrap” the bounds on the differences of the nA

.0/

Coulomb potentials associated with the PJj ˚˛ 0 , we have to invoke energy estimates for covariant wave equations of the form u C 2i @ uA D 0. (4) Bahouri–Gérard II, applied to the first atomic frequency component. In Section 9.6, assuming that we have constructed the first “low frequency ap.0/

nA

proximation” ˚˛ 0 in the previous step, we need to filter out the concentration profiles (analogous to the V .j / at the beginning of this subsection) corresponding to the frequency atoms above the minimum energy threshold and at lowest possible frequency. This is where we have to deviate from Bahouri–Gérard: instead of the free wave operator, we need to use the covariant wave operator An D  C 2iAn @ to model the asymptotics as t ! ˙1, where An is the Coulomb potential associated with the low fre.0/

nA0

quency approximation ˚˛

. Thus we obtain the concentration profiles

16

1 Introduction and overview

as weak limits of the data under the covariant wave evolution. Again a lot of effort needs to be expended on showing that the components we obtain are actually the Coulomb derivative components of Schwartz maps from R2 ! H2 , up to arbitrarily small errors in energy. Once we have this, we can then use the result from the stability section in order to construct the time evolution of these pieces and obtain their a priori dispersive behavior. (5) Bahouri–Gérard II; completion. Here we repeat Steps 3 and 4 for the ensuing frequency pieces, to complete the estimate for the ˛n . The conclusion is that upon choosing n large enough, we arrive at a contradiction, unless there is precisely one frequency component and precisely one atomic physical component forming that frequency component. These are the data that then gives rise to the weak wave map with the desired compactness properties.

1.5

The Kenig–Merle agument

In [14], [15], Kenig and Merle developed an approach to the global wellposendess for defocusing energy critical semilinear Schrödinger and wave equations; moreover, their argument yields a blow-up/global existence dichotomy in the focusing case as well, provided the energy of the wave lies beneath a certain threshold. See [7] for an application of these ideas to wave maps. Let us give a brief overview of their argument. Consider u C u5 D 0 in R1C3 with data in HP 1  L2 . It is standard that this equation is well-posed for small data provided we place the solution in the energy space intersected with suitable Strichartz spaces. Moreover, if I is the maximal interval of existence, then necessarily kukL8 .I IL8x .R3 // D 1 and the energy E.u/ is conserved. t Now suppose Ecrit is the maximal energy with the property that all solutions in the above sense with E.u/ < Ecrit exist globally and satisfy kukL8 .RIL8x .R3 // t < 1. Then by means of the Bahouri–Gérard decomposition, as well as the perturbation theory for this equation one concludes that a critical solution uC exists on some interval I  and that kuC kL8 .I  IL8x .R3 // D 1. Moreover, by similar t arguments one obtains the crucial property that the set n 1  o
 3

 K WD  2 .t /u .t / x y.t / ; t ;  2 .t /@ t u .t / x y.t / ; t W t 2I

1.5 The Kenig–Merle agument

17

is precompact in HP 1  L2 .R3 / for a suitable path .t /; y.t /. To see this, one applies the Bahouri–Gérard decomposition to a sequence un of solutions with energy E.un / ! Ecrit from above. The logic here is that due to the minimality assumption on Ecrit only a single limiting profile can arise in (1.20) up to errors that go to zero in energy as n ! 1. Indeed, if this were not the case then due to fact that the profiles diverge from each other in physical space as n ! 1 one can then apply the perturbation theory to conclude that each of the individual nonlinear evolutions of the limiting profiles (which exist due to the fact that their energies are strictly below Ecrit ) can be superimposed to form a global nonlinear evolution, contradicting the choice of the sequence un . The fact that ` D 1 allows one to rescale and re-translate the unique limiting profile to a fixed position in phase space (meaning spatial position and spatial frequency) which then gives the desired nonlinear evolution uC . The compactness follows by the same logic: assuming that it does not hold, one then obtains a sequence uC .
; tn / evaluated at times tn 2 I  converging to an endpoint of I  such that for n ¤ n0 , the rescaled and translated versions of uC .
; tn / and uC .
; tn0 / remain at a minimal positive distance from each other in the energy norm. Again one applies Bahouri–Gérard and finds that ` D 1 by the choice of Ecrit and perturbation theory. This gives the desired contradiction. The compactness property is of course crucial; indeed, for illustrative purposes suppose that uC is of the form 
 1 uC .t; x/ D .t / 2 U .t / x x.t / where .t/ ! 1 as t ! 1, say. Then uC blows up at time t D 1 (in the sense that the energy concentrates at the tip of a cone) and .t /

1 2

uC .t /

1


x C x.t / D U.x/

is compact for 0  t < 1. Returning to the Kenig–Merle argument, the logic is now to show that uC acts in some sense like a blow-up solution, at least if I  is finite in one direction. The second half of the Kenig–Merle approach then consists of a rigidity argument which shows that a uC with the stated properties cannot exist. This is done mainly by means of the conservation laws, such as the Morawetz and energy identities. More precisely, the case where I  is finite at one end is reduced to the self-similar blow-up scenario. This, however, is excluded by reducing to the stationary case and an elliptic analysis which proves that the solution would have to vanish. If I  is infinite, one basically faces the possibility of stationary solutions which are again shown not to exist.

18

1 Introduction and overview

For the case of wave maps, we follow the same strategy. More precisely, our adaptation of the Bahouri–Gérard decomposition to wave maps into H2 leads to a critical wave map with the desired compactness properties. In the course of our proof, it will be convenient to project the wave map onto a compact Riemann surface S (so that we can avail ourselves of the extrinsic formulation of the wave map equation). However, it will be important to work simultaneously with this object as well as the lifted one which takes its values in H2 (since it is for the latter that we have a meaningful well-posedness theory for maps with energy data). The difference from [14] lies mainly with the rigidity part. In fact, in our context the conservation laws are by themselves not sufficient to yield a contradiction. This is natural, since the geometry of the target will need to play a crucial role. As indicated above, the two scenarios that are lead to a contradiction are the self-similar blow-up supported inside of a light-cone and the stationary weak wave map, which is of course a weakly harmonic map (which cannot exist since the target S is compact with negative curvature). The former is handled as follows: in self-similar coordinates, one obtains a harmonic map defined on the disk with the hyperbolic metric and with finite energy (the stationarity is derived as in [14]). Moreover, there is the added twist that one controls the behavior of this map at the boundary in the trace sense (in fact, one shows that this trace is constant). Therefore, one can apply the boundary regularity version of Helein’s theorem which was obtained by Qing [37]. Lemaire’s theorem [27] then yields the constancy of the harmonic map, whence the contradiction (for a version of this argument under the a priori assumption of regularity all the way to the boundary see Shatah–Struwe [42]).

1.6

An overview of the book

The book is essentially divided into two parts: The modified Bahouri–Gérard method is carried out in its entirety starting with Chapter 2, and ending with Chapter 9. Indeed, all that precedes Chapter 9 leads to this section, which constitutes the core of this book. The Kenig–Merle method adapted to Wave Maps is then performed in the much shorter Chapter 10. We commence by describing in detail the contents of Chapter 2 to Chapter 9.

1.6 An overview of the book

19

1.6.1 Preparations for the Bahouri–Gérard process As explained above, we describe admissible wave maps u W R2C1 ! H2 mostly in terms of the associated Coulomb derivative components ˛ . Our goals then are to (1) Develop a suitable functional framework, in particular a space-time norm k kS.R2C1 / , together with time-localized versions k kS.ŒI R2 / for closed time intervals I , which have the property that lim sup k kS.I R2 / < 1 I IQ

for some open interval IQ implies that the underlying wave map u can be extended smoothly and admissibly beyond any endpoint of IQ, provided such exists. (2) Establish an a priori bound of the form k kS.I R2 /  C.E/ for some function C W RC ! RC of the energy E. This latter step will be accomplished by the Bahouri–Gérard procedure, arguing by contradiction. We first describe (1) above in more detail: in Chapter 2, we introduce the norms k
kS Œk , k
kN Œk , k 2 Z, which are used to control the frequency localized components of and the nonlinear source terms, respectively. The norm k
kS is then obtained by square summation over all frequency blocks. The basic paradigm for establishing estimates on then is to formulate a wave equation 

DF

or more accurately typically in frequency localized form P0

D P0 F;

and to establish bounds for kP0 F kN Œ0 which may then be fed into an energy inequality, see Section 2.3, which establishes the link between the S and N -spaces. In order to be able to estimate the nonlinear source terms F , we need to manipulate the right-hand side of (1.14), making extensive use of (1.15). The precise description of the actual nonlinear source terms that we will use for F is actually rather involved, and given in Chapter 3. In order to estimate the collection of trilinear as well as higher order terms, we carefully develop the necessary estimates in Chapters 4, 5, as well as 6. We note that the estimates in [23], while similar,

20

1 Introduction and overview

are not quite strong enough for our purposes, since we need to gain in the largest frequency in case of high-high cascades. This requires us to subtly modify the spaces by comparison to loc. cit. Moreover, the fact that we manage here to build in sharp Strichartz estimates allows us to replace several arguments in [23] by more natural ones, and we opted to make our present account as self-contained as possible. With the null-form estimates from Chapters 4, 5, 6 in hand, we establish the role of k
kS as a “regularity controlling” device in the sense of (1) above in Chapter 7, see Proposition 7.2. The proof of this reveals a somewhat unfortunate feature of our present setup, namely the fact that working at the level of the differentiated wave map system produces sometimes too many time derivatives, which forces us to use somewhat delicate “randomization” of times arguments. In particular, in the proof of all a priori estimates, we need to distinguish between a “small time” case (typically called Case 1) and a “long time” Case 2, by reference to a fixed frequency scale. In the short time case, one works exclusively in terms of the div-curl system, while in the long-time case, the wave equations start to be essential. Chapter 7 furthermore explains the well-posedness theory at the level of the ˛ , see the most crucial Proposition 7.11. We do not prove this proposition in Chapter 7, as it follows as a byproduct of the core perturbative Proposition 9.12 in Chapter 9. Proposition 7.11 and the technically difficult but fundamental Lemma 7.10 allow us to define the “Coulomb wave maps propagation” for a tuple 2 ˛ , ˛ D 0; 1; 2 which are only L functions at time t D 0, provided the latter are the L2 -limits of the Coulomb components of admissible maps. Indeed, this concept of propagation is independent of the approximating sequence chosen and satisfies the necessary continuity properties. We also formulate the concept of a “wave map at infinity” at the level of the Coulomb components, see Proposition 7.15 and the following Corollary 7.16. Again the proofs of these results will follow as a byproduct of the fundamental Proposition 9.12 and Proposition 9.30 in the core Chapter 9. In Chapter 8, we develop some auxiliary technical tools from harmonic analysis which will allow us to implement the first stage of the Bahouri–Gérard process, namely crystallizing frequency atoms from an “essentially singular” sequence of admissible wave maps. These tools are derived from the embedding 1 BP 2;1 .R2 / ! BMO as well as weighted (relative to Ap ) Coifman–Rochberg– Weiss commutator bounds. As mentioned before, Chapter 9 represents the core of this book. In Section 9.2, starting with an essentially singular sequence un of admissible wave maps with deteriorating bounds, i.e., k ˛n kS ! 1 as n ! 1 but with the

1.6 An overview of the book

21

crucial criticality condition limn!1 E.un / D Ecrit , we show that the derivative components ˛n may be decomposed as a sum ˛n

D

A X

˛na C w˛nA

aD1

where the ˛na are derivative components of admissible wave maps which have frequency supports “drifting apart” as n ! 1, while the error w˛nA satisfies lim sup kw˛nA kBP 0 n!1

2;1

< ı;

provided A  A0 .ı/ is large enough. In Section 9.3, we then select a number of “principal” frequency atoms  na , a D 1; 2; : : : ; A0 , as well as a (potentially very large) collection of “small atoms”  na , a D A0 C 1; : : : ; A. We order these atoms by the frequency scale around which they are supported starting with those of the lowest frequency. The idea now is as follows: under the assumption that there are at least two frequency atoms, or else in case of only one frequency atom that it has energy < Ecrit , we want to obtain a contradiction to the essential criticality of the underlying sequence un . To achieve this, we define in Section 9.3 sequences of approximating wave maps, which are essentially obtained by carefully truncating the initial data sequence  na in frequency space. In Section 9.4, we establish an a priori bound for the lowest frequency approximating map which comprises all the minimum frequency small atoms as well as the component of the small Besov error of smallest frequency, see Proposition 9.9. The proof of this follows again by truncating the data suitably in frequency space, and applying an inductive procedure to a sequence of approximating wave maps. This hinges crucially on the core perturbative result Proposition 9.12, which plays a fundamental role in this book. The main technical difficulty encountered in the proof of the latter comes from the issue of divisibility: let us be given a schematically written expression @ A which is linear in the perturbation (so that we cannot perform a bootstrap argument based solely on the smallness on  itself), while A denotes some null-form depending on a priori controlled components . “Divisibility” means the property that upon suitably truncating time into finitely many intervals Ij whose number only depends on k kS , one may bound the expression by k@ A kN.Ij R2 /  kkS :

22

1 Introduction and overview

In other words, by shrinking the time interval, we ensure that we can iterate the term away. While this would be straightforward provided we had an estimate for kA kL1 L1 (which is possible in space dimensions n  4), in our setting, t x the spaces are much too weak and complicated. Our way out of this impasse is to build those terms for which we have no obvious divisibility into the linear operator, and thereby form a new operator A  WD  C 2i @ A with a magnetic potential term. Fortunately, it turns out that if A is supported at much lower frequencies than  (which is precisely the case where divisibility fails), one can establish an approximate energy conservation result, which in particular gives a priori control over a certain constituent of k
kS . With this in hand, one can complete the bootstrap argument, and obtain full control over kkS . Having established control over the lowest-frequency “essentially non-atomic” approximating wave map in Section 9.4, we face the task of “adding the first large atomic component”,  n1 . It is here that we have to depart crucially from the original method of Bahouri–Gérard: instead of studying the free wave evolution of the data, we extract concentration cores by applying the “twisted” covariant evolution associated with An u D 0; which is essentially defined as above. The key property that makes everything work is an almost exact energy conservation property associated with its wave flow. This is a rather delicate point, and uses the Hamiltonian structure of the covariant wave flow. It then requires a fair amount of work to show that the profile decomposition at time t D 0 in terms of covariant free waves is “geometric”, in the sense that the concentration profiles can indeed by approximated by the Coulomb components of admissible maps, up to a constant phase shift, see Proposition 9.24. Finally, in Proposition 9.30 we show that we may evolve the data including the first large frequency atom, provided all concentration cores have energy strictly less than Ecrit . As most of the work has been done at this point, adding on the remaining frequency atoms in Section 9.9 does not provide any new difficulties, and can be done by the methods of the preceding sections. In conjunction with the results of Chapter 7, we can then infer that given an essentially singular sequence of wave maps un , we may select a subsequence of them whose Coulomb components ˛n , up to re-scalings and translations, converge to a limiting object ˛1 .t; x/, which is well-defined on some interval I R2

1.6 An overview of the book

23

where I is either a finite time interval or (semi)-infinite, and the limit of the Coulomb components of admissible maps there. Moreover, most crucially for the sequel, ˛1 .t; x/ satisfies a remarkable compactness property, see Proposition 9.36. This sets the stage for the method of Kenig–Merle, which we adapt to the context of wave maps in Chapter 10.

2

The spaces S Œk and N Œk

Chapters 2–5 develop the functional framework needed to prove the energy and dispersive estimates required by the wave map system (1.12)–(1.14). The Banach spaces which appear in this context were introduced by Tataru [63], but were specified in this form by Tao [58], and developed further by Krieger [23]. We will largely follow the latter reference although there is much overlap with [58]. We emphasize that this section is self-contained. The spatial dimension is two throughout.

2.1

Preliminaries

As usual, Pk denotes a Littlewood–Paley projection1 to frequencies of size 2k . More precisely, let m0 be a nonnegative smooth, even, bump function supported in jj < 4 and set m./ WD m0 ./ m0 .2/. Then X m.2k / D 1 8  2 R2 n f0g k2Z

b

and Pk f ./ WD m.2 k /fO./. In the sequel, we shall call a function f adapted to k, provided its Fourier transform is supported at frequency  2k . The operator Qj projects to modulation 2j , i.e.,
Qj .;  / WD m 2 j .j j jj/ b .; /

b

with O
referring to the space-time Fourier transform. Similarly,
Qj˙ .;  / WD m 2 j .j j jj/ Œ˙ >0b .; /:

1

s;p;q Then the relevant XP k spaces here are defined as X q kkXP s;p;q WD 2sk 2jpq kPk Qj k 2 k

1

j

L t L2x

 q1

:

Strictly speaking, these are not true projections since Pk2 ¤ Pk , but we shall nevertheless follow the customary abuse of language of referring to them as projections. The same applies to smooth localizers to other regions in Fourier space.

26

2 The spaces S Œk and N Œk A B P6 P5 P4 P3 P2 P1 D C

Figure 2.1. Rectangles

If Pk  D , then kkL1 L2x . kk t

1 ;1 0; 2

XP k

as well as kkL1 . kk t;x

1 ;1 1; 2

XP k 1 2`

.

In what follows, C` is a collection of caps   S 1 of size C and finite overlap (uniformly bounded in ` and with CPsome large absolute constant). There is an associated smooth partition of unity 2C` a .!/ D 1 for all ! 2 S 1 , as
well as projections P f ./ WD a b  fO./ where b  WD  . By construction,

b

Pk; WD Pk ı P is a projection to the “rectangle”

jj

˚ Rk; WD jj  2k ; b 2

(2.1)

in Fourier space. For space-time functions F we shall follow the convention that     O Œ >0 FO .;  / _ C a . / O Œ 1, it is well-known that XP s;b spaces do not suffice in the critical case s D 1. Following the aforementioned references, we now develop Tataru’s null-frame spaces which will provide sufficient control over the nonlinear interactions in the wave map system. For fixed2 ! 2 S 1 define p !˙ WD .1; ˙!/= 2; t! WD .t; x/
!C ; x! WD .t; x/ t! !C ; (2.5) 2

Henceforth, ! will always be a unit vector in the plane.

29

2.2 The null-frame spaces

which are the coordinates defined by a generator on the light-cone. Recall that a plane wave traveling in direction ! 2 S n 1 is a function of the form h.x
! C t / (and h sufficiently smooth). We write a free wave  as a superposition of such plane waves: with   S 1 and Pk; the projection to jj  2k and b  2  as defined above, Z e i.t jjCx
/ Pk; f ./ d  Pk; .t; x/ D Œjj2k  Z Z D e i r.x
!Ct / fO.r!/ r drd! k  Œr2  Z D (2.6) k;! .t C x
!/ d!;

1



where

Z k;! .s/

The argument of

k;!

WD

in (2.6) is

p

e i rs fO.r!/r dr:

Œr2k 

2 t! , whence

Z 

k

k;! kL2t L1 ! x!

1

k

d! . jj 2 2 2 kPk; f k2 :

(2.7)

We now define the following pair of norms3 kGkNFAŒ WD inf dist.!; / !622

kkPWAŒ WD inf kkL2

1 t! Lx!

!2

1

kGkL1

2 t! Lx!

(2.8) (2.9)

which are well-defined for general Schwartz functions. The notation here derives from null-frame and plane wave, respectively. The quantities defined in (2.8) and (2.9) are not norms – in fact, not even pseudo-norms – because they violate the triangle inequality due to the infimum. This indicates that we should be using (2.8) and (2.9) to define atomic Banach spaces (which is why we appended “A” in the norms above). First, recall from (2.6) that Z p Pk1 ; .t; x/ D k;! . 2 t! / d!: 

3

The dist.!; / 1 factor in the NFAŒ-norm arises because of a geometric property of the cone, see the proof of Lemma 2.4.

30

2 The spaces S Œk and N Œk

Then (2.7) suggests that we define Z kPk1 ; kPWŒ WD k k;! kL2

Z

1 t! Lx!



d! D



k

k;! kPWAŒ d!:

In other words, PWŒ is the completion of the space of all functions  which can be written in the form X X D j j ; jj j < 1; k j kPWAŒ  1 (2.10) j

j

where j 2 C and j are Schwartz P functions, say. The norm of any such  in PWŒ is then simply the infimum of j jj j over all representations as in (2.10). By Hölder’s inequality we now obtain the simple but crucial estimate kF kNFAŒ  dist.;  0 /

1

kkPWAŒ 0  kF kL2 L2x ; t

provided  is a PWAŒ-atom. This suggests that we also define NFŒ as the atomic space obtained from NFAŒ as usual: the atoms of NFŒ are functions  for which there exists ! 62 2 such that kkL1 L2x  dist.!; /. The previous t! ! estimate then implies the bound kF kNFŒ  dist.;  0 /

1

kkPWŒ 0  kF kL2 L2x :

(2.11)

t

The dual space NFŒ is characterized by the norm kkNFŒ D sup dist.!; /kkL1 L2x < 1: t!

!622

!

We now turn to defining the spaces which hold the wave maps. O  f 2 R2 W jj  2k g. Definition 2.3. Let  be a Schwarz function with supp./ Henceforth, we shall call such a  adapted to k. Define kkS Œk; WD kkL1 L2x C jj

1 2

t

2

k 2

kkPWŒ C kkNF Œ

(2.12)

kkS Œk WD kkL1 L2x C kQkC2 k P 0; 1 ;1 C kQk k P 1 C";1 ";2 (2.13) t X 2 X 2   21 X 3k . 12 "/` 2 4 C sup sup 2 2 kQ<j Pc kL4 L1 (2.14) j 2Z `0

C sup sup

c2Dk;`

sup

˙ ` 100 `m0

 X

X

t

x

˙ kPR QkC2` k2S Œk;

 21

:

2C` R2Rk;˙;m

(2.15) Here P and PR are as above, and " > 0 is a small number (" D

1 10

is sufficient).

31

2.2 The null-frame spaces 1

k

The factors jj 2 2 2 in (2.12) are from (2.7). By inspection, the norm of S Œk;  is translation invariant, and kkS Œk; : kf kS Œk;  kf kL1 tx

(2.16)

One has the following scaling property: kkS Œk D k.
/kS ŒkCm ;

 D 2m ; m 2 Z:

(2.17)

It will be technically convenient to allow noninteger k in Definition 2.3. The only change required for this purpose is to allow j; `; m 2 R in (2.14) and (2.15). In that case one has kkS Œk D k.
/kS ŒkClog2  ;

8 > 0:

(2.18)

Later we will need to address the question whether kPk kS ŒkCh is continuous in h near h DP 0 for a fixed Schwartz function . Henceforth, we shall use the operator I WD k2Z Pk Qk and I c WD 1 I (we will also use QkCC instead of Qk ). Moreover, we refer to functions which belong to the range of I as “hyperbolic” and to those in the range of I c as “elliptic”. Since kQk Pk k

1 ;1 0; 2

XP k

. kQk Pk k

XP k

1 C";1 ";2 2

;

2 one concludes that the energy norm L1 t Lx in (2.13) as well as the Strichartz norm of (2.14) are controlled by the final norm of (2.13) for the case of elliptic functions (for the Strichartz norm use Lemma 2.2). We first verify that temporally truncated free waves lie in these spaces (with an embedding constant that does not depend on the length of the truncation interval).

Lemma 2.4. Let   S 1 be arbitrary. Then kkS Œk; . kPk; k

(2.19)

1

XP 0; 2 ;1

as well as kPk Qk kS Œk . kPk Qk k

1

XP 0; 2 ;1

:

(2.20)

In particular, if f is adapted to k, then k.t =T /e i t

p




f kS Œk  C kf kL2

with a constant that depends on the Schwartz function  but not on T  2

(2.21) k.

32

2 The spaces S Œk and N Œk

1 Proof. We assume that  is an XP 0; 2 ;1 -atom with P0;  D . Then from Plancherel’s theorem and Minkowski’s and Hölder’s inequalities, j

O 2 1 . 2 2 kk 2 2 kkL1 L2x . kk L L L Lx t

t



kkL2

1 t! L x !

1 2

1

O 2 . .2j jj/ kk L

2 ! L!

D .2j jj/ 2 kkL2 L2x t

j 2

kkL1 L2x . t!

!

2 kkL2 L2 : ! ! dist.; !/

(2.22)

In the final estimate (2.22) we used that ^.`! ; T! 0 /  ^.!; ! 0 /2 where `! is the line oriented along the generator parallel to .1; !/ and T! 0 is the tangent plane to the cone which touches the cone along the generator `! 0 . To establish (2.20) we begin with sup

 X

` 100

X

˙ kPR Q2` k2 0; 1 ;1 XP 0

2C` R2Rk;˙;

 12

. kk

0; 1 2 ;1

XP 0

2

which is obvious from orthogonality of the P0;˙ . In view of (2.19), this bound yields the square function in (2.15). The energy is controlled via the embedding kkL1 L2 . kk P 0; 1 ;1 , whereas the Strichartz component of S Œk is controlled X 2 by Lemma 2.2. Finally, the statement concerning the free wave reduces to the case k D 0 for which we need to verify the bound X


j 2 2 kT O T j ˙ jk m 2

j


j ˙ jjj fO./kL2 L2 C 

j 2Z

C

X


22j kT O T j ˙ jjj m 2

j

 21
2 j ˙ jjj fO./kL . kf k2 2 2 L 

j 2Z



which are both clear provided T  1 due to the rapid decay of . O



Naturally, S Œk contains more general functions than just free waves. One way of obtaining such functions is to take  D  1 F , in other words from the Duhamel formula. We will study this in much greater generality in the context of the energy estimate below, but for now we take F to be a Schwartz function. Remark 2.5. The bounded function  defined via its Fourier transform b .; / D 1 ./2 .jj

j j/.jj

jj/

1

33

2.2 The null-frame spaces

belongs to SŒ0 but is not a truncated free wave. Here 1 is a smooth cut-off to jj  1, and 2 .u/ is a smooth cut-off to juj < 1=10. We leave it to the reader to construct other functions which lie in S Œ0 and which are not (truncated) free waves. The following basic estimates will be used repeatedly: ı if  is adapted to k, then .j k/. 12 "/

kQj kL2 L2 . min.2 kQj kL2 L1 . 2k 2

j

k 4

^0

min.2

; 1/2 .j

j 2

k/. 12

kkS Œk ; "/

; 1/2

(2.23) j 2

kkS Œk :

(2.24)

This follows from the XP s;b;q components of the S Œk-norms, as well as the improved Bernstein’s inequality of Lemma 2.1. ı The duality between NFŒ and NF Œ implies jh; F ij . kkS Œk; kF kNFŒ :

(2.25)

In what follows,  WD sign. /b , and for any ! 2 S 1 , ˘! denotes the orthogonal ? projection onto NP.!/ WD ! (the null-plane of !). Lemma 2.6. The projection ˘! satisfies the following properties: ı Let F  C` be a collection of disjoint caps. Suppose that ! 2 S 1 satisfies dist.!; / 2 Œ˛; 2˛ for any  2 F where ˛ > 2` is arbitrary but fixed. Define4 ˚ T;˛ WD .; / W jj  1;  2 ; jjj j jj . ˛2` : (2.26) Then f˘! .T;˛ /g2F  NP.!/ have finite overlap, i.e., X ˘! .T;˛ /  C 2F

where C is some absolute constant. ı Let ˚ S WD .˙jj; / W  2 R2 ; b  2 ˙ be a sector on the light-cone where   S 1 is any cap. Furthermore, let ! 62 2 and e S WD ˘! .S/. Then on e S the Jacobian @@! satisfies ˇ @ ˇ ˇ ˇ (2.27) ˇ ˇ  d.!; / 2 : @! 4

An important detail here is that these dimensions deviate from the usual wave-packets of dimension 1  2`  22` .

34

2 The spaces S Œk and N Œk X

Y

Z

C

D0

D

X0 Y0

Z0

C0

A0

P0 Q0 R0

B0 A

P Q R

B Figure 2.2. The projected sectors

The same holds on ˘! .Sa / where ˚
Sa WD ˙ jj C a;  W  2 R2 ; jj  1; b  2 ˙ ; provided a is fixed with jaj  jjd.!; /. Proof. Denote ˚ S WD s.1; ! 0 / C .1; ! 0 / W ! 0 2 ; 1  s  2; jj < h where h will be determined. Then ˚ ˘! .S /g2F D fs vE C w E W ! 0 2 ; 1  s  2; jj < h where vE WD vE.!; ! 0 / D .1; ! 0 /

.1; !/;

w E WD .1; ! 0 /

.1; !/;

with  D 12 .1C!
! 0 /,  D 12 .1 !
! 0 /. Recall that dist.!; /  dist.!;  0 / DW ˛ where  2 F is arbitrary. Moreover, diam./  2` DW ˇ. One checks that q p jE v j D 2.1 2 /    ˛:

35

2.2 The null-frame spaces

Furthermore, @E v WD .0; ! 0 ? / 21 !
! 0 ? .1; !/ denotes the derivative @ 0 vE where 0 we have written ! 0 D e i . Then j@E v j  1 and vE ^ @E v D .; ! 0

?

! C !/ ^ .0; ! 0 /

1 ? !
! 0 .1; ! 0 / ^ .1; !/ 2

satisfies jE v ^ @E v j  ˛ 2 . In conjunction with jE v j  ˛ this implies that j^.E v ; @E v /j  ˛. Since ˇ ˇ ˇvE.!; ! 0 / vE.!; ! 00 /ˇ & j! 0 ! 00 j 8 ! 0 ; ! 00 2  0 ; it follows that
dist  .!; ! 0 /;  .!; ! 00 / & ˛j! 0

! 00 j

(2.28)

where ˚  .!; ! 0 / WD s vE.!; ! 0 / W 1  s  2 : Therefore, one needs to take h D ˛ˇ to insure the property of finite overlap of the projections. This is optimal, since one can check that vE and w E always satisfy j cos.^.E v ; w/j E  21 . In Figure 2.2 the left-hand side depicts four sectors as they would appear on the light-cone, whereas the right-hand side is the projected configuration in NP.!/ with A0 WD ˘! .A/ etc. Note that the segments A0 B 0 as well as A0 P 0 , P 0 Q0 , Q0 R0 , R0 B 0 have lengths comparable to the corresponding ones on the left, i.e., AB etc., whereas the lengths of A0 D 0 , B 0 C 0 are those of AD and BC contracted by the factor ˛. Finally, we have shown that ^.A0 B 0 C 0 /  ˛ (and similarly for the angles at the points P 0 , Q0 , R0 ) so that the height of the parallelogram A0 P 0 X 0 D 0 is proportional to ˛ times the length of A0 P 0 , see (2.28). The second statement of the lemma follows from the consideration of the preceding paragraph.  As a consequence of Lemma 2.6, we now show that the square function in (2.15) can always be refined in terms of the angle. 0 0 Lemma 2.7. S Let F 0 C` be a collection of disjoint caps and let  2 C` be a cap with F    . Suppose further that for every  2 F there is a Schwartz function  adapted to k 2 Z and which is supported on n o 0 T;k WD  WD sign. /b  2 ; jjj j jj . 2`C` Ck

with some k 2 Z. Then

X



S Œk; 0 

2F

with some absolute constant C .

C

X 2F

k k2S Œk;

 21

36

2 The spaces S Œk and N Œk

Proof. First, one may take k D 0 and  > 0 (the latter by conjugation symmetry). The L1 L2 -component of (2.12) satisfies the required property due to orthogonality, whereas the PWŒ-component is reduced to Cauchy–Schwarz (via 1 2 the jj 2 -factor). For the final NF Œ (i.e., L1 t! Lx! )-component one exploits orthogonality relative to x! via the preceding lemma. Here ! 2 S 1 n .2 0 / is arbitrary but fixed.  Later we will prove bi- and trilinear estimates involving S and N space. The following bilinear bounds will be a basic ingredient in that context. Lemma 2.8. One has the estimates 1

k0

kF kNFŒ

j 0 j 2 2 2 . kkS Œk 0 ; 0  kF kL2 L2x ; t dist.;  0 /

k kL2 L2x

jj 2 2 2 . kkS Œk; k kS Œk 0 ; 0  : dist.;  0 /

1

t

(2.29)

k

(2.30)

For the final two bounds we require that 2 \ 2 0 D ;. Proof. The second one follows from the definition of the spaces, whereas (2.30) follows from (2.29) and the duality bound (2.25).  Note that both of these estimates have a dispersive character, as they involve space-time integrals. By applying ideas from the energy estimate, we will improve on (2.30) in the high-high case, see Lemma 4.5. Next, we define the spaces which will hold the nonlinearities. These spaces differ from those used for example in [23] as far as the “elliptic norm” k
k 1 C"; 1 ";2 is concerned. Here the extra XP k

2

" ensures that we achieve exponential gains in the maximal frequencies for certain high-high-low interactions. Definition 2.9. N Œk is generated by the following four types of atoms: With F being k-admissible, either ı kF kL1 L2x  2k , t ı FO is supported on jjj jjj  2j  2k and kF k 1; 1 ;1  1, XP k

ı F D Qk F , kF k

XP k

1 C"; 1 ";2 2

2

 1 where " > 0 is as in the S Œk spaces,

ı F P is the sum of wave-packets F : there exists `  100 such that F D c c 2C` F with all supp.F / supported on either  > 0 or  < 0, with F supported on jj  2k , jjj j jj  C 1 2kC2` ,  WD sign. /b  2  and to

37

2.2 The null-frame spaces

that the bound X

kF k2NFŒ

 12

 2k



holds. We refer to these types as energy, XP s;b;q , and wave-packet atoms, respectively. In what follows, we refer to functions  adapted to some k 2 Z as “elliptic” iff Pk Qk  D , whereas those satisfying Pk Qk  D  as “hyperbolic”. This terminology has to do with the behavior of the wave operator  in these respective regimes. We now record a fundamental duality property of N Œk. Lemma 2.10. For any  2 S Œk and F 2 N Œk with F D Pk Qk F jh; F ij . 2k kkS Œk kF kN Œk ; kF k

XP k

1 ;1 2

1;

. kF kN Œk . kF k

XP k

1;

(2.31) 1 ;1 2

:

(2.32)

Proof. The duality relation (2.31) is proved by taking F to be an atom; for the wave-packet atom use (2.25). By definition of N Œk, one has kF kN Œk . kF k 1; 1 ;1 . For the left-hand bound in (2.32) use (2.19) and (2.31).  XP k

2

As an application of the geometric considerations of Lemma 2.6 we now show that refining a wave-packet atom yields another wave-packet atom. Lemma 2.11. Let F D Then

sup

sup

`0 ` `0 j 0

P

X X

2C`

F be a wave-packet atom as in Definition 2.9.

X

  0 2C`0 R2Rk; 0 ;j  0 

kPR P 0 Q 0, ˇ ˇ ˇkPk kS Œk.Œ T ";T C"R2 / kPk kS Œk.Œ T;T R2 / ˇ ˇ ˇ D ˇkPkC  kS ŒkC.Œ T;T R2 / kPk kS Œk.Œ T;T R2 / ˇ ˇ ˇ  ˇkPkC kS ŒkC.Œ T;T R2 / kPk kS Œk.Œ T;T R2 / ˇ C kPkC .

 /kS ŒkC.Œ T;T R2 / :

By the energy estimate, kPkC .

 /kS ŒkC.Œ T;T R2 /

. kPkC . . k.

 /kS ŒkC  /Œ0kL2 HP

C kPkC . . k.

 /kN ŒkC

 /Œ0kL2 HP

C kPkC .

1

1

 /kL1 HP t

1 .R1C2 /

as  ! 1. By Corollary 2.27, lim kPkC

!1

 kS ŒkC.Œ T;T R2 /

D kPk kS Œk.Œ

T;T R2 /

!0

60

2 The spaces S Œk and N Œk

which implies that lim kPk kS Œk.Œ

"!0C

T ";T C"R2 /

D kPk kS Œk.Œ

T;T R2 /

as claimed. The case of T D 0 follows directly from the energy estimate. The case of jjj
jjj is essentially the same.  We define localized N Œk-norms similarly, i.e., k kN Œk.Œ

T;T R2 /

WD

kPk Q kN Œk

inf

Q jŒ

T;T  D

jŒ

T;T 

for Schwartz functions. In particular, one has a localized version of (2.67) jjjjjjS Œk.Œ

T;T R2 /

WD kkL2

I IL2 .R2 /


C kk

N Œk.Œ T;T R2 / :

Furthermore, later we will also need localized norms on asymmetric time intervals Œ T 0 ; T  for which the results here of course continue to hold. Finally, in the perturbative steps to follow, we will need to piece together solutions of time-localized wave equations to solutions on larger time intervals. To justify this procedure we rely on the following lemma. Lemma 2.30. Let I  R be a closed interval, with a covering I D [jND1 Ij by closed intervals; assume that the Ij overlap at most two at a time, and that consecutive intervals have intersection with non-empty interior. Then if we are given k-adapted j with k such that

j jIj \Il

D

j kS Œk.Ij R2 / ` jIj \I` ,

 cj ;

j D 1; 2; : : : ; N;

then defining

k kS Œk.I R2 / .

N X j D1

k

via

jIj WD

j kS Œk.Ij R2 /



N X

j,

we have

cj

j D1

where the implied constant is universal (independent of the decomposition of I or N ). The same applies to the norms jjj
jjjS Œk.Ij R2 / . Proof. Chose a partition of unity fj g subordinate to the cover fIj g, such that suppj  Ij . We shall select the j in such fashion that jsuppj0 j  jIj \ Ik j, provided the latter is non-zero (which happens only for at most two other k). We

61

2.4 A stronger S Œk-norm, and time localizations

first deal with the k
kS Œk -norms. By assumption, we can find Schwartz extensions Qj of j , 8j , such that k Qj kS Œk.R2C1 /  2cj . We now define Q WD

N X

j Qj

j D1

P and verify the desired bound k Q kS Œk.R2C1 / . cj . For simplicity, consider a single interval half-infinite I1 with neighboring half-infinite I2 , and the corresponding expression 1 Q 1 C 2 Q 2 : Note that 01 C 02 D 0 on the overlap of the intervals. It is easy to see that the only potential difficulty in controlling k1 Q 1 C 2 Q 2 kS Œk comes from the “elliptic portion” of k
kS Œk , as we have introduced the cutoffs whose derivatives we do not a priori control. By scaling invariance, it suffices to consider k D 0. Hence consider now P0 Qj Œ1 Q 1 C 2 Q 2  for some j  1. We decompose this by applying a frequency trichotmoy P0 Qj Œ1 Q 1 C 2 Q 2  DP0 Qj ŒQŒj

Q 10;j C10 .1 / 1

C P0 Qj ŒQ<j

10 .1 / Q 1

Q 10;j C10 .2 / 2 

C QŒj

C Q<j

10 .2 / Q 2 

C P0 Qj ŒQ>j C10 .1 / Q 1 C Q>j C10 .2 / Q 2 : (2.69) We start by estimating the last line: We have

P0 Qj ŒQ>j C10 .1 / Q 1 C Q>j C10 .2 / Q 2  2 L

t;x



X

kP0 Qj ŒQr .1 /QŒr

5;rC5

Q 1 C Qr .2 /QŒr

5;rC5

Q 2 k

L2t;x

r>j C10

.

X

X

2

.1 /r

kQŒr

5;rC5

Q `k

`D1;2 r>j C10

From here we easily obtain

X

P0 Qj ŒQ>j C10 .1 / Q 1 C Q>j C10 .2 / Q 2 

XP 0

j >O.1/

.

XP 0

1 2 C;1 ;2

:

1 2 C;1 ;2

X `D1;2

k Q `k

XP 0

1 2 C;1 ;2

:

62

2 The spaces S Œk and N Œk

The second line in (2.69) is estimated similarly, and so we reduce to estimating   P0 Qj QŒj 10;j C10 .1 / Q 1 C QŒj 10;j C10 .2 / Q 2 : We may assume that F .1;2 / decay rapidly away from frequency scale 2R  1, say. Write

D P0 Qj ŒQŒj

C QŒj 10;j C10 .2 / Q 2  Q Q 10;j C10 @ t .1 / 1 C QŒj 10;j C10 @ t .2 / 2 

C P0 Qj ŒQŒj

10;j C10 .1 /@ t

@ t P0 Qj ŒQŒj

Q 10;j C10 .1 / 1

Q 1 C QŒj

10;j C10 .2 /@ t

(2.70)

Q 2 :

We start by estimating the second row: We will consider the case j D R C O.1/, since in the other cases one obtains additional exponential gains from the frequency localization of 1;2 . But then we can write QŒj

10;j C10 @ t .1 /

D Q 1 QŒj

10;j C10 @ t .1 /

N

C OL2 .R t

1 2

where Q localizes to an interval around supp01 of length R for 2 . By picking R large enough, we may assume that

/

, say, and similarly

Q 1 Q 1 D Q 2 Q 2 : Thus we obtain  P0 Qj QŒj 10;j C10 @ t .1 / Q 1 C QŒj

Q 10;j C10 @ t .2 / 2 D



X

OL2 .R

N

t

/ Q`

`D1;2

and from here we infer

P0 Qj ŒQŒj 10;j C10 @ t .1 / Q 1 C QŒj

Q 10;j C10 @ t .2 / 2 



D O.R

N0

/

XP 0

X

1 2 C;1 ;2

k Q ` kS Œ0.R2C1 /

`D1;2

where we recall the assumption j D R C O.1/. The remaining cases j  R, j  R are lead to a similar bound. Next, consider the last line of (2.70); here we write P0 Qj ŒQŒj D P0 Qj ŒQŒj

10;j C10 .1 /@ t

Q 1 C QŒj

10;j C10 .1 /@ t Q<j C20

10;j C10 .2 /@ t

Q 1 CQŒj

Q 2

10;j C10 .2 /@ t Q<j C20

Q 2 : (2.71)

2.5 Solving the inhomogeneous wave equation in the Coulomb gauge

63

But this we can estimate by



P0 Qj QŒj 10;j C10 .1 /@ t Q<j C20 Q 1 C QŒj

10;j C10 .2 /@ t Q<j C20

.

X

2

 Q2

.1 /j

XP 0

1 C;1 ;2 2

k@ t Q<j C20 Q ` kL2 : (2.72) t;x

`D1;2

One can now perform the square summation over j > O.1/, and as a result obtains the upper bound X . k Q ` k 1 C;1 2 XP 0

`D1;2

2

where the implied constant is universal. The argument for controlling the jjj
jjjS Œk -norm is similar. One uses 

N X

j Qj D

j D1

N X

j  Qj ;

j D1



as well as Lemma 2.19.

Remark 2.31. In the sequel, we shall use the preceding lemma freely without an explicit reference.

2.5

Solving the inhomogeneous wave equation in the Coulomb gauge

Consider the wave equation (1.14), i.e.,  ˛ D F˛ . Here F˛ is a nonlinear expression in , but we will not pay attention to this now. In the sequel, we shall require a priori bounds on ˛ in the S Œk-space. To do so, we reduce matters to the energy estimates of Section 2.3 as follows: Writing (suppressing ˛ for simplicity)  one concludes (with both to frequency 1),

D IF C I c F

and F global space-time Schwartz functions adapted Z

.t/ D S.t

t0 /.I /Œt0  C

t

U.t t0

s/IF .s/ ds C 

1 c

I F

(2.73)

64

2 The spaces S Œk and N Œk

where the final term is obtained by division by the symbol7 of , and the first two terms represent the free wave and the Duhamel integral, respectively. Note that the first term here implicitly depends on all of , not just Œt0 , and so in order to actually obtain a bound on k kS , one needs to implement a bootstrap argument. Specifically, assume that we a priori have a bound on k jŒ

T0 ;T0  kS

P for some T0 > 0. Also, assume that we define I D k2Z Pk Qj k j=2

.2 2

2

. 2k 2j=2 2 . 2k 2

j

k 4

C

j

k 4

j

k 4

^0

^0 ^0




j. 12 "/




jjjjjjS Œk1  jjj jjjS Œk2  :

/ L2 L2 t

x

kPk Qj .Q>j kQ>j

min.1; 2

C

C kL2 L2 k . 12

"/j




/kL2 L1x t

kL1 L2

/kkS Œk1  k kS Œk2 

which is admissible. So it suffices to estimate Pk Qj .Qj C 
Qj C /. As usual, we perform a wave-packet decomposition by means of Lemma 4.1. Note

91

4.2 An algebra estimate for S Œk

that (4.1) holds here. We begin with (4.3) where we choose r 0 WD 2k . Thus, k < C and j D O.1/, and in view of (2.30)

Pk Qj .Qj C  C
Qj C C / 2 2 L t Lx X kP Qj C  C
P  Qj C C kL2 L2x . t

2Ck

X

.

1

jj 2 kP Qj

C

C

kS Œk1 ; kP

 Qj C

C

kS Œk2 ;



2Ck

. 2k=2C kkS Œk1  k kS Œk2  where we invoked Lemma 2.12 in the final step. The same estimate applies to  and . It therefore suffices to assume that j  k C O.1/; but then Lemma 4.7 applies. Next, we consider the low-high case k D k2 C O.1/ D 0, k1 < C . We need to prove that

j j 2 2 P0 Qj . / L2 L2 . 2k1 2 t

k1 4 ^0

x

min.1; 2

j. 21 "/

/kkS Œk1  k kS Œ0 :

In view of Lemma 4.7 we can assume that j  k1 . From the XP s;b;q components of the SŒk norm,

2j=2 P0 Qj .Qj C 
/ L2 L2 . 2j=2 kQj C kL2 L1 k kL1 L2 t

. 2k1 2

j

k1 4 ^0

min.1; 2

. 12

"/j

Finally, it remains to bound

2j=2 P0 Qj .Q<j

x

/kkS Œk1  k kS Œk2  :

C


Qj

C

/ L2 L2 t

x

which will be done using the usual angular decomposition. Lemma 4.1, and provided j  C , and with ` D m 2k1 ^ 0,

2j=2 P0 Qj .Q<j X . 2j=2

C


Qj X

mj C

; 0 2C` ; 0

X

X

C

In fact, from

/ L2 L2 t

x

kP0 Qj .Pk1 ; Q<j

C


Pk2 ; 0 Qm /kL2 L2x t

(4.38) .

mj C

2j=2

; 0 2C` ; 0

kPk1 ; Q<j

C kL1 L1 kPk2 ; 0 Qm

kL2 L2x t

92

4 Bilinear estimates involving S and N spaces

X

.

j

2 k1 2

m k1 ^0 4

kPk1 ; kL1 L2 kPk2 ; 0 Qm kL2 L2x t

; 0 2C` ; 0

mj C

. 2 k1 2

X

2j=2 k1 4 ^0

kkS Œk1  k kS Œk2 

where we used Corollary 2.16 in the final inequality. If j  C , then only m D j C O.1/ contributes to the sum in (4.38). The XP 0;1 ";2 component of the S Œk1  norm then leads to a gain of min.1; 2 . 2 "/j / and we are done. Corollary 4.12. Under the same conditions as in the previous lemma and provided k1  k2 one has

Pk .Qj C

X

.

2

` 2

2

` k 4

kPk Q` .F /kL2 L1x C 2 t

3k 2

kPk Qk .F /kL2 L2x t

k`>j C

.2

j

k 4

kkS Œk1  kF kN Œk2 

as desired. It therefore remains to consider X Pk Qj C .Q<j C 
F / D Pk Qj ˙

C .Q<j C 

˙


F ˙/

96

4 Bilinear estimates involving S and N spaces

where all four possibilities .CC/; .C /; . C/; . / are allowed on the righthand side. We first dispose of the contributions “opposing waves” as described by (4.3). This occurs only if k < C and j D O.1/, in fact, Pk Qj

C .Q<j C 

C


F C / D Pk Q

C j Ck2 
F kL1 L2 . kQ>j Ck2 kL2 L2 kF kL2 L1 .2

.j Ck2 /=2 k2

.2 2

j

kkS Œk1  2k2 2

k2 4 ^0

kkS Œk1  2

j 2

j

k2 4 ^0

kF kL2 L2x t

2

k2

kF kL2 L2x t

4.3 Bilinear estimates involving both S Œk1  and N Œk2  waves

99

k2

which is acceptable with a factor of 2 2 to spare. The reason for using Q>j Ck2 rather than Q>j will become clear momentarily. Next, kQj Ck2 
F kN Œk . kQj Ck2

C ŒQj Ck2 

C kQ<j Ck2 As usual, (4.46) is controlled in the XP

C ŒQj Ck2 

1 2 ;1

1;


F kN Œk

(4.46)


F kN Œk

(4.47)

norm whence

(4.46) . 2

.j Ck2 /=2

.2

.j Ck2 /=2

kQj Ck2 kL1 L2 kF kL2 L1

.2

.j Ck2 /=2

kQj Ck2 kL1 L2 2k2 2

kQj Ck2 
F kL2 L2x t

. 2 k2 2

j

k2 4 ^0

kkS Œk1  2

j 2

2

k2

j

k2 4 ^0

kF kL2 L2x t

kF kL2 L2x t

which is again acceptable. Finally, we perform a wave-packet decomposition on (4.47) via Lemma 4.1 in the imbalanced case and duality. Thus, one has X C Q<j Ck2 C .Qj Ck2 
F / D Pk; Q<j .Pk1 ; 0 Qj Ck2 
F / Ck2 C ; 0

where the sum runs over pairs of caps ;  0 of size C 1 2` with ` WD .j C k2 /=2 and dist.;  0 /  2` . Moreover, j  k2 C O.1/ since the only other possibility j D O.1/ allowed by (4.3) contributes a vanishing term (as does Q<j Ck2 C ). Therefore, with  0   denoting the admissible pairs, X X C

. Pk; Q<j Ck2 

.2

` 2

C

.Pk1

; 0

2  21 Qj Ck2 
F / N Œ

 0 

2j=2 2k2 kkS Œk1  kF kN Œk2 

. 2k2 2.j

k2 /=4

kkS Œk1  kF kN Œk2  

as desired.

There is the following general estimate that does not require (4.41) since we restrict ourselves to k  k1 C O.1/. Corollary 4.14. For  and F which are k1 and k2 -adapted, respectively, one has kPk .F /kN Œk . 2k1 ^k2 2

j

k^k1 ^k2 ^0 4

kkS Œk1  kF kN Œk2 

provided Pk2 Qj F D F and k D k1 _ k2 C O.1/.

(4.48)

100

4 Bilinear estimates involving S and N spaces



Proof. This is an immediate consequence of Lemma 4.13.

Another important technical variant of Lemma 4.13 has to do with an additional angular localization of the inputs. This will be important later in the trilinear section. Its statement is somewhat technically cumbersome, but this is precisely the form in which we shall use it later. Corollary 4.15. Let  be k1 -adapted, and assume that for some m0  100, for every  2 Cm0 there is a Schwarz function F which is adapted to k2 and so that Pk2 Qj F D F . Then X

kPk .Pk1 ;  F /kN Œk . jm0 j 2

2Cm0

k1

j

2

k1 4 ^0

kkS Œk1 

 X

kF k2N Œk2 

 12

2Cm0

(4.49) provided we are in the low-high case k D k2 C O.1/  k1 . The sum here runs over caps with dist.1 ; 2 / . 2m0 . Proof. For this, one simply repeats the P proof of the2 low-high case of Lemma 4.13 with one additional twist: since  kPk1 ; kS Œk cannot be controlled by kkS Œk1  , one has to check carefully that the square summation – which (4.49) leads to after Cauchy–Schwarz – is compatible with the estimates we are making (the norm for F is always L2t L2x ). This is the case if we place Pk1 ;  2 P s;b -norm. In the latter case one does not incur any loss in L1 t Lx or an X due to orthogonality, whereas in the former case there is a loss of jm0 j, see Lemma 2.18. The only place where one cannot use either of these norms is (4.45). Indeed, if k1 C 2m0  j C , then the caps of sizes 2m0 are smaller than j

k1

those of size 2` D 2 2 in the wave-packet decomposition of (4.45). In this case, however, one considers a wave-packet decomposition induced by the projections Pk1 ; Q j C , then this issue does not arise at all and the estimate (4.45) is performed essentially as in Lemma 4.13 – the only difference being that the caps in the wave-packet decomposition are grouped together inside the larger Cm0 – caps. 

4.4 Null-form bounds in the high-high case

4.4

101

Null-form bounds in the high-high case

Henceforth, k
kS Œk will mean the stronger norm jjj
jjjS Œk . The following definition introduces the basic null-forms as well as the method of “pulling out a derivative”. Definition 4.16. The null-forms Q˛ˇ for 0  ˛; ˇ  2, ˛ ¤ ˇ, are defined as Q˛ˇ .; / WD R˛ Rˇ

Rˇ R˛

whereas Q0 .; / WD R˛ R˛ : By “pulling out a derivative from” from Q˛ˇ we mean writing Q˛ˇ .; / D @˛ .jrj

1

Rˇ /

or the analogous expression with  and

@ˇ .jrj

1

R˛ /

interchanged.

Recall the L2 -bound (4.13) of Lemma 4.5 for Q˛ˇ -null-forms. We separate the null-form bounds according to high-high vs. high-low and low-high interactions. The high-high case is slightly more involved due to the possibility of opposing .CC/ or . / waves with comparable frequencies and very small modulations which produce a wave of small frequency but very large modulation. Lemma 4.17. For any `  k CO.1/, and j adapted to kj with k1 D k2 CO.1/, ` k

k

kPk Q` Q˛ˇ .1 ; 2 /kL2 L2x . 2 4C 2 2 2

k k1 2

t

k1 kS Œk1  k2 kS Œk2  :

(4.50)

In particular, k1 2

kPk QkCC Q˛ˇ .1 ; 2 /kL2 L2x . 2k t

Finally, for any m0 

k1 kS Œk1  k2 kS Œk2  :

(4.51)

10,

 X 1

Pk QkCC Q˛ˇ .Pk ; 1 ; 2 / 2 2 2 2 1 L L t

x

2Cm0

. jm0 j 2

k1 2

k1 kS Œk1  k2 kS Œk2  : (4.52)

102

4 Bilinear estimates involving S and N spaces

Proof. We can take k1 D k2 C O.1/ D 0. First, by (4.13), kPk Q` Q˛ˇ .QkC`

C 1 ; QkC` C 2 /kL2t L2x

.2

`Ck 4

Second, by an angular decomposition into caps of size 2 X kPk Q` Q˛ˇ .Qm 1 ; Qm 2 /kL2 L2x

k

2 2 k1 kS Œk1  k2 kS Œk2  :

`Ck 2

,

t

`Ck C m`

X

.

2

`Ck 2

2

` k 4

2k kQm 1 kL2 L2x kQm 2 kL1 L2x t

(4.53)

t

`Ck C m`

.2

`C3k 4

k1 kS Œk1  k2 kS Œk2  :

To pass to (4.53) one uses the improved Bernstein inequality, which yields a factor `Ck ` k of 2k 2 4 , whereas the 2 2 corresponds to the angular gain from the null-form (note that the error coming from the modulation is at most 2m  2` which is less than this gain). And third, by the improved Bernstein inequality and a decompomCk sition into caps of size 2 2 , X kPk Q` Q˛ˇ .Qm 1 ; Qm 2 /kL2 L2x t

`mC

X

.

2

` k 4

2k 2

mCk 2

2k 2

mCk 2


C 2m kQm 1 kL2 L2x kQm 2 kL1 L2x t

t

`mC

X

.

2

` k 4


C 2m 2

m 2

k1 kS Œk1  k2 kS Œk2 

`mC ` k

. 2 4C 2k k1 kS Œk1  k2 kS Œk2  : mCk

The factor 2 2 C 2m here is made up out of the angular gain 2 of 2m in modulation (in case ˇ D 0). And finally, due to " < 21 ,

mCk 2

and the loss

`

kPk Q` Q˛ˇ .QC 1 ; 2 /kL2 L2x . 2k 2 2 kQ˛ˇ .QC 1 ; 2 /kL1 L1x t t X k 2` m . 2 2 2 kQm 1 k 2 2 kQQ m 2 k L t Lx

L2t L2x

mC `

. 2k 2 2

X

2m 2

2m.1 "/

k1 kS Œk1  k2 kS Œk2 

mC

.2 as desired.

` k 4

2k k1 kS Œk1  k2 kS Œk2 

103

4.4 Null-form bounds in the high-high case

Next, we consider (4.52). Here one essentially repeats the proof of (4.51) verbatim. The only difference being that instead of Lemma 4.5 one uses Corollary 4.6, in fact the null-form version of (4.24). Note that this loses a factor of jm0 j. To sum over the caps one also needs to invoke Lemma 2.18 in case 2 of a L1  t Lx -norm, which incurs the same loss. We shall also require the following technical variant of the estimate of Lemma 4.17. It obtains an improvement for the case of angular alignment in the Fourier supports of the inputs. Lemma 4.18. Let ı > 0 be small and L > 1 be large. Then there exists m0 D m0 .ı; L/ < 0 large and negative such that for any j adapted to kj for j D 1; 2, X

kPk QkCC Q˛ˇ .Pk1 ;1 1 ; Pk2 ;2 2 /kL2 L2x t

1 ;2 2Cm0 dist.1 ;2 /2m0

 ı2

k1 2

k1 kS Œk1  k2 kS Œk2  ; (4.54)

provided maxj D1;2 jk kj j  L. The constant C is an absolute constant which does not depend on L or ı. Proof. Set k D 0. We first note that summing (4.50) over `  B already yields an improvement over (4.51) provided B is large enough (in relation to ı and L). Hence it suffices to consider the contribution of P0 Q` Q˛ˇ .Pk1 ;1 1 ; Pk2 ;2 2 / with B  `  O.1/ fixed. First, if we choose m0 to be a sufficiently large negative integer, then X P0 Q` Q˛ˇ .Q` C Pk1 ;1 1 ; Q` C Pk2 ;2 2 / D 0 1 ;2 2Cm0 dist.1 ;2 /2m0 `

by Lemma 4.1. Second, by an angular decomposition into caps of size 2 2 , X X

P0 Q` Q˛ˇ .Qm Pk ; 1 ; Qm Pk ; 2 / 2 2 1 1 2 2 L L t

x

1 ;2 2Cm0 ` C mC dist.1 ;2 /2m0

 C.L; ı/

X

X

1 kQm Pk1 ;1 1 kL2 L2x kQm Pk2 ;2 2 kL1 t Lx t

1 ;2 2Cm0 ` C mC dist.1 ;2 /2m0

 C.L; ı/ jm0 j 2

m0 2

k1 kS Œk1  k2 kS Œk2   ı k1 kS Œk1  k2 kS Œk2  :

104

4 Bilinear estimates involving S and N spaces

To pass to the last line we applied Cauchy–Schwarz to the sum over the caps as well as Lemma 2.18. The case dealing with Qm Pk1 ;1 1 and Qm Pk2 ;2 2 is analogous. And finally, due to " < 21 , X kP0 Q` Q˛ˇ .QC Pk1 ;1 1 ; Pk2 ;2 2 /kL2 L2x t

1 ;2 2Cm0 dist.1 ;2 /2m0

X

.

kP0 Q` Q˛ˇ .QC Pk1 ;1 1 ; Pk2 ;2 2 /kL1 L2x t

1 ;2 2Cm0 dist.1 ;2 /2m0

 C.L; ı/

X mC

 C.L; ı/2

m0 2

X

2m

kQm Pk1 ;1 1 kL2 L2x kQQ m Pk2 ;2 2 kL2 L1 x t

t

1 ;2 2Cm0 dist.1 ;2 /2m0

X

2m 2

2m.1 "/

k1 kS Œk1  k2 kS Œk2   ık1 kS Œk1  k2 kS Œk2 

mC



as desired.

In case the output has “elliptic” rather than hyperbolic character, there is the following bound. Lemma 4.19. For any j adapted to kj with k1 D k2 C O.1/, X

k 2 "` Pk Q` Q˛ˇ .1 ; 2 /kL2 L2x . 2 2 2 "k1 hk1 ki2 1 kS Œk1  k2 kS Œk2  : t

`kCC

Furthermore, X kPk Q` Q˛ˇ .Qk1 CC 1 ; Qk2 CC 2 /kL2 L2x t

`kCC k

. 2 2 hk1

ki2 k1 kS Œk1  k2 kS Œk2  : (4.55)

Proof. We set k1 D k2 C O.1/ D 0. One has the decomposition kPk Q` Q˛ˇ .1 ; 2 /kL2 L2x t

. kPk Q` Q˛ˇ .Q`

C 1 ; Qk1 CC 2 /kL2t L2x

(4.56)

C kPk Q` Q˛ˇ .Q`

C 1 ; Q>k1 CC 2 /kL2t L2x

(4.57)

C kPk Q` Q˛ˇ .Q m0 . In the former case, the caps in Ck are smaller than those in Cm0 and (4.63) applies directly (one organizes the caps in Ck into subsets of the larger Cm0 -caps). In the latter case, however, the Cm0 –caps are smaller which forces us to write Q
k1 CC Pk2 ;2 2 /kL2t L2x

(4.68)

C kPk Q` Q˛ˇ .Qm



1 L2 L2 t x

C

mk1 CC 0`mCC 1

PQk
1

@j QQ m Qˇj .



2;

3 / L2 L2 t x

X

C

mk1 CC

X

"`

2

QQ `

PQk
1 L2 L2

1

1

x

t

@j QQ m Qˇj .



2;

3 / L1 L2 : x t

`mCC

In the second to last line we applied Bernstein’s inequality in the time variable to switch from L2t to L1t . We now replace the L1 t on the right-hand side of the last m line by an L2t at the expense of a factor of 2 2 . Together with Lemma 4.19 this yields X X 1 (5.5) . 2 k1 C. 2 "/` kQ>m C 1 kL2 L2x t

mk1 CC 0`mCC

PQk QQ m Qˇj . 1



2;

3 / L2 L2 t x

X

C

X

2

k1 "`

m

22

mk1 CC `mCC

kQQ ` X

.

Q Q 1 kL2 L2x Pk1 Qm Qˇj . t

2

. 21 C"/k1

1 2m

2

k

2;



3 / L2 L2 t x

1 kS Œk1 

PQk QQ m Qˇj . 1

m

.1 "/` . 12 "/k1

2;



3 / L2 L2 t x

mk1 CC

X

C

X

2

k1 "`

222

2

k

1 kS Œk1 

mk1 CC `mCC

PQk QQ m Qˇj . 1 .2

k1 2

2

"k2

hk2



2;

k1 i

3 / L2 L2 t x 2

3 Y

k

i kS Œki  :

i D1

Next, we consider the case where both A0 D I c and A1 D I c . If ˛ ¤ 0, then one can drop R˛ altogether so that the previous analysis applies. Otherwise, if ˛ D 0,

119

5.1 Reduction to the hyperbolic case

then by assumption AQ1 D I and

P0 Q0 @ˇ ŒQk CC R˛ 1
1 @j PQk Qk CC Qˇj .A2 2 ; A3 3 / 1 1 1 N Œ0 X

ˇ Q 1

Q  P0 Qm @ ŒQm R˛ 1
@j Pk1 Qk1 CC Qˇj .A2 2 ; A3 3 / N Œ0 mk1 C10C

(5.6)

C P0 Q0
k

1 C10C

ŒQ0
k1 C10C R˛

@

ˇ 1

1


@j PQk1 Qk1 CC Qˇj .A2

2 ; A3



3 / N Œ0 :

(5.7)

By Lemma 4.17, (5.7) is bounded by

P0 Q0
k C10C @ˇ ŒQ0
k C10C R˛ 1 1

1


1

@j PQk1

Qk1 CC Qˇj .A2

. Q0
k

1 C10C

R˛

. kQ0
k1 C10C R˛ .2

k2 2

3 Y

k

2 ; A3



3 / XP 0; 0

1

@j PQk1 Qk1 CC Qˇj .A2 2 ; A3

1

Q 1 kL1 L2x
@j Pk1 Qk1 CC Qˇj .A2 1




t

1 ";2

3 / L2 L1 t

2 ; A3

x



3 / L2 L2 t x

i kS Œki  :

i D1

On the other hand, (5.6) is estimated as follows: X

kP0 Qm @ˇ ŒQQ m R˛

1


1

@j PQk1

mk1 C10C

.

X

2

m"

kQQ m R˛

Qk1 CC Qˇj .A2 2 ; A3 3 /kXP 0; 1 ";2 0

1

Q 1 kL2 L2x
@j Pk1 Qk1 CC Qˇj .A2 2 ; A3 3 / L1 L2 t

t

x

mk1

.2 .2

. 12 C"/k1

"k1

k2 2

k

1 kS Œk1  2

3 Y

k

k1 2

kPQk1 Qk1 CC Qˇj .A2

2 ; A3

3 /kL2 L2x t

i kS Œki 

i D1

where we applied Bernstein’s inequality relative to t as well as Lemma 4.17 to pass to the last line. Now suppose A0 D I (in fact, A0 D Q0 ), but at least one of A1 or AQ1 equals I c . But then the modulations of 1 and Qˇj essentially agree, whence

120

5 Trilinear estimates

˛ ¤ 0 and X

P0 Q0 @ˇ ŒQm R˛

1


1

@j QQ m Qˇj .

2;



3 / N Œ0

mk1 CC

X

.

P0 Q0 @ˇ ŒQm R˛

1


1

@j QQ m Qˇj .



2;

3 / L1 tx

mk1 CC

X

.

kQm

k1

1 kL2 L2x 2 t

PQk1 QQ m Qˇj .

2;



3 / L2 L2 t x

mk1 CC

X

.

1

2. 2

"/k1

2

m.1 2"/

k

1 kS Œk1  2

k1

2

m"

kPQk1 QQ m Qˇj .

2;

3 /kL2 L2x t

mk1 CC

.2

k1 2

"k2

k1 i2

hk2

3 Y

k

i kS Œki  :

i D1

The final estimate here uses Lemma 4.24. The last case which we need to consider is A0 D A1 D AQ1 D I and either one of A2 ; A3 equal to I c . But then necessarily A2 D A3 D I c , whence

P0 @ˇ I ŒIR˛ 1
1 @j I Qˇj .Qk CC 2 ; Qk CC 3 / 2 2 N Œ0

1

. IR˛ 1
@j I Qˇj .Qk2 CC 2 ; Qk2 CC 3 / L1 L1 t x X

PQk Qk CC Qˇj .Qm 2 ; QQ m 3 / 1 1 . k 1 kL1 L2x 1 1 L L t

t

x

mk2 CC

.k

1 kL1 L2x

X

t

2m

k2

2

2m.1 "/ .1 2"/k2

2

k

2 kS Œk2  k 3 kS Œk3 

mk2 CC

.2

k2

3 Y

k

i kS Œki 

i D1

which concludes Case 1. Case 2: 0  k1 D k2 C O.1/; k3  k2 C. We again begin with A0 D I c , A1 D I and the representation (5.2) and (5.3) (dropping IR˛ from 1 as before). By Lemma 4.23, (5.2) is bounded by

1


1

@j Qk1 CC Qˇj . .k

2;



3 / L2 L1 t x

1 kL1 L2x 2 t

. 21 "/k3

2

.1 "/k1

k

2 kS Œk2  k 3 kS Œk3  ;

121

5.1 Reduction to the hyperbolic case

whereas X

(5.3) .

P0 Qm @ˇ ŒQm

1

1


C

@j QQ m Qˇj .



2;

3 / N Œ0

(5.8)

mk1 CC

X

C

P0 Q0 @ˇ ŒQm

1

1


C

@j Qm Qˇj .



2;

3 / N Œ0 :

mk1 CC

(5.9) Lemma 4.24 yields the following bound on (5.8): X

P0 Qm @ˇ ŒQm

1

1


C

@j QQ m Qˇj .

2;



3 / N Œ0

k1 CC m

X

.

P0 Qm @ˇ ŒQm

1

1


C

@j QQ m Qˇj .



2;

3 / XP 0; 0

1 ";2

k1 CC m

X

.

2

m"

2

k1

k

k1

1 kL1 L2x 2 t

PQk1 QQ m Qˇj .

2;



3 / L2 L2 t x

k1 CC m 1

. 2. 2

"/k3

3 Y

k kS Œki  :

i D1

The bound on (5.9) proceeds similarly: X

(5.9) .

P0 Q0 @ˇ ŒQ>m

1

1


C

@j QQ m Qˇj .

2;



3 / N Œ0

mk1 CC

X

.

X

P0 Q` @ˇ ŒQ>m

C

1


1

@j

mk1 CC 0`mCC

C

QQ m Qˇj . 2 ; 3 / XP 0; 1 ";2 0 X X

P0 Q` @ˇ ŒQQ ` 1


1

@j QQ m Qˇj .

2;



3 / XP 0; 0

1 ";2

mk1 CC `mCC

X

.

1

X

2. 2

"/`

kQ>m

C

Q

1 kL2 L2x kPk1
t

1

@j

mk1 CC 0`mCC

C

QQ m Qˇj . 2 ; X X 2

3 /kL2 L2x

(5.10)

t

"`

kQQ `

Q 1 kL2 L2x Pk1
t

1

@j

mk1 CC `mCC

QQ m Qˇj .

2;



3 / L1 L2 : x t

(5.11)

122

5 Trilinear estimates

To pass to (5.10) we used Bernstein’s inequality to switch from L2t to L1t , which ` 2 costs 2 2 . We now replace the L1 t on the right-hand side of the last line by an L t m at the expense of a factor of 2 2 . In view of Lemma 4.24 one concludes that

X

(5.9) .

X

k1 C. 12 "/`

2

kQ>m

C

1 kL2 L2x t

mk1 CC 0`mCC

PQk QQ m Qˇj . 1



2;

3 / L2 L2 t x

X

C

X

2

k1 "`

m

22

mk1 CC `mCC

Q Q 1 kL2 L2x Pk1 Qm Qˇj .

kQQ ` X

.

t

. 12 C"/k1

2

1 2m

2

k

1 kS Œk1 



2;

3 / L2 L2 t x

PQk QQ m Qˇj . 1

2;



3 / L2 L2 t x

mk1 CC

C

X

X

k1 "`

2

m

222

.1 "/` . 12 "/k1

2

k

1 kS Œk1 

mk1 CC `mCC

kPQk1 QQ m Qˇj . .2

3 Y

k1 . 12 "/k3

2

2;

k

3 /kL2 L2x t

i kS Œki  :

i D1

Next, we consider the case where both A0 D I c and A1 D I c . If ˛ ¤ 0, then one can drop R˛ altogether so that the previous analysis applies. Otherwise, if ˛ D 0, then by assumption AQ1 D I and as in Case 1 one obtains (5.6) and (5.7). By Lemma 4.23, (5.7) is bounded by

P0 Q0
k C10C @ˇ ŒQ0
k C10C R˛ 1 1

1


1

@j PQk1 Qk1 CC Qˇj .A2

. kQ0
k1 C10C R˛ . 12

.2 .2

k1 2

"/k1

1

2. 2

1




1

@j PQk1 Qk1 CC Qˇj .A2

kQ0
k1 C10C R˛ "/k3

3 Y i D1

1 kL2 L2x
t

1

2 ; A3

i kS Œki  :



3 / XP 0; 0

3 /kL2 L1x t

@j PQk1 Qk1 CC Qˇj .A2

k

2 ; A3

2 ; A3



3 / L2 L2 t x

1 ";2

123

5.1 Reduction to the hyperbolic case

On the other hand, (5.6) is estimated as follows: X

P0 Qm @ˇ ŒQQ m R˛ 1
1 @j PQk Qk 1

1 CC

mk1 C10C

X

.

2 m" QQ m r t;x jrj

1

Qˇj .A2 2 ; A3 3 / XP 0; 0

k1 Q 2 P Q 2 k k CC

1

L2t Lx

1

1 ";2

1

mk1

Qˇj .A2 .2

. 21 C"/k1

.2

k1 C. 12 "/k3

k

1 kS Œk1  2 3 Y

k

k1 2



2 ; A3

3 / L1 L2 x t

PQk Qk CC Qˇj .A2 1 1

2 ; A3



3 / L2 L2 t x

i kS Œki 

i D1

where we applied Bernstein’s inequality relative to t as well as Lemma 4.23. We now turn to the case where A0 D I , but at least one of A1 or AQ1 equals c I . But then the modulations of 1 and Qˇj essentially agree whence ˛ ¤ 0. Bounding N Œ0 by L1t L2x and invoking Lemma 4.24 yields X

P0 Q0 @ˇ ŒQm

1


1

@j QQ m Qˇj .



2;

3 / N Œ0

mk1 CC

X

.

kQm

1 kL2 L2x t

2

k1

PQk1 QQ m Qˇj .

2;



3 / L2 L2 t x

mk1 CC 3 Y

. 23 "/k1 C. 12 "/k3

.2

k

i kS Œki 

.2

k1 . 12 "/k3

2

iD1

3 Y

k

i kS Œki  :

iD1

The last case which we need to consider is A0 D A1 D AQ1 D I and either one of A2 ; A3 equal to I c . We begin with A2 D I c . But then necessarily A2 D A3 D I c , whence

P0 @ˇ I ŒI

. I 1


1


1

@j I Qˇj .Qk2 CC



2 ; Qk2 CC

3 / N Œ0

1

@j I Qˇj .Qk2 CC 2 ; Qk2 CC 3 / L1 L1 t x X k1

Q . k 1 kL1 L2x 2 Pk1 Qk1 CC Qˇj .Qm 2 ; QQ m t



3 / L1 L2 t x

mk2 CC

.k

1 kL1 L2x

X

t

mk2 CC

2m

k2

2

2.1 "/m . 12 "/k2 . 12 "/k3

2

2

k

2 kS Œk2  k 3 kS Œk3 

124

5 Trilinear estimates

.2

3 Y

. 32 "/k1 C. 12 "/k3

k

k1 . 12 "/k3

i kS Œki  . 2

2

i D1

3 Y

k

i kS Œki  :

i D1

It remains to consider the case A2 D I and A3 D I c . We begin by reducing the modulation of the entire output. Indeed, by Lemma 4.23,

P0 @ˇ Q.1 1 2 .1

.2

3"/k3 
C ŒQk1 CC

3"/k3 1 2 .1

.2

k1

.2

.1 "/k1

2

k

1 kL1 L2x t

3"/k3

"

2 2 k3

k

3 Y

2

k1

@j Qk1 CC Qˇj .I

kI Qˇj .I

2; I

. 12 "/k3 "k2

1 kL1 L2x

2

t

k

1

1


2

c

k

2; I

c



3 / N Œ0

3 /kL2 L2x t

2 kS Œk2  k 3 kS Œk3 

i kS Œki  :

iD1

Next, we reduce the modulation of

1:

P0 @ˇ Q.1 3"/k ŒQ.1 3"/k k 3 3 1

ˇ

. P0 @ Q.1 3"/k3 ŒQ.1 3"/k3 Qˇj .I . kQ.1 .2 .2

k1 2

2

1 kL2 L2x

3"/k3 k1 1 2 .1

. 12 "/k1

t

3"/k3 "

2 2 k3

k

3 Y

2

k1 c

2; I

1

@j Qk1 CC Qˇj .I 1

1


2; I

c



3 / N Œ0

@j Qk1 CC

3 /

L1t L1x

k1

I Qˇj .I

1 kS Œk1  2

k

1


2; I

. 21 "/k3 "k2

2

k

c



3 / L2 L2 t x

2 kS Œk2  k 3 kS Œk3 

i kS Œki  :

i D1

Finally, we reduce the modulation of the interior null-form using Lemma 4.13:

P0 @ˇ Q.1

3"/k3 ŒQ.1 3"/k3 k1

1


1

@j Qk3 
k1 CC Qˇj .I

.k

X

1 kS Œk1 

` 4

2

2

k1

hk1 i Pk1 Q` Qˇj .I

2; I 2; I

c

c



2 3/

3 / N Œ0 L t L2x

k3 `k1 CC

.2

.1 2"/k1 . 41 "/k3

2

3 Y

k

i kS Œki 

i D1

which is again admissible. After these preparations, we are faced with the follow-

125

5.1 Reduction to the hyperbolic case

ing decomposition: P0 @ˇ Q.1

D P0 @ˇ Q.1



Q.1 3"/k3 k1  3"/k3 Q.1 3"/k3

3"/k3

1

1


Qˇj .Qk3 
k2 CC X  P0; @ˇ Q.1 3"/k3 Pk1 ; 0 Q.1 D ; 0 2C`

3/

2 ; Qk3 CC 
k2 CC

3/

1

1


k1

c

@j Qk3 Qˇj .I

2; I



@j Qk3 

1

@j Qk3  2 ; Qk3 CC 
k2 CC 3 /

3"/k3 k1

Qˇj .Qk3 
k2 CC

1


where ` D 21 .1 3"/k3 and dist.;  0 / . 2` . Placing the entire expression in L1t L2x and using Bernstein’s inequality results in the following estimate: With J WD Qk3 
k2 CC ,

P0 @ˇ Q.1

3"/k3 ŒQ.1 3"/k3 k1

1


1

@j Qk3

Qˇj .I

 X

P0; ŒPk ; 0 Q.1  1

2; I 1

1


3"/k3 k1

c

@j Qk3

; 0 2C`

Qˇj .J

 X `

P0; ŒPk ; 0 Q.1  22 1

3"/k3 k1



3 / L1 L2 t x

2; J

1


1

2  12

2 /

3 L

@j Qk3

; 0 2C`

Qˇj .J

 X `  22 kPk1 ; 0 Q.1  0 2C

2; J

2 1k 2


3"/k3 k1

1

Lx

`

L1t

x

2

3 / L1 x

 12

L1t

@j Qk3

2  21

Qˇj .J 2 ; J 3 / L2

L1t

x

` 2

 2 kQ.1 .2 .2

1 4 .1 1 4 .1

3"/k3 3"/k3

3"/k3 k1

k k

1 kS Œk1  1 kS Œk1 

1 kL1 L2x
t

2 2

k1 k1

1 @j Qk3 Qˇj .J 2 ; J 3 / L1 L2

kr t;x jrj 2

k3 2

k

x

t

1

J

2 kL2 L2x kr t;x jrj t

. 12 "/k3 "k2

2 kS Œk2  2

2

k

1

J

3 kL2 L1 x t

3 kS Œk3 

which is again admissible for small " > 0. Case 3: 0  k1 D k3 C O.1/; k2  k3 one.

C: This case is symmetric to the previous

126

5 Trilinear estimates

Case 4: O.1/  k2 D k3 C O.1/; k1  C. This case proceeds similarly to Case 1. We again begin with A0 D I c and A1 D I . Then we can drop IR˛ from 1 and estimate

P0 Q0 @ˇ Œ 1
1 @j Qˇj . 2 ; 3 / N Œ0

ˇ 1 e 0 Qm

(5.15) .

C

1

1


@j QQ m Qˇj .

2;



3 / N Œ0

mC

.

X

X

P0 Q` @ˇ ŒQ>m

C

1


1

X

P0 Q` @ˇ ŒQQ `

1


1

@j QQ m Qˇj .

2;

3 / XP 0; 0

@j QQ m Qˇj .

2;

3 / XP 0; 0



mC 0`mCC

C

X



mC `mCC

.

X

1

X

"/`

2. 2

kQ>m

C

1 kL2 L1 x

"`

1 kL2 L1 x

t

1 ";2

e P 0 QQ m Qˇj .

1 ";2



2;

3 / L2 L2 t x

mC 0`mCC

C

X

X

2

kQQ `

t

e P 0 QQ m Qˇj .



2;

3 / L1 L2 : x t

mC `mCC

In the second to last line we applied Bernstein’s inequality in the time variable to switch from L2t to L1t . We now replace the L1 t on the right-hand side of the last m line by an L2t at the expense of a factor of 2 2 . Together with Lemma 4.19 this yields

(5.15) .

X

X

1

2. 2

"/` k1

2 kQ>m

1 kL2 L2x

C

t

e P 0 QQ m Qˇj .

2;



3 / L2 L2 t x

mC 0`mCC

C

X

X

2

m 2 2 2 kQQ `

"` k1

1 kL2 L2x t

e P 0 QQ m Qˇj .

2;



3 / L2 L2 t x

mC `mCC

. 2 k1

X

2

1 2m

k

1 kS Œk1 

e P 0 QQ m Qˇj .

2;



3 / L2 L2 t x

mC

C 2k1

X

X

2

"`

m

222

.1 "/`

mC `mCC

k .2

. 23 "/k1

2

"k2

hk2 i

2

3 Y

k

1 kS Œk1 

e P 0 QQ m Qˇj .

2;



3 / L2 L2 t x

i kS Œki 

i D1

which is admissible. Next, we consider the case where both A0 D I c and A1 D I c . If ˛ ¤ 0, then one can drop R˛ altogether so that the previous analysis

128

5 Trilinear estimates

applies. Otherwise, if ˛ D 0, then by assumption AQ1 D I and

P0 Q0 @ˇ ŒQC R˛ 1
1 @j P e 0 QC Qˇj .A2 2 ; A3 3 / N Œ0 X

ˇ 1

P0 Qm @ ŒQQ m R˛ 1
@j P e 0 QC Qˇj .A2 2 ; A3 3 /  N Œ0 m10C

C P0 Q0
10C @ˇ ŒQ0
10C R˛

1

1


e 0 QC Qˇj .A2 @j P

2 ; A3

(5.16)

3 / N Œ0 : (5.17)

By Lemma 4.17, (5.17) is bounded by

P0 Q0
10C @ˇ ŒQ0
10C R˛

1


1

@j

e 0 QC Qˇj .A2 P

. Q0
10C R˛

1

k1

.2 2

3 Y

k2 2

k



3 / XP 0; 0

1 ";2

1

e 0 QC Qˇj .A2 2 ; A3 3 / 2 2 @j P L t Lx

1

e 1
@j P 0 QC Qˇj .A2 2 ; A3 1 kL1 t Lx




. kQ0
k1 C10C R˛

2 ; A3



3 / L2 L2 t x

i kS Œki  :

i D1

On the other hand, (5.16) is estimated as follows: X

P0 Qm @ˇ ŒQQ m R˛

1


1

e 0 QC Qˇj .A2 @j P

2 ; A3

m10C

.

X

2

m"

QQ m r t;x jrj

1

m0

. 2. 2

1

"/k1

1

"/k1

. 2. 2

k

1 kS Œk1  k2 2

3 Y

k



1 L2 L1 t x

e P 0 Qk1 CC Qˇj .A2

e P 0 QC Qˇj .A2

2 ; A3



3 / XP 0; 0 2 ; A3

1 ";2



3 / L1 L2 x t



3 / L2 L2 t x

i kS Œki 

i D1

where we applied Bernstein’s inequality relative to t as well as Lemma 4.17. Now suppose A0 D I (in fact, A0 D Q0 ), but at least one of A1 or AQ1 equals I c . If AQ1 D I c , then the modulations of 1 and Qˇj essentially agree,

129

5.1 Reduction to the hyperbolic case

whence ˛ ¤ 0 and X

P0 Q0 @ˇ ŒQm R˛

1


1

@j QQ m Qˇj .

2;



3 / N Œ0

mC

X

.

P0 Q0 @ˇ ŒQm R˛

1


1

@j QQ m Qˇj .

2;



3 / L1 L2 t x

mk1 CC

.

X

kQm

1 kL2 L1 x t

e P 0 QQ m Qˇj .

2;



3 / L2 L2 t x

mC

.

X

3

"/k1

2. 2

2

m.1 2"/

k

1 kS Œk1  2

m"

e 0 QQ m Qˇj . kP

2;

3 /kL2 L2x t

mC 3

. 2. 2

"/k1 "k2

hk2 i2

3 Y

k

i kS Œki  :

i D1

The final estimate here uses Lemma 4.24. Now suppose that AQ1 D I and A1 D I c . Then

P0 Q0 @ˇ ŒI c R˛ 1
1 @j I Qˇj . 2 ; 3 / N Œ0

c 1

e . I R˛ 1
@j P 0 I Qˇj . 2 ; 3 / L1 L2 t x

c

1

e

. I r t;x jrj 1 L2 L1 P 0 QC Qˇj . 2 ; 3 / L2 L2 t

. 21

"/k1

.2

k

1 kL2 L2x 2 t

x

k2 2

t

k

x

2 kS Œk2  k 3 kS Œk3  :

The last case which we need to consider is A0 D A1 D AQ1 D I and either one of A2 ; A3 equal to I c . But then necessarily A2 D A3 D I c whence

P0 @ˇ I ŒI 1
1 @j I Qˇj .Qk CC 2 ; Qk CC 3 / 2 2 N Œ0

. I 1
1 @j I Qˇj .Qk2 CC 2 ; Qk2 CC 3 / L1 L1 t x X

e Q 1 P 0 QC Qˇj .Qm 2 ; Qm 3 / L1 L1 . k 1 kL1 t Lx t

x

mk2 CC

. 2k1 k

1 kL1 L2x

X

t

mk2 CC

. 2k1

k2

3 Y

k

i kS Œki 

i D1

which concludes Case 4.

2m

k2

2

2m.1 "/ .1 2"/k2

2

k

2 kS Œk2  k 3 kS Œk3 

130

5 Trilinear estimates

Case 5: O.1/ D k1 ; k2 D k3 C O.1/. We begin with A0 D I c and AQ1 D I (in fact, A0 D Q0 suffices here as usual). Moreover, we will drop R˛ from 1 which amounts to excluding the case A1 D I c and ˛ D 0 but nothing else. Then, from Lemma 4.17,

P0 I c @ˇ Œ

1

@j I Qˇj . 2 ; 3 / N Œ0 X

1
1 @j IPk Qˇj .

1


.

2;



3 / L2 L2 t x

kk2 ^0CO.1/

X

.

k

1 kL1 L2x

k

1 kL1 L2x 2

t

2

k

IPk Qˇj .

2;



3 / L2 L1 t x

kk2 ^0CO.1/

X

.

k

k2 2

t

k

2 kS Œk2  k 3 kS Œk3 

kk2 ^0CO.1/

.2

3 Y

jk2 j 2

k

i kS Œki 

i D1

which is better than needed. Now suppose ˛ D 0 and A0 D A1 D I c , which implies that AQ1 D I . Then

P0 I c @ˇ ŒI c R0 X .

1

@j I Qˇj . 2 ; 3 / N Œ0 X

2 "m P0 Qm ŒQQ m R0 1


1


1

@j Pk I Qˇj .



2;

3 / L2 L2 t x

kk2 ^0CO.1/ m0

X

.

X

2.1

"/m

kQQ m

1 kL2 L2x 2 t

k

Pk I Qˇj .

2;

kk2 ^0CO.1/ m0

X

.

k

k

1 kS Œk1  2 2

IPk Qˇj .

2;



3 / L1 L1 x t



3 / L2 L2 t x

kk2 ^0CO.1/ k

X

.

22 k

1 kS Œk1  2

k

k2 2

k

2 kS Œk2  k 3 kS Œk3 

kk2 ^0CO.1/

.2

jk2 j 2

3 Y

k

i kS Œki  :

i D1

Next, consider the case A0 D I c , and AQ1 D I c . Since I c R0

1

is now excluded,

131

5.1 Reduction to the hyperbolic case

we may drop A1 R˛ altogether. Then

P0 I c @ˇ Œ 1
1 @j I c Qˇj . 2 ; 3 / N Œ0 X

1

P0 Q0 Π1
@j Qk
C Pk Qˇj . .



2;

3 / L2 L2 t x

(5.18)

kk2 ^0CO.1/

C

X

X

2

"m

kk2 ^0CO.1/ mC

P0 Qm ŒQm C

X

X

P0 Q0 ŒQ>m

1

1


C

@j QQ m Pk Qˇj .



2;

3 / L2 L2 t x

(5.19)

1

Q 1
@j Qm Pk Qˇj . 2 ; 3 / L2 L2 :

C

t

x

kk2 ^0CO.1/ mC

(5.20) First, by Lemma 4.19, X (5.18) .

k

kQk
C Pk Qˇj .

k

1 kL1 L2x

2

k

1 kL1 L2x

kQk
C Pk Qˇj .

k

1 kL1 L2x

22 2

t

2;

3 /kL2 L1 x t

kk2 ^0CO.1/

X

.

t

2;

3 /kL2 L2x t

kk2 ^0CO.1/

X

.

t

k

"k2

hk2

ki2 k

2 kS Œk2  k 3 kS Œk3 

kk2 ^0CO.1/

.2

"jk2 j

hk2 i

3 Y

k

i kS Œki  :

i D1

Second, again by Lemma 4.19, X X (5.19) . 2

"m

P0 Qm ŒQm

C

1


1

@j QQ m Pk

kk2 ^0CO.1/ mC

Qˇj . X

.

X

2

"m

k

1 kL1 L2x t

QQ m Pk Qˇj .

kk2 ^0CO.1/ mC k 2

X

.

2 2

"k2

hk2

.2

hk2 i

3 Y i D1

ki

3 Y i D1

kk2 ^0CO.1/ "jk2 j

2

k

i kS Œki 

k

i kS Œki 

2; 2;



3 / L2 L2 t x



3 / L2 L2 t x

132

5 Trilinear estimates

and third, X

(5.20) .

X

kQ>m

kk2 ^0CO.1/ mC

X

.

X

2

X

X

"jk2 j

hk2 i

3 Y

k

2;

2

. 12 2"/m



3 / L1 L1 x t m

k

1 kS Œk1  2 2

QQ m Pk Qˇj .

kk2 ^0CO.1/ mC

.2

.1 "/m

k

t

QQ m Pk Qˇj .

kk2 ^0CO.1/ mC

.

1 kL2 L2x 2

C

k

2;



3 / L2 L2 t x

1 kS Œk1  2

QQ m Pk Qˇj .

2;

"m



3 / L2 L2 t x

i kS Œki 

i D1

where one argues as in the previous two cases to pass to the last line. Thus, A0 D Q0 for the remainder of Case 5. If A1 D I c , then necessarily AQ1 D I c which implies ˛ ¤ 0. Therefore,

P0 Q0 @ˇ ŒI c X .

@j I c Qˇj . 2 ; 3 / N Œ0 X

Qm 1
1 @j Pk QQ m Qˇj .

1


1

2;



3 / L1 L2 t x

kk2 ^0CO.1/ mC

X

.

X

kQm

1 kL2 L2x t

Pk QQ m Qˇj .

2;



3 / L2 L2 t x

kk2 ^0CO.1/ mC

X

.

X

2

.1 2"/m

k

1 kS Œk1  2

"m

Pk QQ m Qˇj .

2;



3 / L2 L2 t x

kk2 ^0CO.1/ mC

X

.

k

1 kS Œk1 

k

22 2

"k2

hk2

ki2 k

2 kS Œk2  k 3 kS Œk3 

kk2 ^0CO.1/

.2

"jk2 j

hk2 i

3 Y i D1

k

i kS Œki 

(5.21) which is again admissible. So we may assume also that A1 D I which means that we can drop R˛ from 1 . First, consider the case AQ1 D I c , whence we now face the expression   P0 Q0 @ˇ 1
1 @j I c Qˇj . 2 ; 3 /

133

5.1 Reduction to the hyperbolic case

with the implicit frequency constraints of Case 5. We write this as

P0 Q0 @ˇ Œ 1
1 @j I c Qˇj . 2 ; 3 / X D P0 Q0 @ˇ Œ 1
1 @j Q>kCC Pk Qˇj .

2;

3/

kkCC

C

X

X

P0 Q0 @ˇ ŒQl

1

1


C

@j Ql Pk Qˇj .

2;

3 /:

kO.1/

(5.22) To estimate the first term of (5.22), we use

X

X

P0 Q0 @ˇ ŒQkCC



X

X

P0 QlCO.1/
0 @ˇ ŒQkCC

Ql Pk Qˇj . .

X

X

2

. 21 "/l

kQkCC

X



@j Ql Pk Qˇj .



1

kl>kCC

X

.

X

kQl

C

1 kL2

t;:x




1

@j Ql Pk Qˇj .

2;



3 / L2 L1 t x

kl>kCC

(5.25) and from here the estimate continues as for (5.23). The third term of (5.22) is handled identically, and we note here that one actually gains exponentially in l. Finally, suppose at least one choice of j D 2; 3 satisfies Aj D I c . Then necessarily, A2 D A3 D I c and

P0 I @ˇ ŒI 1
1 @j I Qˇj .I c 2 ; I c 3 / N Œ0 X

1 c

. I 1
@j IPk Qˇj .I 2 ; I c



3 / L1 L2 t x

kk2 ^0CO.1/

X

.

k

kIPk Qˇj .I c

k

1 kL1 L2x

2

k

1 kL1 L2x

2k kIPk Qˇj .I c

t

2; I

c

3 /kL1 L1 x t

(5.26)

kk2 ^0CO.1/

X

.

t

2; I

c

Q

3 /kL1 L1x

3 /kL1 L1x t

kk2 ^0CO.1/

. 2k2 ^0 k

1 kL1 L2x

X

t

kQˇj .Qm

2 ; Qm

t

mk2 CC

. 2k2 ^0 k

1 kL1 L2x

X

t

2m

k2

2

2.1 "/m .1 2"/k2

2

k

2 kS Œk2  k 3 kS Œk3 

mk2 CC

.2

k2 _0

3 Y

k

i kS Œki 

(5.27)

i D1

as claimed. Case 6: O.1/ D k1  k2 C O.1/  k3 C C. This case proceeds similarly to Case 5. We begin with A0 D I c and AQ1 D I (in fact, A0 D Q0 suffices here as usual). Moreover, we will drop R˛ from 1 which amounts to excluding

135

5.1 Reduction to the hyperbolic case

the case A1 D I c and ˛ D 0 but nothing else. Then, from Lemma 4.23,

P0 I c @ˇ Œ .k

1

1


1 kL1 L2x 2 t

@j I Qˇj . k2



2;

3 / N Œ0

I PQk2 Qˇj .

2;

.

1


1

@j I Qˇj .

.1

3 / L2 L1 . 2 2



2;

"/k3 C"k2

3 / L2 L2 t x

3 Y

x

t

k

i kS Œki 

i D1

which is better than needed. Now suppose ˛ D 0 and A0 D A1 D I c , which implies that AQ1 D I . Then

P0 I c @ˇ ŒI c R0 1
1 @j I Qˇj . 2 ; 3 / N Œ0 X

"m 1 . 2 P0 Qm ŒQQ m R0 1
@j PQk2 I Qˇj .



2;

3 / L2 L2 t x

m0

.

X

2.1

"/m

kQQ m

1 kL2 L2x 2 t

k2

PQk2 I Qˇj .

m0

.2

k2 2

1

2. 2

"/k3 C"k2

3 Y

k

2;



3 / L1 L1 x t

i kS Œki  :

iD1

Next, consider the case A0 D I c , and AQ1 D I c . As before, we can drop A1 R˛ in this case. Then

P0 I c @ˇ Œ 1
1 @j I c Qˇj . 2 ; 3 / N Œ0

1 . P0 Q0 Π1
@j Qk
C PQk2 Qˇj . 2 ; 3 / L2 L2 t x X

"m 1 Q Q C 2 P0 Qm ŒQm C 1
@j Qm Pk2 Qˇj .

(5.28)

2 ; 3 / L2 L2 (5.29) t

x

mC

C

X

P0 Q0 ŒQ>m

C

1


1

@j QQ m PQk2 Qˇj .

2;



3 / L2 L2 : t x

mC

First, by Lemma 4.24,

2 k2 Qk2 O.1/
C PQk2 Qˇj . 2 ; 3 / L2 L1 t t x

. k 1 kL1 L2x Qk2 O.1/
C PQk2 Qˇj . 2 ; 3 / L2 L2

(5.28) . k

1 kL1 L2x t

1

. 2. 2

"/k3

t

3 Y iD1

k

i kS Œki  :

x

(5.30)

136

5 Trilinear estimates

Second, again by Lemma 4.24, (5.29) .

X

2

"m

2

"m

P0 Qm ŒQm

C

1


1

@j QQ m PQk2 Qˇj .



2;

3 / L2 L2 t x

mC

.

X

k

1 kL1 L2x t

QQ m PQk Qˇj . 2

2;



3 / L2 L1 t x

mC 3 Y

. 21 "/k3

.2

k

i kS Œki 

i D1

and third, (5.30) .

X

kQ>m

1 kL2 L2x 2

C

t

k2

QQ m PQk2 Qˇj .

2;



3 / L1 L1 x t

mC

.

X

2

.1 "/m

m

k

1 kS Œk1  2 2

QQ m PQk Qˇj . 2

2;



3 / L2 L2 t x

mC

.

X

2

. 12 2"/m

k

1 kS Œk1  2

"m

QQ m PQk2 Qˇj .

2;



3 / L2 L2 t x

mC 1

. 2. 2

"/k3

3 Y

k

i kS Œki 

i D1

where one argues as in the previous two cases to pass to the last line. Thus, A0 D Q0 for the remainder of Case 6. If A1 D I c , then necessarily AQ1 D I c which implies ˛ ¤ 0. Therefore,

P0 Q0 @ˇ ŒI c 1
1 @j I c Qˇj . 2 ; 3 / N Œ0 X

1

Q Q . Qm 1
@j Pk2 Qm Qˇj . 2 ; 3 / L1 L2 t

x

mC

.

X

kQm

1 kL2 L2x t

PQk QQ m Qˇj . 2

2;



3 / L2 L2 t x

(5.31)

mC

.

X

2

.1 2"/m

k

1 kS Œk1  2

"m

PQk2 QQ m Qˇj .

2 ; 3 / L2 L2 t

x

mC . 21 "/k3

.2

3 Y

k

i kS Œki 

i D1

which is again admissible. So we may assume also that A1 D I which means that

137

5.1 Reduction to the hyperbolic case

we can drop R˛ from

1.

Next, assume AQ1 D I c . Then write

  P0 Q0 @ˇ 1
1 @j I c Qˇj . 2 ; 3 / X  D P0 Q0 @ˇ Qk2 CC

C

X

 P0 Q0 @ˇ Q`

C

1


1

@j Pk2 CO.1/ Q` Qˇj .

2;

 :

3/

`>k2 CC

(5.32) The first term we estimate by using Lemma 4.24: We obtain

X

P0 Q0 @ˇ ŒQk2 CC

X



P0 Q`

C 
0 @

ˇ

ŒQ`>k2 CC

C

t

X

.

2

. 21 "/` . 21 "/k3

2

.2

kPkj

x

j kS Œkj 

j D1

O.1/>`>k2 CC . 21 "/.k2 k3 /

3 Y

3 Y

kPkj

j kS Œkj  :

j D1

The second term in (5.32) is more of the same, and estimated using

P0 Q0 @ˇ ŒQ`

C

1


. kŒQ`

C

1

@j Pk2 CO.1/ Q` Qˇj . 1 kL2

t;x




1

2;



3 / N Œ0

@j Pk2 CO.1/ Q` Qˇj .

2;



3 / L2 L1 t x

from which point the estimate is concluded as in the preceding case. This leaves the cases A2 D I c or A3 D I c to be considered. In the former case, necessarily

138

5 Trilinear estimates

A2 D A3 D I c and

P0 I @ˇ ŒI 1
1 @j I Qˇj .I c 2 ; I c 3 / N Œ0

1 c c

Q . I 1
@j I Pk2 Qˇj .I 2 ; I 3 / L1 L2

x

t

.k

1 kL1 L2x t

k2

2

kI PQk2 Qˇj .I c

2; I

c

3 /kL1 L1 x t

. k 1 kL1 L2x kI PQk2 Qˇj .I c 2 ; I c 3 /kL1 L2x t t X . k 1 kL1 L2x kQˇj .Qm 2 ; QQ m 3 /kL1 L2x t

t

mk2

.k

X

1 kL1 L2x

kr t;x jrj

t

1

Qm

Q

2 kL2 L2x kQm t

3 kL2 L1 x t

mk2

.k


C kQm 2 kL2 L2x kr t;x jrj 1 QQ m 3 kL2 L1 t x t X 1 1 2 .1 2"/m 2. 2 "/k2 2. 2 "/k3 k 2 kS Œk2  k 3 kS Œk3 

1 kL1 L2x t

mk2 1

3 Y

"/.k3 k2 /

. 2. 2

k

i kS Œki 

i D1

which is acceptable. The one remaining case is A0 D A1 D AQ1 D A2 D I and A3 D I c . Of course one may also assume that 3 D Qk2 CC 3 . Then we write   P0 I @ˇ I 1
1 @j I Qˇj .I 2 ; I c 3 /   D P0 I @ˇ I 1 PQk2
1 @j I Qˇj .I 2 ; I c 3 / (5.33)   1 ˇ Q c C P0 I I 1
@j @ Pk2 I Qˇj .I 2 ; I 3 / : (5.34) The term on the right-hand side of (5.33) is difficult. More specifically, the methods that we have employed up to this point do not seem to yield the necessary bound. However, Tao’s trilinear estimate (5.1) implies that ˇ

k@ P0

1 Rˇ

3

2 kN Œ0

.2

 .k3 k2 / k2

2

3 Y

k

i kS Œki 

(5.35)

i D1

for some constant  > 0 as well as k@ˇ P0

1 Rˇ

2

k3 3 kN Œ0 . 2

3 Y i D1

k

i kS Œki  :

(5.36)

139

5.1 Reduction to the hyperbolic case

Since 2k2 PQk2
1 @j I can be replaced by the convolution by a measure and all norms involved are translation invariant, these estimates imply (5.33). The analysis of (5.34) is easier and similar to the considerations at the end of Case 2. More precisely, we first reduce the modulation of the entire output by means of Lemma 4.23:

P0 Q.1

@j @ˇ PQk2 I Qˇj .I 2 ; I c 3 / N Œ0

3"/k3 k 1 kL1 L2x I Qˇj .I 2 ; I c 3 / L2 L1

3"/k3 
C ŒI

.2 .2

1 2 .1

1

1


t

1 2 .1

3"/k3

k

"

. 2.1C"/k2 2 2 k3

1 kS Œk1  2 3 Y

x

t

. 12 "/k3 .1C"/k2

k

2

k

2 kS Œk2  k 3 kS Œk3 

i kS Œki  :

i D1

Next, we reduce the modulation of

1:



P0 Q.1 3"/k ŒQ.1 3"/k 1
1 @j @ˇ I PQk Qˇj .I 2 ; I c 3 / 2 3 3 N Œ0

1 ˇ . P0 Q.1 3"/k3 ŒQ.1 3"/k3 1
@j @ I PQk2 Qˇj .I 2 ; I c 3 / L1 L2 t

k2

. 2 kQ.1 1 2 .1

.2

Q 1 kL2 L2x kPk2 I Qˇj .I

3"/k3

3"/k3

k

"

. 2 2 k3 C.1C"/k2

t

. 21

1 kS Œk1  2 3 Y

k

"/k3 .1C"/k2

2

2; I

k

c

x

3 /kL2 L2x t

2 kS Œk2  k 3 kS Œk3 

i kS Œki  :

i D1

Finally, we reduce the modulation of the interior null-form using Corollary 4.14:

P0 Q.1

3"/k3 ŒQ.1 3"/k3

1


1

@j @ˇ PQk2 Qk3 
k2 CC Qˇj .I

.k

X

1 kS Œk1 

2

` k2 4

2

` 2

PQk Q` Qˇj .I 2

2; I

c

2; I



3 / N Œ0 c



3 / L2 L2 t x

k3 `k2 CC 1

. 2. 4

"/.k3 k2 /

3 Y

k

i kS Œki 

i D1

which is again admissible. After these preparations, we are faced with the follow-

140

5 Trilinear estimates

ing decomposition: P0 Q.1

3"/k3

 Q.1

D P0 Q.1

3"/k3

3"/k3

1

1


 Q.1

@j @ˇ Qk3 Qˇj .I 1


3"/k3

2; I

c



3/

1 ˇ

@ @j Qk3

 Qˇj .Qk3 
k2 CC 2 ; Qk3 CC 
k2 CC 3 / X  P0; Q.1 3"/k3 Pk1 ; 0 Q.1 3"/k3 1
1 @j @ˇ Qk3 D  ; 0 2C` Qˇj .Qk3 
k2 CC 2 ; Qk3 CC 
k2 CC 3 / where ` D 21 Œk2 C .1 3"/k3  and dist.;  0 / . 2` . Placing the entire expression in L1t L2x and using Bernstein’s inequality results in the following estimate: With J WD Qk3 
k2 CC ,



P0 Q.1 3"/k Q.1 3"/k k 3 3 1

 X

P0; ŒPk ; 0 Q.1  1

1


1

@j @ˇ Qk3 Qˇj .I

3"/k3 k1

1


1

2; I

Qk3 Qˇj .J 3"/k3 k1

; 0 2C`

3/



L1t L2x

@j @ˇ

; 0 2C`

 X `

P0; ŒPk ; 0 Q.1  22 1

c

1


1

2  21

2 /

3 L

2; J

L1t

x

@j @ˇ

2  21

Qk3 Qˇj .J 2 ; J 3 / L1

L1t

x

 X

Pk ; 0 Q.1 2 1 ` 2

2 1 ˇ 1 kL2 k
@j @

3"/k3 k1

x

 0 2C`

Qk3 Qˇj .J

2  12

/

3 L2

2; J

x

L1t

` 1 @j @ˇ Qk3 Qˇj .J 2 ; J 3 / L1 L2  2 2 kQ.1 3"/k3 k1 t t x

` 1 1 . 2 2 k 1 kS Œk1  r t;x jrj J 2 kL2 L2x r t;x jrj J 3 L2 L1

1 kL1 L2x


t

.2

1 4 .1

3"/k3 C

k2 4

k

1 kS Œk1 

2

k3 2

k

t

. 12 "/k3 "k2

2 kS Œk2  2

2

k

x

3 kS Œk3 

which is again admissible for small " > 0. Case 7: k1 D O.1/  k3 C O.1/  k2 C C. This case is symmetric to the previous one. Case 8: k2 D O.1/; max.k1 ; k3 /  C. We begin with A0 D Q0 and AQ1 D I , and we drop R˛ from 1 excluding the case A1 D I c and ˛ D 0 but nothing

141

5.1 Reduction to the hyperbolic case

else. Then, from Lemma 4.23,

P0 I c @ˇ Œ 1
1 @j I Qˇj . 2 ;



3 / N Œ0

.

1


1

@j I Qˇj .

k1 . 1

2 ; 3 / L2 L2 . 2 2 2

Q0 Qˇj . 1 I P . k 1 kL1 t Lx

t

"/k3



2; 3 Y

3 / L2 L2 t x

k

x

i kS Œki 

i D1

which is better than needed. Now suppose ˛ D 0 and A0 D A1 D I c , which implies that AQ1 D I . Then by Lemma 4.23

P0 I c @ˇ ŒI c R0 1
1 @j I Qˇj . 2 ; 3 / N Œ0 X

"m 1 . 2 P0 Qm ŒQQ m R0 1
@j PQ0 I Qˇj . 2 ; 3 / L2 L2 t

m0

.

X

2

"m

1

QQ m r t;x jrj



1 L2 L1 t x

m0

.

X

2.1

"/m

kQQ m

1 kL2 L2x t




PQ0 I Qˇj .

1

@j PQ0 I Qˇj .

2;

3 / L2 L2 t x

m0 . 21 "/.k1 Ck3 /

.2

3 Y

k

2;

x



3 / L1 L2 x t



i kS Œki  :

i D1

Next, consider the case A0 D I c , and AQ1 D I c . As before, we can drop A1 R˛ in this case. Then

P0 I c @ˇ Œ 1
1 @j I c Qˇj . 2 ; 3 / N Œ0 X

"m . 2 P0 Qm ŒQm C 1
1 @j QQ m PQ0 Qˇj . 2 ; 3 / L2 L2 (5.37) x

t

mC

C

X

P0 Q0 ŒQ>m

1

1


C

@j QQ m PQ0 Qˇj .



3 / L2 L2 : t x

2;

(5.38)

mC

First, by Lemma 4.24, X

(5.37) . 2 "m P0 Qm ŒQm

C

1


1

@j QQ m PQ0 Qˇj .

2;

mC

.

X

2

"m k1

2 k

1 kL1 L2x t

QQ m PQ0 Qˇj .

mC

.2

k1 C. 21 "/k3

3 Y i D1

k

i kS Œki 

2;



3 / L2 L2 t x



3 / L2 L2 t x

142

5 Trilinear estimates

and second, (5.38) .

X

2k1 kQ>m

1 kL2 L2x

C

t

QQ m PQ0 Qˇj .



2;

3 / L1 L2 x t

mC

.

3

X

"/k1

2. 2

.1 "/m

2

k

m

1 kS Œk1  2 2

QQ m PQ0 Qˇj .

2;



3 / L2 L2 t x

mC

. 2k1

X

2

. 12 2"/m

k

1 kS Œk1  2

"m

QQ m PQ0 Qˇj .

2;



3 / L2 L2 t x

mC k1 C. 12 "/k3

.2

3 Y

k

i kS Œki 

i D1

where one argues as in the previous two cases to pass to the last line. Thus, A0 D Q0 for the remainder of Case 8. If AQ1 D I c , then necessarily A1 D I c which implies ˛ ¤ 0. Therefore,

P0 Q0 @ˇ ŒI c 1
1 @j I c Qˇj . 2 ; 3 / N Œ0 X

1

Q Q . Qm 1
@j P0 Qm Qˇj . 2 ; 3 / L1 L2 t

x

mC

.

X

2k1 kQQ m

1 kL2 L2x t

PQ0 Qm Qˇj .

2;



3 / L2 L2 t x

mC

X

. 2k1

2

.1 2"/m

k

1 kS Œk1  2

"m

PQ0 QQ m Qˇj .

2;



3 / L2 L2 t x

mC k1 . 12 "/k3

.2 2

3 Y

k

i kS Œki 

i D1

which is again admissible. So we may assume also that AQ1 D I . Now suppose that A1 D I c . Then we can take A1 D Qk1 
C whence

P0 Q0 @ˇ ŒQk 
C R˛ 1
1 @j I Qˇj . 2 ; 3 / 1 N Œ0

1

Q . Qk1 
C R˛ 1
@j P0 I Qˇj . 2 ; 3 / L1 L2 t x

PQ0 I Qˇj . 2 ; 3 / 2 2 . kQk1 
C r t;x jrj 1 1 kL2 L1 L L x t

. 21 "/k1

.2

1

. 2. 2

k

. 12 "/k3

1 kS Œk1  2

"/.k1 Ck3 /

3 Y i D1

k

t

k

i kS Œki  :

2 kS Œk2  k 3 kS Œk3 

x

143

5.1 Reduction to the hyperbolic case

So we may assume for the remainder of this case that A1 D I which means that we can drop R˛ from 1 . This leaves the cases A2 D I c or A3 D I c to be considered. In the former case, necessarily A2 D A3 D I c and

P0 I @ˇ ŒI 1
1 @j I Qˇj .I c 2 ; I c 3 / N Œ0

1 c c

Q . I 1
@j I P0 Qˇj .I 2 ; I 3 / L1 L2 t x

k1 c c . 2 k 1 kL1 L2x I PQ0 Qˇj .I 2 ; I 3 / L1 L2 x t

t k1 c c

Q . 2 k 1 kL1 L2x I P0 Qˇj .I 2 ; I 3 / L1 L2 t t x X

k1

Q . 2 k 1 kL1 L2x Qˇj .Qm 2 ; Qm 3 / L1 L2 t

t

m0

. 2k1 k

X

1 kL1 L2x

kr t;x jrj

t

1

x

Q

2 kL2 L2x kQm

Qm

t

m0

C kQm . 2 k1 k

X

1 kL1 L2x

2

t

2 kL2 L2x kr t;x jrj t

.1 2"/m . 21 "/k3

2

k

3 kL2 L1 x t

1

QQ m

3 kL2 L1 x




t

2 kS Œk2  k 3 kS Œk3 

m0 1

. 2k1 C. 2

"/k3

3 Y

k

i kS Œki 

i D1

which is acceptable. The one remaining case is A0 D A1 D AQ1 D A2 D I and A3 D I c . Of course one may also assume that 3 D QC 3 . The analysis in this case is similar to the considerations at the end of Case 2. More precisely, we first reduce the modulation of the entire output by means of Lemma 4.23:

P0 @ˇ Q.1 3"/k 
C ŒI 1
1 @j PQ0 I Qˇj .I 2 ; I c 3 / 3 N Œ0

1 .1 3"/k c 3

1 IQ .2 2 k 1 kL1 3 / L2 L2 ˇj .I 2 ; I t Lx t

1 2 .1

k1

.2 2 "

. 2k1 2 2 k3

3"/k3

3 Y

k

k

. 21 "/k3

1 kS Œk1  2

k

x

2 kS Œk2  k 3 kS Œk3 

i kS Œki 

i D1

Next, we reduce the modulation of 1 :

P0 Q.1 3"/k @ˇ ŒQ.1 3"/k 1
1 @j I PQ0 Qˇj .I 2 ; I c 3 / 3 3 N Œ0

ˇ 1

Q . P0 @ Q.1 3"/k3 ŒQ.1 3"/k3 1
@j I P0 Qˇj .I 2 ; I c 3 / L1 L2 t x

. 2k1 kQ.1 3"/k 1 k 2 2 PQ0 I Qˇj .I 2 ; I c 3 / 2 2 3

L t Lx

L t Lx

144

5 Trilinear estimates

. 2k1

1 2 .1

"

3"/k3

. 2 2 k3 Ck1

3 Y

k

k

1 kS Œk1  2

. 21 "/k3

k

2 kS Œk2  k 3 kS Œk3 

i kS Œki 

i D1

Finally, we reduce the modulation of the interior null-form using Corollary 4.14:

P0 Q.1 3"/k ŒQ.1 3"/k 1
1 @j PQ0 Qk 
C Qˇj .I 2 ; I c 3 / 3 3 3 N Œ0 . 2k1 k

1 kS Œk1 

X

2

` k1 4

2

` 2

PQ0 Q` Qˇj .I

2; I

c



3 / L2 L2 t x

k3 `C

.2

3k1 1 4 C. 4

"/k3

3 Y

k

i kS Œki 

i D1

which is again admissible. After these preparations, we are faced with the following decomposition:   P0 @ˇ Q.1 3"/k3 Q.1 3"/k3 1
1 @j Qk3 Qˇj .I 2 ; I c 3 /  D P0 Q.1 3"/k3 @ˇ Q.1 3"/k3 1
1 @j Qk3  Qˇj .Qk3 
k2 CC 2 ; Qk3 CC 
k2 CC 3 / X P0; Q.1 3"/k3 @ˇ ŒPk1 ; 0 Q.1 3"/k3 1
1 @j Qk3 D ; 0 2C`

Qˇj .Qk3 
k2 CC

2 ; Qk3 CC 
k2 CC

3 /

where ` D 21 Œ.1 3"/k3 k1 ^0 and dist.;  0 / . 2` . Placing the entire expression in L1t L2x and using Bernstein’s inequality results in the following estimate:

P0 Q.1 3"/k ŒQ.1 3"/k 1
1 @j Qk Qˇj .I 2 ; I c 3 / 1 2 3 3 3 L t Lx

 X

P0; ŒPk ; 0 Q.1 3"/k 1
1 @j Qk  1 3 3 ; 0 2C`

2  12

Qˇj .Qk3 
C 2 ; Qk3 CC 
C 3 / L2 1 x Lt

 X

1

Pk ; 0 Q.1 3"/k 1
@j Qk  1 3 3 0 ; 2C`

2  21

Qˇj .Qk3 
C 2 ; Qk3 CC 
C 3 / L2 1 x Lt

 X

2 1  kPk1 ; 0 Q.1 3"/k3 1 kL @j Qk3 1
x  0 2C`

145

5.1 Reduction to the hyperbolic case

Qˇj .Qk3 
C `

 2 2 2k1 kQ.1

3"/k3

1 kL1 L2x t

`

1 kS Œk1 

r t;x jrj .2

3k1 1 4 C 4 .1

3"/k3

1

kr t;x jrj

k

1 kS Œk1 

1

1




Qˇj .Qk3 
C . 2 k1 C 2 k

2 ; Qk3 CC 
C

2 ; Qk3 CC 
C

Qk3 CC 
C

2

k

L1t

x

@j Qk3

Qk3 
C k3 2

2  12

/

3 L2



3 / L1 L2 t x

2 kL2 L2x

t

3

L2t L1 x

. 1 "/k3 k 3 kS Œk3  2 kS Œk2  2 2

which is again admissible for small " > 0. Case 9: k3 D O.1/; max.k1 ; k2 / 



C. Symmetric to Case 8.

It is important to realize that Lemma 5.1 yields the following statement, which is really a corollary of its proof rather than its lemma. Corollary 5.2. Let i be Schwarz functions adapted to ki for i D 0; 1; 2. Then for any ˛; ˇ D 0; 1; 2, and j D 1; 2,

P0 r t;x A0 ŒA1 R˛ 1
1 @j AQ1 Qˇj .A2 2 ; A3 3 / N Œ0 . w.k1 ; k2 ; k3 /

3 Y

k

i kS Œki 

i D1

where Ai and AQ1 are either I or I c , with at least one being I c . Moreover, we impose the following restrictions: ı If A1 D AQ1 D I c then ˛ D 0 is excluded, ı if k1 D O.1/ > k2  k3 C C , then A0 D A1 D AQ1 D A2 D I , A3 D I c is excluded, ı if k1 D O.1/ > k3  k2 C C , then A0 D A1 D AQ1 D A3 D I , A2 D I c is excluded. In particular, kP0 r t;x Œ

1


1

c

@j I Qˇj .

2;

3 /kN Œ0

. w.k1 ; k2 ; k3 /

3 Y i D1

k

i kS Œki  :

(5.39)

Proof. Note that the first exclusion in our list is precisely the exclusion in Lemma 5.1. The only real difference between this statement and that of Lemma 5.1

146

5 Trilinear estimates

lies with the fact that we no longer require the outer most derivative to be @ˇ . But this mattered only in one case, namely when we applied Tao’s bound (5.1) in Cases 6 and 7 above. Moreover, inspection of the argument in those cases reveals that the @ˇ @ˇ null-form was needed only in those instances which are excluded as the second and third conditions of our above list (in fact, the modulations were narrowed down much more before any need for (5.1) arose). The final statement is an immediate consequence of the first one, since we removed R˛ altogether (which eliminates the first exclusion) and since the other two exclusions do not arise due to AQ1 D I c . Therefore, one simply sums over all choices of A0 ; A1 ; A2 and A3 .  In fact, the proof of Lemma 5.1 makes no use of the fact that
1 @j contains the same index as the null-form Qˇj . But the strengthening resulting from replacing
1 @j by jrj 1 , say, is of no benefit to us so we do not carry it out. The following variant of Lemma 5.1 covers the other two types of trilinear nonlinearities arising in the Coulomb gauged wave map system. Lemma 5.3. Let i be Schwarz functions adapted to ki for i D 0; 1; 2. Then for any ˛ D 0; 1; 2, j D 1; 2,

P0 @ˇ A0 ŒA1 Rˇ 1
1 @j I Q˛j .A2 2 ; A3 3 / N Œ0

. w.k1 ; k2 ; k3 /

3 Y

k

i kS Œki 

(5.40)

i kS Œki 

(5.41)

i D1

P0 @˛ A0 ŒA1 Rˇ

1


1

@j I Qˇj .A2

2 ; A3

3 / N Œ0

. w.k1 ; k2 ; k3 /

3 Y

k

i D1

where Ai are either I or I c , with at least one being I c . Proof. Both these bounds follow from Corollary 5.2 provided we are not in those cases described as Items 2 and 3 in the list of exclusions (observe that the first exclusion does not arise due to our limitation to AQ1 D I ). So let us consider the second exclusion k1 D O.1/ > k2  k3 C C and A0 D A1 D AQ1 D A2 D I , A3 D I c (the third one being symmetric to this case). Then (5.41) is an immediate consequence of (5.1), see (5.35) and (5.36) above. As for (5.40), observe that due to the analysis of (5.34) we may assume that the outer @ˇ derivative hits 1 . Hence, it suffices to bound

P0 I ŒI @ˇ Rˇ 1
1 @j I Q˛j .I 2 ; I c 3 / : N Œ0

147

5.1 Reduction to the hyperbolic case

However, due to the property that kIP0 kL2 L2x . kkS Œ0 and @ˇ @ˇ D , this t is easy:

P0 I ŒQC @ˇ Rˇ 1
1 @j I Q˛j .I 2 ; I c 3 / N Œ0

ˇ 1

. P0 I ŒQC @ Rˇ 1
@j I Q˛j .I 2 ; I c 3 / L1 L2 t x

1

c ˇ

Q . kQC @ Rˇ 1 kL2 L2x
@j Pk2 I Q˛j .I 2 ; I 3 / L2 L1 t

.k

1 kS Œk1  2

1 2 .1

t

3"/k3 "k2

2

k

x

2 kS Œk2  k 3 kS Œk3 



as desired. The following technical corollary will be important later.

Corollary 5.4. For some absolute constant 0 > 0, and arbitrary Schwartz functions i , 2 X

P0 r t;x Œ

1


1

@j I c Qˇj .

2;



3 / N Œ0

j D1

. K 2 sup max 2

0 jkj

k2Z i D1;2;3

kPk

i kS Œk

(5.42)

P provided maxi D1;2;3 k2Z kPk i k2S Œk  K 2 and with an absolute implicit constant. Moreover, given any ı > 0 there exists a constant L D L.ı/  1 such that X0 k1 ;k2 ;k3

2 X

P0 r t;x ŒPk 1

1


1

@j I c Qˇj .Pk2



2 ; Pk3

3 / N Œ0

j D1

 ı K 2 sup

max 2

0 jkj

k2Z i D1;2;3

where the sum

P0

k1 ;k2 ;k3

Further, if X00

k1 ;k2 ;k3

k1 ;k2 ;k3

X

i kS Œk

extends over all k1 ; k2 ; k3 outside of the range

jk1 j  L; P00

kPk

k2 ; k3  L;

jk2

k3 j  L:

(5.43)

denotes the sum over this range, then

2 X

P0 r t;x ŒPk 1

1


1

@j I c Pk Qˇj .Pk2

2 ; Pk3



3 / N Œ0

kk2 L0 j D1

 ı K 2 sup max 2 k2Z i D1;2;3

0 jkj

kPk

i kS Œk

148

5 Trilinear estimates

where L0 D L0 .L; ı/ is a large constant. Finally, given ı > 0, there exists C > 1 large enough such that we have X X

P0 r t;x ŒPk 1
1 @j I c P< k Q>kCC Qˇj .Pk 2 ; Pk 3 / 1 2 3 N Œ0 k1 ;2;3 k< C

 ı K 2 sup max 2

0 jkj

k2Z i D1;2;3

kPk

i kS Œk :

P Proof. Write i D ki 2Z Pki i for 1  i  3. In view of the definition of the weights w.k1 ; k2 ; k3 /, summing (5.39) over all choices of k1 ; k2 ; k3 yields (5.42). The second statement follows immediately from the fact that the weights w.k1 ; k2 ; k3 / gain some smallness outside of the range (5.43) (namely 2 ıL ). For the third statement one needs to observe that in Case 5 – which is the one specified by (5.43) but of course with a range specified by the constant L – an extra gain can be obtained by restricting k to sufficiently small values compared to k2 ; k3 . 

5.2

Trilinear estimates for hyperbolic S-waves

The following lemma finally proves the trilinear estimates in the “hyperbolic” case. The argument will rely on the following trilinear null-form expansion from [23]: 2@ˇ

1

D .

1




@j Qˇj .

1 /jrj

C .

1

2;

2 jrj 1

1 .jrj 1 1 /
@j

3/ 1

3

2 /jrj

 C

1 jrj

1

2 jrj 1

1

jrj

1


1

@j .Rj

1 jrj

1

C


2 jrj

Rj

as well as its “dual” form  2@ˇ 1
1 @j Qˇj . 2 ; D

.

3/ 1

3

3 1

1

3

1

2 /jrj 1

1


3

@j .Rj 1

1 


2 jrj

@j Rj

1




3/ 1

2 jrj

3




(5.44)






C


2 jrj

1

2 jrj

1

1 jrj

3




3/

1

3

.

1 /


C

1 



jrj 1

1

@j Rj

1

@j Rj

2 2 jrj 2 jrj

1 1

3




3


: (5.45)

Strictly speaking, we shall want to apply these identities to the trilinear expression   @ˇ 1
1 @j IPk Qˇj . 2 ; 3 /

149

5.2 Trilinear estimates for hyperbolic S -waves

for some Pk . In the case of (5.45) the operator IPk can be inserted in front of any product involving 2 and 3 which is the case for all but the second term on the right-hand side of (5.45), i.e., . 1 jrj 1 3 /jrj 1 2 (and similarly for (5.44)). Since IPk is disposable, it takes the form of convolution with a measure k with mass kk k . 1. Thus, the second term needs to be replaced by the convolution Z
 1 jrj 1 3 .
y/ jrj 1 2 .
y/k .dy/: (5.46) The logic will be that any estimate that we make on . 1 jrj 1 3 /jrj 1 2 in the context of the S Œk and N Œk spaces will equally well apply to this convolution since all norms are translation invariant. We shall use this observation repeatedly in what follows without any further comment. Finally, the weights w.k1 ; k2 ; k3 / are those specified at the beginning of this section. Lemma 5.5. Let

j

be adapted to kj , for j D 1; 2; 3. Then

2

X 

P0 I @ˇ IR˛

1


1

@j I Qˇj .I

2; I



3/

N Œ0

j D1

. w.k1 ; k2 ; k3 /

3 Y

k

i kS Œki 

(5.47)

k

i kS Œki 

(5.48)

k

i kS Œki 

(5.49)

iD1 2

X 

P0 I @˛ IRˇ

1


1

@j I Qˇj .I

2; I



3/

N Œ0

j D1

. w.k1 ; k2 ; k3 /

3 Y iD1

2

X 

P0 I @ˇ IRˇ

j D1

1


1



@j I Q˛j .I 2 ; I 3 /

N Œ0

. w.k1 ; k2 ; k3 /

3 Y i D1

for any ˛ D 0; 1; 2. Proof. We begin with (5.47). Due to the I in front of 1 we shall drop the R˛ operator. Also, it will be understood in this proof that i D Qki CC i for 1  i  3 and we will often drop the I -operator in front of the input functions.

150

5 Trilinear estimates

Case 1: 0  k1  k2 C O.1/ D k3 C O.1/. By Lemma 4.17,

 

ˇ 1

P0 I @ Q0 1
@j I Qˇj .I 2 ; I 3 / N Œ0

 

. P0 I @ˇ Q0 1
1 @j I Qˇj .I 2 ; I 3 / 1 2 L L

t x

k1 Q . kQ0 1 kL2 L2x 2 Pk1 I Qˇj .I 2 ; I 3 / 2 2 t

(5.50)

L t Lx

.2

k2 2

3 Y

k

i kS Œki  :

i D1

So it suffices to consider  P0 I @ˇ Q 0. The sixth and final term is estimated by means of (4.40) and Lemma 4.11:

 

P0 I 1 
1 @j .Rj 2 jrj 1 3 / N Œ0




. 2k1 k 1 kS Œk1  PQ0 Qk1 
1 @j Rj 2 jrj 1 3 0; 1 ;1 XP k

. 22k1 k 1 kS Œk1  PQ0 Rj . 22k1 2

3 Y

k2 4

k




3

1

2 jrj

2

1

1 ;1 0; 2

XP 0

i kS Œki 

i D1

which concludes Case 4. Case 5: O.1/ D k1 ; k2 D k3 C O.1/. We start with the decomposition P0 @ˇ



1

1


@j I Qˇj .

2;



3/

X

D

P0 @ˇ



1

1


@j Pk I Qˇj .

 : (5.55)

2;

3/

kk2 ^0CO.1/

We first limit the modulation of 1 :

X 

P0 @ˇ Q>k I Q>kCC I kk2 ^0CO.1/

.

X

X




1

.Rˇ



P0 @ˇ Q>k I Q>kCC I

kk2 ^0CO.1/

.

1

k 2

2

kQ>kCC

kk2 ^0CO.1/

@j Pk I 2 Rj

1

Œ


Rj

3



2 Rˇ

3/

1 2 @jˇ Pk I.jrj 1

2 Rj



Pk I.jrj 1 2 Rˇ 3 / 0; 1 ;1 2 XP

1 2 1 2 Rj 3 / 1 kL2 L2x
@jˇ Pk I.jrj t

Pk I.jrj 1 2 Rˇ 3 / 1 1 L t Lx

.

X

2 k 1 kS Œk1 
k

1 2 @jˇ Pk I.jrj 1

2 Rj

3/

2 Rˇ

/ 3

kk2 ^0CO.1/

Pk I.jrj .

X kk2 ^0CO.1/

2k

k2

k

N Œ0

1

1 kS Œk1  k 2 kL1 L2x k 3 kL1 L2x t t

1 L1 t Lx

3/

159

5.2 Trilinear estimates for hyperbolic S -waves

.2

k2 _0

3 Y

k

i kS Œki  :

(5.56)

i D1

Hence, if the inner output has frequency  2k then we may assume that 1 has modulation . 2k . As usual, we apply (5.45). First, by the Strichartz component (2.14),

X


P0 I  Qk 1 Pk I Œjrj 1 2 jrj 1 3  N Œ0

kk2 ^0CC

X

.



P0 QkCC  Qk

1 Pk I Œjrj

k 2 2 Qk

1

1

1

2 jrj




3

kk2 ^0CC

X

.

 1 Pk I jrj

1

2 jrj

X

2

k k2 2

L2t L2x

1 2 k2 _0

k

i kS Œki 

i D1

kk2 ^0CC

.2

3 Y

1 ;1 2



3

kk2 ^0CC

.

0;

XP 0

3 Y

k

i kS Œki  :

i D1

For the second term, we can assume that 1 D Qk2 ^0CC by (4.40) of Lemma 4.13 and Lemma 4.11,

 

P0 I . 1 jrj 1 3 /jrj 1 2 N Œ0

X j k2 ^0

. 2k2 ^0 2 4 PQk2 _0 Qj 1 jrj

1,

1

see above. Then,

3




j k2 ^0CC

jrj

PQ0 Qk2 ^0CC

.2

k2 _0

.2

2k2 _0

3 Y

k

1

1 jrj

3




i kS Œki  :

i D1

Third, by (4.42) and Lemma 4.11,



P0 I Qk2 ^0CC 1 .jrj 1 2 /jrj

1

3



1 ;1 0; 2 2 _0

XP k

N Œ0

k

1



1 2 ;1 2 _0

0;

XP k

2 S Œk2  2 kS Œk2 

160

5 Trilinear estimates j

X

.

2

k2 ^0 4

Q

Pk2 _0 Qk2 ^0CC

1 jrj




3

1



Qj jrj k2 _0

.2

3 Y

k

0; 1 2 ;1 2 _0

XP k

j k2 ^0CC

1

2




0;

XP k

1 ;1 2

2

i kS Œki  :

iD1

Fourth, again by (4.42) and Lemma 4.11,

X 

P0 I .QkCC 1 /


1

@j Pk I.Rj

2 jrj

1

2 jrj

1



3/

kk2 ^0CC

X

.

X

`

2 4 kQ`

1k

kk2 ^0CC `C

1 2 ;1

0;

XP k

1



Q

Pk QkCC Rj X

.

k

1k

kk2 ^0CC

.

3 Y

k

0; 1 ;1 XP k 2 1

2

k k2 2

k

N Œ0



3

0; 1 2 ;1

XP k

1 2 k2 _0

2 kS Œk2  k 3 kS Œk3  2

i kS Œki  :

iD1

Fifth, with ` D k k2 ,

X 

P0 I  QkCC

1

1


@j Pk I.Rj

2 jrj

1



3/

kk2 ^0CC

X

.

k 2 2 QkCC

1


1

2 jrj

@j Pk I Rj

1

N Œ0




3

L2t L2x

kk2 ^0CC

.k

1 kL1 L2x

X

t

2

k 2

1 kL1 L2x

X

t

2 jrj

1

P



c

3 L2 L1 t x

c2Dk2 ;`

kk2 ^0CC

.k

X

Pc Rj

2

k 2

k3

 X

kPc Rj

2 2 kL4 L1

 X

x

t

c2Dk2 ;`

kk2 ^0CC

kP

c

2 3 kL4 L1

c2Dk2 ;`

.k

1 kL1 L2x

X

t

kk2 ^0CC

1

2. 2

2"/.k k2 /

k

 21

2 kS Œk2  k 3 kS Œk3 

t

x

 21

161

5.2 Trilinear estimates for hyperbolic S -waves

.2

3 Y

. 12 2"/k2 _0

k

i kS Œki 

i D1

which is admissible for small " > 0. The sixth and final term is estimated by means of (4.40) and Lemma 4.11:



P0 I QkCC

X

1 


1

2 jrj

@j Pk I.Rj

1



/ 3

kk2 ^0CC

.k

2k Pk QkCC 


X

1 kS Œk1 

1

@j Rj

N Œ0 1

2 jrj




3

kk2 ^0CC

.k

2 Pk QkCC Rj

X

1 kS Œk1 

k

2 jrj

1

kk2 ^0CC

.k

X

1 kS Œk1 

2k 2

k k2 2

k




3

1 ;1 2

0;

XP k

0; 1 2 ;1

XP k

2 kS Œk2  k 3 kS Œk3 

kk2 ^0CC

.2

1 2 k2 _0

3 Y

k

i kS Œki 

i D1

which concludes Case 5. Case 6: O.1/ D k1  k2 C O.1/  k3 C C. Since Lemma 4.23 implies that



P0 @ˇ Q>k2 . kQ>k2 1

. 2. 2



@j I Qˇj . 2 ; 3 / 1 2 L t Lx

k2 I Qˇj . 2 ; 3 / L2 L1 1 kL2 L2x 2

1


1

t

"/.k3 k2 /

t

3 Y

k

x

i kSŒki 

i D1

we may assume that precisely, P0 I @ˇ



1

D Qk2

1.

Next, we reduce matters to (5.1). More

1

 @j I Qˇj . 2 ; 3 /   D P0 I @ˇ 1 PQk2
1 @j I PQk2 Qˇj . 2 ; 3 /   C P0 I I 1
1 @j @ˇ PQk2 I Qˇj . 2 ; 3 / : 1


(5.57) (5.58)

The term in (5.57) satisfies the bounds (5.35) and (5.36), whereas (5.58) is ex-

162

5 Trilinear estimates

panded further:  P0 I I 1


1

 @j @ˇ PQk2 I Qˇj . 2 ; 3 /  D P0 I I 1
1 @j PQk2 I.jrj 1 C Rˇ

2@

ˇ

Rj

3

@ˇ Rj

2 Rj

2 Rˇ

Rj

3

2 jrj

1

(5.59)

3

 3/ :

(5.60)

The two terms in (5.60) are again controlled by (5.1). Consider the first term on the right-hand side of (5.59). Replacing
1 @j PQk2 by 2 k2 as usual, one obtains from Lemmas 4.13 and 4.11,

1 jrj 1 2 Rj 3 N Œ0 X

j k2 . 2k2 2 4 Qj jrj 1 2 0; 1 ;1 k 1 Rj 3 k 0; 1 ;1 XP k

j k2 CC

. 2k2 k

2k

0; 1 2 ;1

XP k

2k3 hk3 ik

2

2

XP 0

2

1 kS Œk1  k 3 kS Œk3 

2

which is more than enough. The second term in (5.59) is estimated similarly:

1 jrj 1 3 Rj 2 N Œ0 X

j k3 . 2k3 2 4 Qj jrj 1 3 0; 1 ;1 k 1 Rj 2 k 0; 1 ;1 XP k

j k3 CC

. 2k3 k

2k

0; 1 2 ;1

XP k

2k2 hk2 ik

2

3

XP 0

2

1 kS Œk1  k 2 kS Œk2 

2

which concludes Case 6. Case 7: k1 D O.1/  k3 C O.1/  k2 C C. This case is symmetric to the previous one. Case 8: k3 D O.1/; max.k1 ; k2 /  C. By Lemma 4.23,

 

ˇ 1 P I @ Q
@ I Q .I ; I /

0 j 2 3 k1 C.1 3"/k2 1 ˇj N Œ0

 

. P0 I @ˇ Qk1 C.1 3"/k2 1
1 @j I Qˇj .I 2 ; I 3 / 1 2 L L

t x

Q

k1 . 2 kQk1 C.1 3"/k2 1 kL2 L2x P0 I Qˇj .I 2 ; I 3 / 2 2 t L t Lx

1

. 2 2 "k2 2

k1 2

3 Y i D1

k

i kS Œki  :

163

5.2 Trilinear estimates for hyperbolic S -waves

A similar calculation shows that one can place Qk1 C.1 entire output. So it suffices to consider P0 Qk1 C.1

in front of the

  Qm

C

1


1

@j QQ m Pk Qˇj .

2;



3/

mC

L2t L2x

:

(5.89) First, by Lemma 4.19, and with M large but finite and

(5.87) .

k2 CO.1/ X kDk2 L0

k

p 1 kL1 t Lx

Qk
C Pk Qˇj .

2;

1 p

C



1 M

D 12 ,

3 / L2 LM t x

183

5.3 Improved trilinear estimates with angular alignment

.

k2 CO.1/ X

1

2m0 . 2

1 p/

k

k

1 kL1 L2x

22 2

t

2 "k2 jk2 j M

2

k

2 kS Œk2  k 3 kS Œk3 

kDk2 L0 3 Y

ı

k

i kS Œki  :

i D1

Since p > 2 one can take m0 large and negative to obtain the final estimate here. Second, again by Lemma 4.19, (5.88) .

k2 CO.1/ X

X

2

"m

kDk2 L0 mC



P0 Qm Qm .

k2 CO.1/ X

X

2

"m

1

1 p/

k

1

1


C

p 1 kL1 t Lx

@j QQ m Pk Qˇj . 2

k

2;

QQ m Pk Qˇj .



3/



2;

3 / L2 LM t x

kDk2 L0 mC

.

k2 CO.1/ X

2m0 . 2

k

22 2

3 Y

2 "k2 jk2 j M

2

L0

kDk2

k

i kS Œki   ı

i D1

L2t L2x

3 Y

k

i kS Œki 

i D1

and third, (5.89) .

k2 CO.1/ X

X

kQ>m

C

2 1 kL2 Lp x t

k

QQ m Pk Qˇj .

2;

kDk2 L0 mC 2

. 2jk2 j M

k2 CO.1/ X

1

2m0 . 2

1 p/

kDk2 L0

X

2

.1 "/m

k

m

1 kS Œk1  2 2

mC

QQ m Pk Qˇj . m0 . 12

.2

1 p/

2

2 jk2 j M



3 / L1 LM x t

k2 CO.1/ X

X

2

. 12 2"/m

k

2;

1 kS Œk1  2



3 / L2 LM t x "m

kDk2 L0 mC

QQ m Pk Qˇj . ı

3 Y

k

2;



3 / L2 L2 t x

i kS Œki 

i D1

where one argues as in the previous two cases to pass to the last line. Next,

184

5 Trilinear estimates

suppose the output is limited by Q0 . Then



P0 Q0 @ˇ I c .

k2 CO.1/ X

1

1


@j I c Qˇj .

X

Qm

1

1




2;

3/

N Œ0

@j Pk QQ m Qˇj .



2;

3 / L1 L2 t x

kDk2 L0 mC

.

k2 CO.1/ X L0

kDk2

.2

m0 . 21

X

kQm

1 kL2 Lp x t

k

2

kPk QQ m Qˇj .

2;

3 /kL2 LM x t

mC k2 CO.1/ X

1 p/

X

2

.1 2"/m

k

2 "m jk2 j M

1 kS Œk1  2

2

kDk2 L0 mC

Pk QQ m Qˇj . 1

. 2m0 . 2

k2 CO.1/ X

1 p/

kDk2

ı

3 Y

k

k

1 kS Œk1 

k

22 2

2 "k2 jk2 j M

2

k



2;

3 / L2 L2 t x

2 kS Œk2  k 3 kS Œk3 

L0

i kS Œki 

i D1

which is again admissible. On the other hand, assume now that 1 as we may suppose that k D k2 C O.1/ D k3 C O.1/, we obtain

 

P0 Q0 @ˇ I 1
1 @j I c Qˇj . 2 ; 3 / N Œ0



ˇ 1 c  P0 Q0 @ I 1
@j I Qˇj . 2 ; X

.

2

` 2

2


kQ`k2 CC1

X

C

2

` 2

t




1

@j Q` Qˇj .



2;

3 / L2 L1 t x

O.1/>`k2 CC1

C

X k2 CC k I Q>kCC I 1
1 @j Pk I.Rˇ 2 Rj 3 Rj 2 Rˇ 3 / N Œ0



ˇ 1 2 1 . P0 @ Q>k I Q>kCC I 1 Œ
@jˇ Pk I.jrj 2 Rj 3 / 

Pk I.jrj 1 2 Rˇ 3 / 0; 1 ;1 P .2

.2 ı

k 2

k

kQ>kCC

k2 m0 k 1 kS Œk1  k 2 kL1 L2x k 3 kL1 L2x 1 kS Œk1  2 2 t t

k

3 Y

2 X 1 2 1 2 Rj 3 / 1 kL2 L2x k
@jˇ Pk I.jrj t 1 Pk I.jrj 1 2 Rˇ 3 /kL1 t Lx

k

i kS Œki 

i D1

where the gain is a result of Bernstein’s inequality. Summation over 2 ; 3 is admissible here in view of Lemma 2.18. Hence, if the inner output has frequency  2k then we may assume that 1 has modulation . 2k . Next, we apply (5.45) and bound the six terms on the right-hand side of that identity one by one. Previously, we estimated the first term by means of the Strichartz component (2.14).

189

5.3 Improved trilinear estimates with angular alignment

However, this does not seem to yield the angular improvement so we use a different argument:

P0 I .Qk 1 Pk I Œjrj 1 2 jrj 1 3 / N Œ0 X

1 1

P0 Qa .Qk 1 Pk I Œjrj

1 . 2 jrj 3 / 0; 2 ;1

XP 0

akCC a

X

.

2 2 kQk

1 kL1 L2x t



2k Pk Qj jrj

1

2 jrj

1

2 jrj

1

3



L2t L2x

aj kCC

X

C

2k 2

a k 4



kQk

1 kS Œk1  Pk Qj

1

 jrj

3



j akCC

1 ;1 0; 2

XP k

:

(5.92) Lemma 4.11 was used to pass to the last line. By Corollary 4.10 one can continue as follows: X j j k2 3k2 2 2 k 1 kL1 L2x 2k ı2 3 2 2 k 2 kS Œk2  k 3 kS Œk3  . t

j kCC

X

C

2k k

j

1 kS Œk1 

ı2

k2 3

2

3k2 2

j

22 k

2 kS Œk2  k 3 kS Œk3 

(5.93)

j kCC

ı

3 Y

k

i kS Œki  :

i D1

Moreover, Corollary 4.10 shows that this bound allows for summation over the caps. For the second term, we can assume that 1 D Qk2 CC 1 , see above. Then, by Corollary 4.15 as well as Corollary 4.10, and some large constant M ,

X  

1 1

P0 I . 1 jrj Pk3 ;3 3 /jrj Pk2 ;2 2 N Œ0

2 ;3 2Cm0 dist.2 ;3 /2m0

X

. 2k2 jm0 j

j

2

k2 4

j k2 CC

 X

Q

P0 Qj

1 jrj

1

Pk3 ;3

3 2Cm0

.2

m0 M

jm0 j

X j k2 CC

j

2

k2 4

2

k2 j 3

2

k2 2

j 2

2 2




2 3 0; P X0

k2

3 Y i D1

1 2 ;1

k

 21

jrj

i kS Œki 

1



2 S Œk2 

190

5 Trilinear estimates

.ı

3 Y

k

i kS Œki  :

i D1

Third, by Lemma 4.13 and (4.33) of Corollary 4.10,

X 

P0 I Qk2 CC 1 .jrj 1 Pk2 ;2 2 /jrj

1



3

Pk3 ;3

N Œ0

2 ;3 2Cm0 dist.2 ;3 /2m0

X

.

j

2

k2 4

Q

P0 Qk2 CC

X

1

1 jrj




3

Pk3 ;3



Qj jrj

1




2

Pk2 ;2

1 ;1 0; 2

XP 0

2 ;3 2Cm0 dist.2 ;3 /2m0

j k2 CC

1 2 ;1

0;

XP k

2

.

 X

kPQ0 Qk2 CC .

1 jrj

1

Pk3 ;3

3 /k

3 2Cm0

X

.

 X

X

1 jrj

1

Pk3 ;3

3 2Cm0

`k2 CC

.

kPQ0 Q` .

`

ı2 2 2

 12

2 1 ;1 0; 2

XP 0

2 3 /k 0; 1 ;1 XP 2

k

2 kS Œk2 

 12

k

2 kS Œk2 

0

k2 ` 3

2

k3 2

2

k3

3 Y

k

i kS Œki  . ı

i D1

`k2 CC

3 Y

k

i kS Œki  :

i D1

The summation over the caps was carried out explicitly for the second and third terms since it requires some care. Fourth, by (4.42) and Corollary 4.10,

 

1 1 P I .Q /
@ P I.R jrj /

0 j k j 2 3 kCC 1 N Œ0

X `  

. 2 4 kQ` 1 k 0; 1 ;1 PQk QkCC Rj 2 jrj 1 3 0; 1 ;1 2 2 XP k

`kCC

X

.ı

X

`

24 k

`kCC mkCC

. ı2

k2 4

3 Y

k

XP k

1

1k

0; 1 ;1 XP k 2 1

2

k j 3

2

k2 2

m

22k

2 kS Œk2 

jrj

i kS Œki  :

i D1

Since k D k1 C O.1/ D k2 C O.1/, the fifth term



P0 I  QkCC 1
1 @j Pk I.Rj 2 jrj

1



3/

N Œ0

1



3 S Œk3 

191

5.3 Improved trilinear estimates with angular alignment

is bounded exactly like the first, see (5.92), (5.93). The sixth and final term is estimated by means of (4.40) and Corollary 4.10:

 

P0 I QkCC 1 
1 @j Pk I.Rj 2 jrj 1 3 / N Œ0

2 kPk QkCC 
@j .Rj 2 jrj 1

X


. k 1 kS Œk1  2k Pk Qm Rj 2 jrj 1 3 .k

k

1 kS Œk1 

1

X

1 kS Œk1 

1 2 ;1

k

0; 1 2 ;1

XP k

mkCC

. ık

3 /k 0; XP

2k 2

k m 3

m

222

k2 2

k

2 kS Œk2  k 3 kS Œk3 

mkCC

. ı2

3 Y

k2

k

i kS Œki 

i D1

as claimed. We now repeat this analysis for the case of alignment between 1 and 3 (the remaining case being symmetric). We again begin with the reduction of various modulations. Using the notation of Lemma 5.1, if A0 D I c , then A1 D I c . By (4.52) of Lemma 4.17 and with 12 D p1 C q1 where q < 1 is very large,

X  c 

c ˇ 1 P I @ I r P
@ I Q . ; P /

0 t;x k1 ;1 1 j 2 ˇj k3 ;3 3 N Œ0

1 ;3 2Cm0 dist.1 ;3 /2m0

X

.

X

2



P0 Qm QQ m r t;x Pk1 ;1

"m

1 ;3 2Cm0 m0 dist.1 ;3 /2m0

.

X

2.1

m0

m0 . 21

.2

Qˇj .

X

"/m

kQQ m Pk1 ;1

1 ;3 2Cm0 dist.1 ;3 /2m0 1 p/

X

2

1


.1 "/m

t

Pk I Qˇj .

 X

m0

1 kL2 Lp x

kPk1 ;1 QQ m

1 2Cm0

 X

IPk Qˇj .

2

1

@j Pk I

2 ; Pk3 ;3



/ 3

L2t L2x

k

2 ; Pk3 ;3

2 1 kL2 L2 t x

 21

2 ; Pk3 ;3



3 / L1 Lq x t 1

2. 2

2 q /k

 21

2

/ 3 L2 L2 t

x

3 2Cm0 1

. jm0 j 2m0 . 2

1 p/

2.1

2 q /k

k

1 kXP 0;1 0

";2

kPk2

2 kS Œk2  kPk3

3 kS Œk3 

192

5 Trilinear estimates

.ı

3 Y

k

i kS Œki  :

i D1

Hence, we can assume that A0 D I as well as A1 D I . If A2 D I c , then also A3 D I c and

 

P0 I @ˇ IPk1 ;1 1
1 @j I Qˇj .I c 2 ; I c Pk3 ;3 3 / N Œ0 X 1 c c

. IPk1 ;1 1
@j IPk Qˇj .I 2 ; I Pk3 ;3



3 / L1 L2 t x

mk2 CC

. kPk1 ;1

1 kL1 L2x

. kPk1 ;1

1 kL1 L2x

t

2

t

k

IPk Qˇj .I c 2 ; I c Pk3 ;3 3 / L1 L1 t x X

k2 2 Qˇj .Qm 2 ; QQ m Pk3 ;3 3 / L1 L1 x

t

mk2 CC

. kPk1 ;1

X

1 kL1 L2x

2m

t

2k2

Q

kQm

kPk3 ;3 Qm 2 kL2 L1 x

2m

2

t

3 kL2 L1 x t

mk2 CC

. 2m0 Ck2 kPk1 ;1

X

1 kL1 L2x t

k2

2.1 "/m .1 2"/k2

2

mk2 CC

kQm . 2m0 kPk1 ;1

2k

XP k

1 2 C";1 ";1 2

1 kL1 L2x k 2 k t XP k

kPk3 ;3 QQ m

1 C";1 ";2 2 2

kPk3 ;3

3k

3k

XP k

XP k

1 C";1 ";1 2 3

1 2 C";1 ";2 3

:

Summing over the caps 1 ; 3 and k1 D O.1/, k2 D k3 C O.1/ yields the desired gain. For 1 one uses Lemma 2.18. As before, this reduces us to the trilinear nullform expansion (5.45). By the estimate (5.92), it suffices to consider PkCC 1 if the inner output has frequency  2k . Beginning with the first term on the right-hand side of (5.45), one has

P0 I  Qk Pk1 ;1

X

Pk I Œjrj

1

1 Pk I Œjrj

1

1

Pk3 ;3




3

Pk3 ;3

3

2 jrj

1

2 jrj

1

1 ;3 2Cm0 dist.1 ;3 /2m0

.

X

X

akCC

1 ;3 2Cm0 dist.1 ;3 /2m0



P0 Qa Qk Pk1 ;1




N Œ0

0; 1 2 ;1

XP 0

193

5.3 Improved trilinear estimates with angular alignment a

X

.

2 2 2k

X

2 1 kL1 L2

kQk Pk1 ;1

x

t

1

aj kCC

X 

Pk Qj jrj

1

 21

2 jrj

1

Pk3 ;3

3

1

X

C

2

3k 4

a

24 k

 21



2

2

L t L2x

1 kS Œk1 

j akCC

X 

Pk Qj jrj

1

2 jrj

1

Pk3 ;3

3

 12 

2 1 :

;1 0; 3 P 2

(5.94)

Xk

Corollary 4.8 was used to pass to the last line. By Lemma 2.18 and Corollary 4.10 one can continue as follows: .ı

j

X

22 k

1 kL1 L2x t

2k 2

j

k2 3

3k2 2

2

k

2 kS Œk2  k 3 kS Œk3 

j kCC

X

Cı

j

2k k

1 kS Œk1 

k2 3

2

2

3k2 2

j

22 k

2 kS Œk2  k 3 kS Œk3 

j kCC

ı

3 Y

k

i kS Œki  :

(5.95)

i D1

Moreover, Corollary 4.10 shows that this bound allows for summation over the caps. For the second term, we can assume that 1 D Qk2 CC 1 , see above. Then, by Lemma 4.13 as well as Corollary 4.10,



P0 I .Pk1 ;1

X

1 jrj

1

Pk3 ;3

3 /jrj

1



2

N Œ0

1 ;3 2Cm0 dist.1 ;3 /2m0

. 2k2

j

X

2

k2 4

1 ;3 2Cm0 dist.1 ;3 /2m0

j k2 CC

Q

P0 Qj Pk1 ;1 .ı

X j k2 CC

j

2

X

k2 4

2

1 jrj

k2 j 3

2

k2 2

1

Pk3 ;3

j

22 2

k2




3

0; XP 0

3 Y i D1

k

1 2 ;1

jrj

i kS Œki  . ı

1

3 Y i D1



2 S Œk2 

k

i kS Œki  :

194

5 Trilinear estimates

Third, by Lemma 4.13 and (4.33) of Corollary 4.10,

X 

P0 I Qk2 CC Pk1 ;1 1 .jrj 1 2 /jrj

1

Pk3 ;3

3



1 ;3 2Cm0 dist.1 ;3 /2m0 j

X

.

k2 4

2

Q

P0 Qk2 CC Pk1 ;1

X

1 ;3 2Cm0 dist.1 ;3 /2m0

j k2 CC

1

Qj .jrj

1

1 jrj



2/

N Œ0

Pk3 ;3

3




1 ;1 0; 2

XP 0

1 2 ;1

0;

XP k

2

X

.

Q

P0 Qk2 CC Pk1 ;1

1 jrj

1

Pk3 ;3




3

Q

P0 Q` Pk1 ;1

1 jrj

1

Pk3 ;3




3

1 ;3 2Cm0 dist.1 ;3 /2m0

X

.

X

`k2 CC

`

X

.

1 ;3 2Cm0 dist.1 ;3 /2m0

ı2 2 2

k2 ` 3

2

k3 2

2

k3

3 Y

k

i kS Œki  . ı

i D1

`k2 CC

3 Y

k

1 ;1 0; 2

k

2 kS Œk2 

0; 1 2 ;1

k

2 kS Œk2 

XP 0

XP 0

i kS Œki  :

i D1

Fourth, by Lemma 4.13, Cauchy–Schwarz applied to the cap-sum, and Corollary 4.10, X  

1 1

P0 I .QkCC Pk1 ;1 1 /
@j Pk I.Rj 2 jrj Pk3 ;3 3 / N Œ0

1 ;3 2Cm0 dist.1 ;3 /2m0

. jm0 j

`

X

2 4 kQ`

1k

`kCC

0;

XP k

1 ;1 2

1

 X 

Q

Pk QkCC Rj

2 jrj

1

Pk3 ;3

3

3 2Cm0

X

.ı

X

`

24 k

`kCC mkCC

. ı2

k2 4

3 Y

k

1k

0; 1 ;1 XP k 2 1

2

k m 3

2

k2 2

m

22k

2 kS Œk2  kjrj

i kS Œki  :

i D1

Since k D k1 C O.1/ D k2 C O.1/, the fifth term



P0 I  QkCC 1
1 @j Pk I.Rj 2 jrj

1 

2 1 2

P 0; 2 ;1

1



3/

N Œ0

Xk

1

3 kS Œk3 

195

5.3 Improved trilinear estimates with angular alignment

is bounded exactly like the first, see (5.92), (5.93). The sixth and final term is estimated by means of Corollary 4.15 and Corollary 4.10: X  

1 1

P0 I QkCC Pk1 ;1 1 
@j Pk I.Rj 2 jrj Pk3 ;3 3 / N Œ0

1 ;3 2Cm0 dist.1 ;3 /2m0

. jm0 jk

1 kS Œk1 

X

2k

mkCC

 X

Pk Qm 


1

2 jrj

@j Rj

1

Pk3 ;3

3 2Cm0

. ık

1 kS Œk1 

X

2k 2

k m 3

m

222

k2 2

k




2 3 0; P Xk

1 2 ;1

 12

2 kS Œk2  k 3 kS Œk3 

mkCC

. ı2

k2

3 Y

k

i kS Œki 

i D1

as claimed. The other two types of trilinear null-forms are similar and left to the reader.  Remark 5.12. The proof of the preceding estimates actually leads to a slightly better result: letting P0 F .Pk1 1 ; Pk2 2 ; Pk3 3 / be a frequency localized trilinear null-form as above, then given any ı > 0, there exists some `0  100 such that we can write P0 F .Pk1 1 ; Pk2 2 ; Pk3 3 / D F1 C F2 where F1 is a sum of energy, XP s;b;q , as well as wave-packet atoms of scale `  `0 (where scale refers to the size 2` of the caps  used), with the bound 3 Y

kF1 kN Œ0 . w.k1 ; k2 ; k3 /

kPkj

j kS Œkj 

j D1

and universal implied constant (independent of ı), while we also have kF1 kN Œ0 . ıw.k1 ; k2 ; k3 /

3 Y

kPkj

j kS Œkj  :

j D1

The reason for this is that whenever a wave-packet atom of extremely fine scale is being used to estimate some constituent of P0 F , one gains a small exponential power in that scale.

6

Quintilinear and higher nonlinearities

Here we detail the estimates needed in order to control the higher order error terms generated by the process described in Chapter 3. This section is quite technical but the main point here is that the higher order terms, while still somewhat complicated, are much easier to estimate than the trilinear null-forms, and only require a very mild null-structure. We start with the lowest order errors, of quintilinear type. These are either of first or second type, see the discussion in Chapter 3. We commence with those of the first type, which can be schematically written as h 
i rx;t r 1 R r 1 r 1 Qj . ; / where not both ;  are simultaneously zero. Assume that  D 0;  ¤ 0, the remaining cases being treated analogously. The following lemma is then representative for the higher order errors, for a universal ı > 0. Lemma 6.1. We have the estimates

h

rx;t P0 0 r 1 Pr1  R0 Pk1 1 r 1 Pr2 Pk2 . 2ıŒminj ¤0 frj ;kj g

1

2r

maxj ¤0 frj ;kj g

4 Y

Pr3 Qj k .Pk3

kPki

i kS Œki  ;

3 ; Pk4

r1
10:

N Œ0

198

6 Quintilinear and higher nonlinearities

All implied constants are universal. Proof. All three inequalities are proved similarly, and we treat here the high-low case in detail, i.e., the first of them. We first deal with the elliptic cases: (i): Output in elliptic regime. This is the expression (we have included the gratuitous cutoff PŒ 5;5 in light of r1 < 10) h Q P0 0 r 1 Pr1 >10 5;5  R0 Pk1 1 r 1 Pr2 Pk2 2 r 1 Pr3 Qj k .Pk3 h X D rx;t PŒ 5;5 Ql P0 0 r 1 Pr1

rx;t PŒ

3 ; Pk4


i

4/

l>10



R0 Pk1

1

1r

Pr2 Pk2

1

2r

Pr3 Qj k .Pk3


i

3 ; Pk4

4/

:

Now distinguish between further cases: (i1): maxfk1 ; : : : ; k4 g  l, R0 Pk1 1 D R0 Pk1 Ql 10 2 (the other cases being similar), so we now reduce to estimating X

rx;t PŒ

5;5 Ql

h P0

1

0r

 Pr1 R0 Pk1 Q10

Pk2 Q>l

10

2r

1

Pr3 Qj k .Pk3

3 ; Pk4


i

4/

where we also make the further assumptions of case (i1). Freezing l for now, we estimate this expression as follows: First, note that

r

1

Pr2 Pk2 Q>l

10

2r

1

Pr3 Qj k .Pk3

3 ; Pk4




4/

.1 /.k2 l/ Œminfr2;3 ;k2;3;4 g maxfr2;3 ;k2;3;4 g

.2

2

1

L2t HP x2 4 Y

kPkj

j kS Œkj  :

j D2

This follows by straightforward usage of Bernstein’s inequality and the definition of SŒk, as well as exploiting the null-structure of Qj k . Furthermore, we have kR0 Pk1 Q 0 is as in the definition of S Œk, which implies that kR0 Pk1 Ql

1

1 kL1 L2x t

 Pr1 R0 Pk1 Ql

10

. 2.l

2r

1

1

l

2

100

 Pr1 R0 Pk1 Ql

10

2r

k1 2

kPk1 kS Œk1  :

1

Pr3 Qj k .Pk3

3 ; Pk4

minfminfr1;2 g k1 ;0g 2

r1

k1 /

100

4/

1r

1

XP 0

1 2 C; 1 ;2

Pr2




/

4

kR0 Pk1 Q 0, there exist M1 D M1 .C0 ; "0 / many intervals Ij as in (7.2) with the following property: for each Ij D .tj ; tj C1 /, there is a decomposition .j / L

jIj D

C

.j / NL ;



.j / L

D0

which satisfies X

kPk

.j / 2 NL kS Œk.Ij R2 /

< "0

(7.4)

 M2 .C0 ; "0 /;

(7.5)

k2Z

krx;t

.j / P L kL1 t H

1

1

where the constant M2 D M2 .C0 ; "0 / satisfies M2 . C03 "0 M with M  100 as .j / .j / in the preceding lemma. Moreover, Pk NL and Pk L are Schwartz functions for each k 2 Z. We also have the bounds

rx;t Pk

.j /

P 1 L Hx

.j /

NL S Œk.Ij R2 /

C Pk

. ck

(7.6)

with implied constant depending on C0 , provided ck is a sufficiently flat frequency envelope with kPk kS Œk  ck . Proof. The nent P0 ˛ .

˛

satisfy the system (1.12)–(1.14). Consider the frequency compo-

Case 1: The underlying time interval I D . T0 ; T1 / is very small, say jI j < "1 with an "1 that is to be determined. As explained in Section 2.5 one uses the div-curl system (1.12), (1.13) in this case. Schematically, this system takes the form   @ t P0 D rx P0 C P0 r 1 . 2 / where we suppress the subscripts and also ignore the null-structure in the nonlinearity. Therefore, kP0 .t/

P0 .0/kL2x

Z t

 rx P0 .s;
/ ds 0

L2x

Z t 

C P0 r 0

1

.

2



/ .s;
/ ds

L2x

: (7.7)

229

7.1 A blow-up criterion

For all j 2 Z define aj WD sup 2

jk j j M

k2Z

kPk kS Œk.I R2 / . C0 :

(7.8)

 "1 kP0 kS Œ0.I R2 /  a0 "1 :

(7.9)

Clearly,

Z t

rx P0 .s;
/ ds

L2x

0

Lemma 7.4 implies

Z t 

 1

P0 r 1 . 2 / .s;
/ ds 1 . C02 a0 "1

2 L t .I ILx / 0

Z t 

3 

2 2 P0 r 1 . 2 / .s;
/ ds 2 . C a "

0 0 1 2 L t .I ILx /

0

1 M

1 M

; :

From the div-curl system (1.12) and (1.13), k@ t P0 kL2 .I IL2x / t

1

  krx P0 kL2 .I IL2x / C P0 r t

.

2

1  / L2 .I IL2 / . C02 a0 "12 t

x

1 M

;

where we assumed without loss of generality that C0  1. We claim that these bounds imply that

Z t

1 2 P Œ r . /  "0 a0 (7.10)

0 2 S Œ0.I R /

0

provided "1 was chosen sufficiently small depending on "0 . To see this, let I 0 WD Œ T 0 ; T   I D . T0 ; T1 / and pick any smooth bump function  supported in I so that  D 1 on I 0 and with 0    1. Moreover, let Q be any smooth compactly supported function with Q D 1 on I (the choice of this function does not depend on I 0 ). Then define Z t h i Q .t / WD .t Q / P0 .0/ C .s/@s P0 .s/ ds : 0

By construction, Q is a global Schwartz function so that Q D by the preceding bounds, k Q kL2 L2x C k@ t Q kL2 L2x  "0 a0 t

t

on I 0 . Moreover,

230

7 Some basic perturbative results

provided "1 was chosen small enough (this smallness does not depend on the choice of I 0 ). This now implies that k Qk

1 ;1 0; 2

XP 0

 "0 a0 ;

whence (7.10). In view of (7.9), (7.10) and (7.7), kP0 .t / We now define P0 t D 0. Clearly

L

P0 .0/kS Œ0.I R2 /  "0 a0 :

to be the free wave with initial data .P0 .0/; 0/ at time kP0 rx;t

P0

(7.11)

L

P0

L kL1 HP

1

. P0 .0/ L2

t

.0/  "0 a0 : L S Œ0.I R2 /

x

The second inequality here implies that kP0

P0

L kS Œ0.Ij R2 /

 "0 a 0 :

Thus in the present situation, we approximate P0 by the free wave P0 L just described and the bounds which we just obtained should be viewed as versions of (7.4) and (7.5) on a fixed dyadic frequency block. Several remarks are in order: First, we shall of course need to construct L and NL for each such dyadic block Pk , and then obtain the global bounds required by (7.4) and (7.5). In this regard, any bound depending on aj can easily be square-summed since X X aj2  C.M / kPk k2S Œk.I R2 /  C.M / C02 : j

k2Z

Second, the construction we just carried out applies to Pk equally well provided jI j  2 k "1 . Moreover, I can be any time interval on which is defined — with any t0 2 I playing the role of t D 0 — and we shall indeed apply this exact same procedure to those intervals Ij which we are about to construct provided they satisfy this length restriction. Case 2: The underlying time interval I D . T0 ; T1 / satisfies jI j > "1 with "1 as in Case 1. To construct the Ij , we shall use the wave equation (1.14) for ˛ . By means of Schwartz extensions and successive Hodge type decompositions of the ˛ -components as explained above, the nonlinearity can be written as 

˛

D F˛ . / D F˛3 . / C F˛5 . / C F˛7 . / C F˛9 . / C F˛11 . /

(7.12)

231

7.1 A blow-up criterion

where the superscripts denote the degree of multi-linearity, see Chapter 3. The contribution of the trilinear null-form F˛3 . / here is in a sense the principal contribution, and causes the main technical S 1 difficulties. We now make the following claim: There exists a cover I D jMD1 Ij by open intervals Ij , 1  j  M1 , M1 D M1 ."0 /, such that X

P` F˛ . / 2 < "0 C06 : (7.13) max N Œ`.I R2 / 1j M1

j

`2Z

This will be enough to ensure the conditions of the lemma, if we replace "0 by C0 6 "0 , whence the number of intervals will then also depend on C0 , the bound on k kS . We verify this for each of the different types of nonlinearities appearing on the right-hand side of (7.12) starting with the trilinear ones. Let us schematically write anyone of these trilinear expressions in the form r t;x Œ 1 jrj 1 I c Q. 2 ; 3 / or r t;x ŒR 1 jrj 1 I Q. 2 ; 3 /, where Q stands for the usual bilinear null-forms and R for a Riesz transform (each of the i D but it will be convenient to view these Pinputs as independent). Break up the inputs into dyadic frequency pieces: i D ki Pki i for i D 1; 2; 3. In view of our discussion in Section 5.3, it suffices to consider the high-low-low case jk2 k3 j < L, k2 < k1 C L for some large L D L."0 /. In addition, it suffices to restrict attention to frequencies k > k2 L0 where Pk localizes the frequency of Q and L0 D L0 .ı/ is large. Finally, one can assume angular separation between the inputs: there exists m0 D m0 ."0 /  1 so that (7.13) reduces to the estimates X 

X max r t;x P` Pk1 ;1 1 jrj 1 Pk I c

1j M1

`

k;k1 ;k2 ;k3 1 ;2 ;3

Q.Pk2 ;2 max

1j M1

X

`

X

2 ; Pk3 ;3



2 3/

N Œ`.Ij R2 /

 r t;x P` Pk1 ;1 R

1 jrj

1

< "0 C06

(7.14)

Pk I

k;k1 ;k2 ;k3 1 ;2 ;3

Q.Pk2 ;2

2 ; Pk3 ;3



2

3/

N Œ`.Ij R2 /

< "0 C06

(7.15)

where the sums extend over integers k; `; k1 ; k2 ; k3 as specified above and further jk1 `j < L, as well as over caps 1 ; 2 ; 3 2 Cm0 with dist.i ; j / > 2m0 for i ¤ j . Let us first consider the case where the entire output is restricted by Qk0

for some (very small) ı3 > 0, we have

rx;t P>k 0

.j /

P 1 L Hx

C

X

Pk k>k0

where the implied constant depends on k kS .


21 .j /

2 NL S Œk.Ij R2 /

. ı3

238

7 Some basic perturbative results

Remark 7.8. The preceding proof can be easily modified to give the following result that will be important later: Let be the gauged derivative components of an admissible wave map. Assume that we have an a priori bound of the form X

X

k1 >k2

1;2 2Cm0 dist.1 ;2 /&2

2

k2

2 kPk1 ;1 Pk2 ;2 kL 2

t;x

m0

C

X 2.1

"/l . 12 "/k

kPk Ql kL2

2

t;x

<  (7.19)

k k2 one has X 1;2 2Cm0 dist.1 ; 2 /&2

Pk1 ;1 Pk2 ;2

D fk1 ;k2 C gk1 ;k2

m0

Pk Q>k

D hk C ik

with for some positive integer  X

2

k2

2 kfk1 ;k2 kL C 2 t;x

k1 >k2

X k1 >k2

2

k2

2 kgk1 ;k2 kL 2 t;x

X

2.1

"/l . 21 "/k

kQl hk kL2

2

0 is some small constant. Also, write jIj D is a number C1 D C1 . L / < 1 with the property that kPk kS Œk.Ij R2 /  C1 ck ;

L

C

NL .

Then there

8 k 2 Z:

Proof. We prove this by splitting the interval Ij into a finite number of smaller intervals depending on L . Thus we shall write Ij D [i Jj i for a finite number of smaller intervals depending on L . The exact definition of these intervals will be given later in the proof. On each Jj i , we now run a bootstrap argument, commencing with the bootstrap assumption: kPk kS Œk.Jj R2 /  A.C0 /ck Here A.C0 / is a number that depends purely on the a priori bound we are making on the wave map. We shall show that provided A.C0 / is chosen large enough, the bootstrap assumption implies the better bound kPk kS Œk.Jj R2 / 

A.C0 / ck : 2

We prove this for each frequency mode. By scaling invariance, we may assume k D 0. As before, one needs to distinguish between jJj j < "1 and the opposite

240

7 Some basic perturbative results

case, where "1 is chosen sufficiently small. In the former case, one directly uses the div-curl system @ t D rx C r 1 . 2 / as in the previous section to obtain the desired conclusion for P0 . Thus we can assume that the interval satisfies jJj j  "1 , which means we can control
P0 .t0 ;
/; P0 @ t .t0 ;
/ for some t0 2 Jj via




P0 .t0 ;
/; P0 @ t .t0 ;
/ 2 P 1 . A1 .C0 /c0 L H x

for some constant A1 .C0 /, which is explicitly computable, independently of A.C0 /. Passing to the wave equation D P0 F˛ . / D

P0

5 X

P0 F˛2iC1 . /

i D1

via Schwartz extensions and Hodge decompositions as before, we first consider the principal terms P0 F˛3 . /. These terms can be schematically written as P0 rx;t



r

1

2

.

 /:

More accurately, they are of the form  r t;x P` Pk1 ;1 1 jrj 1 Pk I c Q.Pk2 ;2  r t;x P` Pk1 ;1 R 1 jrj 1 Pk I Q.Pk2 ;2



2 ; Pk3 ;3

3/

2 ; Pk3 ;3

3/



with a Riesz projection R and a null-form Q. Substituting the decomposition D L C NL into the inner null-form yields P0 rx;t



r

1

2

.

  / D P0 rx;t r 1 .  C P0 rx;t r 1 .

2 L/



L



NL /

C P0 rx;t



r

1

.

2 NL /

 :

Note that the last term automatically has the desired smallness property if we choose "0 smaller than some absolute constant. Indeed, by (7.4), and the trilinear estimates of Chapter 5,



P0 rx;t r

1

.

2 NL /



.

N Œ0 2 "0 sup 2 0 jkj kPk k2Z

kS Œk . "20 A.C0 /c0  A.C0 /c0

241

7.1 A blow-up criterion

for small "0 . Next, for the mixed term P0 rx;t Œ r 1 . L NL /, choosing "0 sufficiently small (depending on C0 ), we can arrange in light of Lemma 7.6 and the trilinear estimates



P0 rx;t r

1

.

L

1



NL /

N Œ0

. A.C0 /C03 "0

1 M

c0  A.C0 / c0 :

The first term P0 rx;t



r

1

.

2 L/



requires a separate argument. In fact, we treat this term by decomposing the interval Ii into smaller ones. In order to select these intervals, first note that upon localizing the frequencies of the inputs according to   P0 rx;t Pk1 r 1 Pk .Pk2 L Pk3 L / one obtains from the trilinear bounds of Chapter 5



P0 rx;t Pk

1

r

1

Pk .Pk2

L Pk3



L/

N Œ0  jk1 j

 "0 2

kPk1

1 kS Œk1 

 A.C0 / c0

in the following two cases: k1 ; k2 ; k3 fall outside the range (5.43) (the high-lowlow case), or, if they do fall in the range (5.43), then k  k2 L0 . Here L and L0 are large constants depending on C0 ; "0 , due to the bounds on L from Lemma 7.6. Thus, denoting by  0 P0 rx;t r 1 . L2 / the sum over all frequency interactions described by these conditions, one then obtains the estimate

 

P0 rx;t r 1 . 2 / 0  A.C0 / c0 : L N Œ0 Employing the notations of Section 5.3, it thus suffices to consider the sum of expressions X00 k1 ;k2 ;k3 2Z

k2 CO.1/ X

 P0 rx;t Pk1 r

1

Pk .Pk2

L Pk3



L/

kDk2 L0

where, of course, the implied constants may be quite large depending on C0 ; "0 . Furthermore, by the results of that section, we may assume that the inputs have

242

7 Some basic perturbative results

pairwise angular separation on the Fourier side, and in particular we make this assumption for the free wave inputs Pk2 L and Pk3 L . Thus we have now reduced ourselves to estimating X00

X

k1 ;k2 ;k3 2Z

1 ;2 ;3 2Cm0 maxi¤j dist.i ;j />2m0

k2 CO.1/ X

1

 P0 rx;t Pk1 ;1 r

Pk .Pk2 ;2

Pk3 ;3

L

 :

L/

kDk2 L0

The next step is to exploit the dispersive properties of the expression 1

r

Pk .Pk2 ;2

First, due to the energy bound for

Pk3 ;3

L /:

there exists some finite set A  Z so that

X00

X 1 ;2 ;3 2Cm0 maxi¤j dist.i ;j />2m0 k2 CO.1/ X

L,

L

k1 ;k2 ;k3 2Z k2 62A



P0 rx;t Pk ; r 1 1

1

Pk .Pk2 ;2

L

Pk3 ;3



L/

N Œ0

kDk2 L0

 "0 2

0 jk1 j

kPk1 kS Œk1  :

On the other hand, assume now that k2 2 A and consider r

1

Pk .Pk2 ;2

L

Pk3 ;3

L/

where k; k3 are chosen as in (5.43). Note that the set A depends on the dyadic frequency of the output, in this case frequency 20 . Changing the frequency localizations of the output amounts to a rescaling of A. Nonetheless, one has the following estimates which are independent under rescaling:

1

r Pk .Pk ; L Pk ; L / 1 1 < C3 . L ; k2 / 2 2 3 3 L L t

In particular, X

r k2 2A

1

Pk .Pk2 ;2

L Pk3 ;3



x

L / L1 L1 t x

< C4 .

L/

< 1:

243

7.1 A blow-up criterion

To prove these bounds, set k2 D 0 by scaling invariance. But then Pk2 L .tj ;
/ is a Schwartz function in the x-variable. Using the angular separation of the inputs it is now straightforward to see that

1

r Pk .Pk ; L Pk ; L / 1 1 < C3 . L ; k2 /: 2 2 3 3 L L t

x

Indeed, this follows from stationary phase and the angular separation of the inputs. We now define the intervals Jj by requiring that X

X

1 ;2 ;3 2Cm0 k2 2A maxi¤j dist.i ;j />2m0 jk2 k3 j 0 such that min.T0n ; T1n / > T0 for all sufficiently large n and lim sup k ˛n kS.. T0 ;T0 /R2 /  C.V / < 1: n!1

Furthermore, there is a constant C1 .V / with the following property: defining the frequency envelope .n/

ck WD max

˛D0;1;2

X

2

jk `j

kP`

n 2 ˛ .0;
/kL2x


12

`2Z

for sufficiently small fixed  > 0, one has for all k 2 Z and all large n max kPk

˛D0;1;2

n ˛ kS Œk.. T0 ;T0 /R2 /

.n/

 C1 .V /ck :

Finally, the wave map propagations of the ˛n converge on fixed time slices t D t0 2 Œ T0 ; T0  in the L2 -topology, uniformly in time. The proof of this lemma will occupy the remainder of this section. Before we begin with the proof, we discuss some related results and implications of Lemma 7.10. Most fundamental is the following stability result:

2

In the usual sense that ˛n j t Dconst is Schwartz on R2 .

7.2 Control of wave maps via a fixed L2 -profile

245

Proposition 7.11. Let u W Œ T0 ; T1   R2 ! H2 be an admissible wave map with gauged derivative components denoted by . Assume that k kS.Œ T0 ;T1 R2 / D A < 1. Then there exists "1 D "1 .A/ > 0 with the following property: any other admissible wave map v defined locally around t D 0 and with gauged derivative components Q satisfying k .0/ Q .0/k2 < " < "1 extends as an admissible wave map to Œ T0 ; T1  and satisfies k Q kS.Œ T0 ;T1 R2 / < A C c."1 / where c."/ ! 0 as " ! 0. We also have the local Lipschitz type bound kQ

kS.Œ

T0 ;T0 R2 /

. .0/

Q .0/

L2

for Q satisfying the above proximity condition, with implied constant depending on k kS.Œ T0 ;T0 R2 / . Proof. The proof will be given in Chapter 9.5, as it follows directly from the proof of Proposition 9.12.  As a consequence, one has the following important continuation result. Corollary 7.12. Let f n g1 nD1 be a sequence of Coulomb components of admissible wave maps un W I ! H2 where I some fixed nonempty closed interval such that for some t0 2 I one has

lim

n!1

n ˛ .t0 ;
/

V˛ L2 D 0 x

with V˛ 2 L2 .R2 / as well as sup k n

n

kS.I R2 / < 1:

Then there exists a true extension IQ of I (meaning that it extends by some positive distance beyond the endpoints of I in sofar as they are finite) to which each un can be continued as an admissible wave map provided n is large. Proof. By Proposition 7.11 we can define limn!1 ˛n .t; :/ in the L2 -sense for t an endpoint of I . By Lemma 7.10 the ˛n extend beyond the (finite) endpoints for n large enough.  We can use the preceding results to define wave maps with L2 data at the level of the Coulomb gauge.

246

7 Some basic perturbative results

Definition 7.13. Assume we are given a family fV˛ g, ˛ D 0; 1; 2, of L2 .R2 /functions, to be interpreted as data at time t D 0. Also, assume we have V˛ D lim

n!1

n ˛

where f ˛n g are Coulomb components of admissible wave maps at time t D 0. Determine I D . T0 ; T1 / D [I1 to be the union of all open time intervals I1 with the property that lim inf k

sup

n!1 IQI1 ; IQ closed

n ˛ kS.IQR2 /

< 1:

Then we define the Coulomb wave maps propagation of fV˛ g to be ˛1 .t; x/ WD lim

n!1

n ˛ .t; x/;

t 2 I:

We call I  R2 the life-span of the (Coulomb) wave maps evolution of fV˛ g. It is of course important that the life-span does not depend on the choice of sequence and, moreover, that the “solutions” V˛ are unique. These statements follow from Proposition 7.11. The aforementioned uniqueness properties are now immediate – indeed, simply mix any two sequences which converge to V˛ . Moreover, we can characterize the life-span as follows. Corollary 7.14. Let V˛ , f that I ¤ . 1; 1/. Then

n ˛ g,

and I be as in Definition 7.13. Assume in addition

sup lim inf k

J I n!1 J closed

n ˛ kS.J R2 /

D 1:

(7.21)

Proof. Suppose not. Let I D . T0 ; T1 / where without loss of generality we assume T1 < 1. Then there exists a number M < 1 with the property that for every closed J  I with 0 2 J one has lim inf k

n ˛ kS.J R2 /

< M:

lim sup k

n ˛ kS.J R2 /

D1

n!1

Now observe that n; J I

7.2 Control of wave maps via a fixed L2 -profile

247

where J ranges over the closed subsets of I . Indeed, if not, we have lim sup k n!1

n ˛ kS.Œ0;T1 R2 /

< 1:

But then by Corollary 7.12 one can extend ˛n beyond the endpoint T1 of I to some interval IQ for n large enough while maintaining the finiteness of k ˛n kS.IQR2 / , contradicting the definition of I . Now pick 1 as in Proposition 7.11, with M replacing A, and pick J  I , n0 large enough such that

m

sup . ˛n k ˛n0 kS.J R2 /  M; ˛ /.0;
/ L2 < 1 : n;mn0

But by our definition of M there exists k0 > n0 with the property that k

k0 ˛ kS.J R2 /

and then applying Proposition 7.11 to proves the corollary.

< M;

k0 ˛ .0;
/

we obtain a contradiction. This 

Another important property is to be able to ensure the a priori existence of wave maps flows “at infinity”, i.e., the solution of the scattering problem. In this regard, we have the following result. Proposition 7.15. Assume we are given admissible data at time t D 0 of the form
t0 /.@ t V; V / C oL2 .1/; ˛ D 0; 1; 2: ˛ D @˛ S.0 Here .@ t V; V / 2 L2  HP 1 is a fixed profile. Then for t0 D t0 .@ t V; V / > 0 large enough and oL2 .1/ small enough, the wave map associated with ˛ exists on . 1; 0, is admissible there, and we have k

˛ kS.. 1;0R2 /

< 1:

Moreover, letting ˛n be a sequence of admissible Coulomb components (i.e., associated with admissible maps) at time t D 0 satisfying
n t0 /.@ t V; V / ˛ ! @˛ S.0 for .@ t V; V / as before and t0 large enough also as before, the limit lim

n!1

n ˛ .t; x/

D ˛1 .t; x/;

t 2 . 1; 0;

248

7 Some basic perturbative results

exists independently of the particular sequence chosen. We call this the Coulomb wave maps evolution of the data
@˛ S.0 t0 /.@ t V; V / at time t D

1. A similar construction applies at time t D 1.

Corollary 7.16. Assume that for a sequence of admissible Coulomb components n ˛ at time t D 0 we have
n t n /.@ t V; V / C oL2 .1/: ˛ D @˛ S.t Then if tn ! 1, the limits lim

n!1

n ˛ .t

C t n ; x/ D ˛1 .t; x/

exist in the L2 -sense on some interval . 1; C /, uniformly on closed subintervals, for C large enough. We have

lim sup ˛n .t C t n ; x/ S.. 1; C0 R2 / < 1 n!1

for C0 > C . We call the maximal interval I D . 1; C / for which these statements hold the life-span of the limiting object ˛1 ; here C may be negative or 1. A similar construction applies when tn ! 1. Both Proposition 7.15 as well as Corollary 7.16 will be proved in Section 9.8. Having defined limiting objects ˛n as in Lemma 7.13 (temporally bounded case) as well as Corollary 7.16, we can now define in obvious fashion the norms k ˛1 kS.J R2 / D lim k n!1

n ˛ kS.J R2 /

for J  I closed, with I the life-span of the limiting object. This is well-defined due to Proposition 7.11. We can then also state the following Lemma 7.17. Let ˛1 be as before, with life-span I . Assume in addition that I ¤ . 1; 1/. Then sup k ˛1 kS.J R2 / D 1:

(7.22)

J I J closed

The same conclusion holds for arbitrary I provided the sequence singular.3 3

Recall the definition in Section 1.4.

n ˛

is essentially

7.2 Control of wave maps via a fixed L2 -profile

249

We now turn to the proof of Lemma 7.10. We begin with the the lower bound on the life-span of the ˛n . In essence, this is a consequence of the fact that n 2 ˛ ! V˛ in L implies a uniform non-concentration property of the energy n of the ˛ . This then allows one to approximate the corresponding wave maps with derivative components ˛n on small discs – with radii depending only on the limiting “profile” V – by small energy smooth wave maps; the small energy theory and finite propagation speed then imply a uniform lower bound on the life-span. Technically speaking, restricting to small scales requires some care since localizing the wave map by applying a smooth cutoff does not necessarily decrease the energy. To see this, let  be a cutoff to a small ball B of size r. Then the first term on the right-hand side of Z Z Z ˇ ˇ ˇ ˇ ˇ ˇ ˇr./.x/ˇ2 dx . r 2 ˇ.x/ˇ2 dx C ˇr.x/ˇ2 dx (7.23) R2

B

B

does in general not become small as r ! 0. Let "0 > 0 be the cutoff such that smooth data with energy less than "0 result in global wave maps. More precisely, we will rely on the following result by the first author, see [23]. Theorem 7.18. Given smooth initial data .x; y/Œ0 W 0  R2 ! H2 which are sufficiently small in the sense that Z 2 h X @˛ x
2 @˛ y 2 i C dx1 dx2 < "20 2 y y 0R ˛D0

where "0 > 0 is a small absolute constant, there exists a unique classical wave 2C1 2 map from R P2 to H extending these data globally in time. Moreover, one has the bound ˛D0 k ˛ kS.R1C2 /  C "0 where C is an absolute constant. Denoting the actual map at time t D 0 giving rise (together with the time derivatives) to ˛n ; ˛n , by .x; y/.0;
/ W R2 ! H2 , where we have omitted the superscript n for simplicity, we now consider a “re-normalized” map, subject to a choice of x0 2 R2 and r0 > 0,  x x0 Œjx x0 j k1 , then





Pj v 1 T .n vn / . 2 j r v 1 T .n vn / n n 2 2
j .2 kn k4 k'n k4 C krn k2 . 2 j R: On the other hand, if j < k0 , then Z 1



1

Pj v t ŒT; .
/ 12 'n .n v t / dt

Pj v T .n vn / . n n n 2 2 0 Z 1


j

Pj v t ŒT; .
/ 12 'n .n v t / 3 dt . 23 n n 2

0

j 3

. 2 k.
/

1 2

j 3

'n k6 kn k2 . 2 r

1 3

:

280

8 BMO, Ap , and weighted commutator estimates

To pass to the final line here, one interpolates between k.
/ 1 k.
/ 2 'n kBMO . 1.

1 2

'n k 2 . r

1

and 

The following result allows us to strip away weights from T ./ provided they result from functions with frequencies which are well-separated from the Fourier support of . In what follows, we use the following terminology from [1]: Given a bounded sequence f WD ffn gn1  L2 , and sequence " WD f"n gn1  RC , we say that f is "-oscillatory iff Z lim lim sup jfn ./j2 d  D 0: R!1 n!1

Œjj"n 2.0;1/n.R

1 ;R/

We say that f is "-singular iff Z lim sup n!1

Œjj"n 2.a;b/

jfn ./j2 d  D 0:

for all b > a > 0. In what follows, we shall freely use the scale selection algorithm from Section III.1 from [1], see in particular Lemma 3.1, Lemma 3.2 part (iii), and Proposition 3.4 in that section. 1 2 2 2 2 Lemma 8.3. Suppose both f'n g1 nD1  L .R / and fn gnD1  L .R / are 11 2 2 oscillatory, whereas f n gnD1  L .R / is 1-singular. Define

vn WD exp .
/

1 2


'n ;

wn WD exp .
/

1 2

n




:

Then .vn wn /

1

T .n vn wn / D vn 1 T .n vn / C oL2 .1/

(8.3)

as n ! 1. Moreover, vn 1 T .n vn / is 1-oscillatory. Proof. By assumption, kn k2 C k'n k2 C k

n k2

A 0 there exists ı > 0 such that

.vn wn / 1 T .n vn wn / v 1 T .n vn / P 0 < " n B2;1

1  1 1

r .vn wn / T .n vn wn / vn T .n vn / 1 < "

(8.23) (8.24)

for all sufficiently large n provided lim sup kP 0

.vn wn / 1 T .n vn wn / v 1 T .n vn / < "2 n 2 for large n. Indeed, combining this bound with the preceding then implies

.vn wn / 1 T .n vn wn / v 1 T .n vn / P 0 . "2 log " n B 2;1

which is more than enough. To this end, fix a large enough a, and let wn D wn;low wn;high where wn;low corresponds to F 1 ŒŒjja cn  and wn;high to

292 F

8 BMO, Ap , and weighted commutator estimates

1 Œ c Œjja n 

.vn wn /

1

(with sharp cut-offs). By (8.28) and Lemma 8.1,

T .n vn wn /

1

.vn wn;high /

T .n vn wn;high / 2  C.T /kF

1

ŒŒjja cn kBP 0

2;1

;

whereas

sup vn 1 T .n vn / n

.vn wn;high /

1

T .n vn wn;high / 2  C.T / a

1 2

by the same argument as in (8.5). Choosing a so that this final bound is < " defines both k0 .T; "/ and ı.  Clearly, one has the following limiting statement. 1 Corollary 8.6. Suppose f'n g1 nD1 ; fn gnD1 ; f Furthermore, assume that

1 n gnD1

lie in the unit-ball of L2 .

c supp.'bn /; supp. n /  fjj  1g and

Z supp. cn /  fjj  1g;

lim

n!1 jja

j cn ./j2 d  D 0

(8.28)

for each a  1. Define vn WD exp .
/

1 2


'n ;

wn WD exp .
/

1 2

n




:

Then

.vn wn / 1 T .n vn wn / v 1 T .n vn / P 0 ! 0 n B2;1

1  1 1

r .vn wn / T .n vn wn / vn T .n vn / 1 ! 0 as n ! 1.

9

The Bahouri–Gérard concentration compactness method

In this section, we execute the scheme that was sketched in the introduction. We shall follow the five individual steps which we outlined there.

9.1

The precise setup for the Bahouri–Gérard method

As far as the concentration compactness method is concerned, our goal is to demonstrate the following main result. Proposition 9.1. Let u D .x; y/ W . T0 ; T1 /  R2 ! H2 be a Schwartz class wave map. Then denoting its energy

@ y 2  X 

@˛ x 2

˛ D E < 1;

2 C

L y y L2x x ˛D0;1;2

there is a an increasing function C.E/ W RC ! RC with the property k kS..

T0 ;T1 /R2 /

 C.E/:

We refer to the derivative components of u with respect to the standard frame .y@x ; y@y / as ˛i , i D 1; 2, ˛ D 0; 1; 2. We also use the complex notation ˛ WD ˛1 C i˛2 . We shall refer to a wave map as admissible, provided its derivative components at time t D 0, ˛i .0;
/ lie in the Schwartz class. Finally, for wave maps of Schwartz class as before, we denote the Coulomb components by ˛

WD ˛ e

i

P

kD1;2

4

1@ 1 k k

:

The energy is then given by E.u/ D

X ˛D0;1;2

2 k˛ kL 2 D x

X ˛D0;1;2

k

2 ˛ kL2 : x

To prove Proposition 9.1, we proceed by contradiction, assuming that the set of energy levels E for which it fails is nonempty. Then it has an infimum Ecrit > 0 by the small energy result. We can then find a sequence of wave maps un D .xn ; yn / W . T0n ; T1n /  R2 ! H2 with the properties

294

9 The Bahouri–Gérard concentration compactness method

ı limn!1 E.un / D Ecrit (these energies approach Ecrit from above) ı limn!1 k n kS.. T0n ;T1n /R2 / D 1. We call such a sequence of wave maps essentially singular. It is now our goal to apply the Bahouri–Gérard method to the derivative components of a sequence of essentially singular data ˛n .0;
/. ı In Section 9.2, we construct decompositions of the form ˛n D

A X

˛na C w˛nA

aD1

where the ˛na correspond to derivative components of admissible maps which are well-frequency localized. ı In Section 9.3, we use these decompositions to approximate the data ˛n by lower-frequency components. The goal is to inductively prove bounds on the Coulomb components of these lower-frequency approximations and finally obtain bounds on the Coulomb components ˛n , unless there is only one frequency atom of maximal energy Ecrit present. ı In Section 9.4, the most involved, we obtain a priori bounds on the lowest nA

.0/

frequency non-atomic components ˛ 0 , by means of a careful induction on low-frequency approximations. ı In Section 9.6, we construct the profile decomposition for the lowest frequency above-threshold energy frequency atoms. Here a lot of work is involved in showing that the profiles, which are obtained as weak limits of the linear covariant wave evolution associated with operators An , can actually be interpreted as Coulomb derivative components of actual maps, up to constant phase shifts. ı In Section 9.7, we then complete the approximate solution which is given by the sum of the profiles and the low-frequency term to an exact solution, via a perturbative argument. This culminates in Proposition 9.30. ı Finally, in Section 9.9 we explain how to add the remaining frequency atoms.

9.2

Step 1: Frequency decomposition of initial data

We consider wave maps u W R2C1 ! H2 , with Schwartz initial data. Here H2 stands for two-dimensional hyperbolic space which we identify with the upper half-plane. More precisely, introducing coordinates .x; y/ on H2 in the standard model as upper half plane, and expressing u in terms of these coordinates, we

295

9.2 Step 1: Frequency decomposition of initial data

assume that x, y; @ t x; @ t y are smooth, decay toward infinity in the sense that lim

jxj!1


x.x/; y.x/ D .x0 ; y0 / 2 H2

and such that the derivative components ˛1 D

@˛ x @˛ y ; ˛2 D ; ˛ D 0; 1; 2; y y

are Schwartz, all at fixed time t D 0. We make the following Definition 9.2. We call initial data fx; y; @ t x; @ t yg W R2 ! H2 T H2 admissible, provided the derivative components ˛k are Schwartz functions for any ˛ D 0; 1; 2 and k D 1; 2. We note here that the property of admissibility is propagated along with the wave map flow on fixed time slices, as long as the wave map persists and is smooth. This follows from finite propagation speed, as well as the small-data well-posedness theory. We recall that the energy associated with given initial data at time t D 0 is given by Z X   E WD .˛1 /2 C .˛2 /2 dx1 dx2 : R2 ˛D0;1;2

We now come to the first step in the Bahouri–Gérard decomposition of a sequence of initial data, cf. [1]. More precisely, we wish to obtain a decomposition of the derivative initial data which is analogous to the one of [1]. An added feature for wave maps, which does not appear in [1], consists of the fact that the decomposition has to be performed in such a way that the individual summands in it are themselves derivatives of admissible maps. This requires some care, as the requisite condition is nonlinear, see Lemma 9.3 below. In what follows we write ˛ WD ˛1 C i˛2 , any additional superscript referring to the index of a sequence. Lemma 9.3. The complex-valued Schwartz functions ˛ , ˛ D 1; 2, correspond to the derivative components of admissible data u W R2 ! H2 iff @k j are satisfied.

@j k D k1 j2

k2 j1 ; k; j D 1; 2

(9.1)

296

9 The Bahouri–Gérard concentration compactness method

Proof. The “only if” part follows from (1.6), (1.7). For the “if” part, note first that @k j2 @j k2 D 0 (9.2) for the imaginary parts of j and k . This implies that @j y ; j D 1; 2 y

j2 D

for a suitable positive function y W R2 ! RC which is unique only up to a multiplicative positive constant. We can rewrite (9.1) in the form @k .yj1 /

@j .yk1 / D 0; k; j D 1; 2

(9.3)

which in turn implies that j1 D

@j x y

for a suitable function x W R2 ! R. To understand the behavior of .x; y/ at infinity, we observe the following:1 from (9.2), Z

1

@2 1

12 .x1 ; x2 / dx1 D 0

which implies that the integral does not depend on x2 and therefore is, in fact, zero. Similarly, Z

1 1

22 .x1 ; x2 / dx2 D 0;

8 x1 2 R:

It follows that y tends to the same constant at infinity irrespective of the way in which we approach infinity. Without loss of generality, we may set this constant equal to 1. From (9.3) one further sees that Z

1 1

y 11 .x1 ; x2 / dx1

Z D

whence x approaches a constant x0 at 1. 1

Here the superscripts are not powers.

1 1

y 21 .x1 ; x2 / dx2 D 0; 

9.2 Step 1: Frequency decomposition of initial data

297

Now for the first step in the concentration compactness method, which is the Metivier–Schochet scale selection process, see [33] and Section III.1 of [1]. As already explained above, the difficulty we face here in contrast to [1] is that we need to make sure that the pieces we decompose the derivative components into are geometric, i.e., they are themselves derivative components of maps R2 ! H2 . Chapter 8 provides us with the tools required for this purpose. Proposition 9.4. Let fxn ; yn ; @ t xn ; @ t yn gn1 be any sequence of admissible data with energy bounded by E and with associated derivative sequence f˛n gn1 , ˛ D 0; 1; 2. Then up to passing to a subsequence the following holds: given ı > 0, there exists a positive integer A D A.ı; E/ and a decomposition ˛n D

A X

˛na C w˛nA

aD1

for ˛ D 0; 1; 2 and n  1. Here the functions ˛na , 1  a  A are derivative components of admissible maps uan W R2 ! H2 , and are an -oscillatory for a sequence of pairwise orthogonal frequency scales fan gn1 while the remainder w˛An is an -singular for each 1  a  A and satisfies the smallness condition

sup w˛nA BP 0 < ı: 2;1

n1

Finally, given any sequence Rn ! 1 one has the frequency localization with an WD log an , 1

sup kPj ˛na k2  E Rn3 2

˛D0;1;2

1 3 jj

a nj

;

8j 2Z

(9.4)

for all 1  a  A and all large n. Proof. We omit the time dependence in the notation, keeping in mind that everything takes place at initial time t D 0. As in Section III.1 of [1] one obtains a decomposition A X ˛n D Q ˛na C wQ ˛nA ; ˛ D 0; 1; 2 (9.5) aD1

where the functions Q ˛na 2 L2 .R2 / are an -oscillatory for suitable pairwise orthogonal frequency scales fan gn1 for all 1  a  A. Moreover, there is the smallness

nA

wQ P 0 < ı: ˛ B 2;1

298

9 The Bahouri–Gérard concentration compactness method

We now restrict to Fourier supports of these functions. Pick a sequence Rn ! 1 growing sufficiently slowly such that the intervals Œ.an / 1 Rn 1 ; .an / 1 Rn  are mutually disjoint for n large enough and different values of a. Then we replace wQ ˛nA by P\A

a aD1 Œn

c log Rn ;a n Clog Rn 

wQ ˛nA C

A X aD1

where an WD PŒan

P\A

a0 D1

0

Œa n

Q na 0 c ˛ log Rn ;a n Clog Rn 

log an , while we replace each Q ˛na , 1  a  A, by log Rn ;a n Clog Rn 

A X

0 Q ˛na C PŒan

wQ nA : log Rn ;a n Clog Rn  ˛

a0 D1

We need to make Rn increase sufficiently slowly so that the second term here remains an -oscillatory. Of course the new Q ˛na now also depend on the cutoff A; in order to remove this dependence, we may replace A by An where An ! 1 suitably slowly. Then the new decomposition, which we again refer to as ˛n

D

A X

Q ˛na C wQ ˛nA ;

aD1

has the same properties as the original one with the added advantage of the sharp frequency localization around the scales .an / 1 . In particular, since the ˛n are Schwartz functions, one concludes that the Q ˛na have the same property which means that the components Q ˛na are admissible, and so is wQ ˛nA . In order to prove the proposition we need to show that we can replace the components Q ˛na by components ˛na which actually belong to admissible maps una W R2 ! H2 up to a small error (which again can be absorbed into wQ ˛nA ). Note that the ˛ D 0 component does not present a problem here. For the ˛ D 1; 2 components, however, we need to ensure that the compatibility relations (9.1) hold. Continuing with the proof of the Proposition 9.4, we notice that P X 2n 1 xn D 4 1 @k Œk1n yn ; yn D e kD1;2 4 @k k kD1;2

for the coordinate functions .xn ; yn /; here we recall that we may impose the normalizations limjxj!1 x.x/ D 0, limjxj!1 y.x/ D 1. In turn, these identities imply that X X j1n D .yn / 1 4 1 @j @k Œk1n yn ; j2n D 4 1 @j @k k2n : kD1;2

kD1;2

299

9.2 Step 1: Frequency decomposition of initial data

These relations shall allow us to replace (9.5) by a “geometric decomposition”. Indeed, we simply substitute the decomposition (9.5) to obtain j1n D

A X

.yn /

1

aD1

j2n D

A X

X

4

1

@k @j ŒQ k1na yn  C .yn /

kD1;2

X

1

4

X

1

4

1

@j @k ŒwQ 1nA yn 

4

1@

kD1;2

@k @j Q k2na C

aD1 kD1;2

X

4

1

@k @j wQ k2nA :

kD1;2

This suggests making the following choices: X

xna WD

1

4

@k ŒQ k1na yna ;

yna WD e

P

kD1;2

Q 2na k k

kD1;2

and then defining j1na WD .yna / wj1nA WD .yn /

1

1

X

1

4

@j @k ŒQ 1na yna ;

X

j2na WD

kD1;2

kD1;2

X

X

4

1

@j @k ŒwQ 1nA yn ;

wj2nA WD

1

4 4

1

@j @k Q k2na

@j @k wQ k2nA

kD1;2

kD1;2

as well as  na WD  1na C i  2na , w nA WD w 1nA C iw 2nA . Clearly the components j1na , j2na are now geometric in the sense that they derive from a map into hyperbolic space; in fact, they are associated with the maps given by the components .xna ; yna /. The proof is now concluded by appealing to Lemma 8.2, Corollary 8.4, and Lemma 8.3. For the final statement, note that by Lemma 8.2, the “geometric” components ˛na are also frequency localized to the interval Œan log Rn ; an C log Rn  up to exponentially decaying errors.  As an immediate consequence of Proposition 9.4 one obtains that jk na ; wjk nA , k D 1; 2, are asymptotically orthogonal (where j1na D Re jna and j2na D Im jna ). We now make some preparations for the second stage of the Bahouri–Gérard procedure. More specifically, we shall have to pass to the Coulomb gauge components, ˛ , and transfer the above decomposition to the level of these components. One can split n ˛

D ˛n e

i

P

kD1;2

4

1@

1n k k

D

A hX aD1

i ˛na C w˛nA e

i

P

kD1;2

4

1@

1n k k

:

300

9 The Bahouri–Gérard concentration compactness method

However, the components ˛na e

i

P

1@

4

kD1;2

1n k k

are not the Coulomb gauge components of a suitable wave map, and should ideally be replaced by P 1na 1 ˛na e i kD1;2 4 @k k : Due to the lack of L1 control over the exponent, this cannot be done without further physical localizations. Nevertheless, we can state the following fact. Lemma 9.5. The components ˛na e

i

P

kD1;2

1@

4

1n k k

w˛nA e

;

i

P

kD1;2

1@

4

1n k k

are an -oscillatory and an -singular, respectively, for each a and we have

nA

w e ˛

i

P

kD1;2

1@

4

1n k k

P0

B2;1

.ı

where ı is as in Proposition 9.4. Proof. We may assume an D 1 by scaling invariance. Given any " > 0, we can choose k0 large enough such that P

na i k0 ;k0 c ˛ e

lim sup PΠn!1

kD1;2

1@

4

1n k k



L2

< ":

Next, for k1 > k0 C C , consider the expressions  PŒ k0 ;k0  ˛na e  P>k1 PŒ k0 ;k0  ˛na e

P
0, each of which will eventually be chosen depending only on the energy of the initial data. Ultimately we wish to show that there can only be a single frequency block, i.e., A D 1, and furthermore, that the energy of this block converges to the critical energy Ecrit as n ! 1. Thus we now use the following dichotomy:

9.3 Step 2: Frequency localized approximations to the data

303

Figure 9.1. Atoms and the Besov error

P ı We have A D 1 and limn!1 ˛D0;1;2 k˛na k2 2 D Ecrit . Lx ı The previous scenario does not occur. Thus, for a suitable subsequence X

 na 2 2 < Ecrit ı2 lim sup ˛ L n!1

x

˛D0;1;2

for some ı2 > 0, and all a. If the first alternative occurs, then continue with Step 4 below. Hence we now assume that the second alternative occurs, in which case we will show that the sequence un cannot be essentially singular. We may of course assume that for each 1  a  A, X

 na 2 > 0; lim inf ˛ L n!1

x

˛D0;1;2

as otherwise we may pass to a subsequence for which the ˛na may be absorbed into the error w˛nA . We may also assume that lim inf n!1

X X

 na 2 2 D lim sup

 na 2 2 ˛ L ˛ L x

˛D0;1;2

n!1

x

˛D0;1;2

by passing to a subsequence. The issue now becomes how to choose the cutoff A. Due to the asymptotic orthogonality of the ˛na as n ! 1, and for each ˛ D 0; 1; 2, X

2 lim lim sup ˛na L2 D 0 A0 !1

aA0

n!1

x

For some absolute "0 > 0 which is small enough only depending on Ecrit , in particular smaller than the cutoff for the small energy global well-posedness theory,

304

9 The Bahouri–Gérard concentration compactness method

we choose A0 large enough such that X

2 lim sup ˛na L2 < "0 ; x

n!1

aA0

and then put A D A0 . Thus we now arrive at the decomposition ˛n D

A0 X

˛na C w˛nA0 :

aD1

We may further decompose w˛nA0

A X

D

˛na C w˛nA ;

aDA0 C1

with the smallness property X

2 lim sup ˛na L2 < "0 :

X

x

n!1

aA0 C1 ˛D0;1;2

By adjusting A, we can further achieve

lim sup w˛nA BP 0

< ı0

2;1

n!1

for any given ı0 > 0. Re-ordering the superscripts if necessary, we may assume that the frequency scales .an / 1 of the ˛na are increasing with 1  a  A0 . The error term w˛nA0 may be written as a sum of constituents .0/

nA0

w˛nA0 D w˛

.A0 /

.1/

nA0

C w˛

nA0

C : : : C w˛

C oL2 .1/

which satisfy the property that .k/

nA0

w˛

.k/

D Pkn

1

nA0

C oL2 .1/ as n ! 1

Ln w˛

CLn A0 is chosen such that w˛nA BP 0

2;1

.0/

ı0 > 0 which is to be determined, while the aj ŒA0 ; A.0/ . Our choice of A0 ensures that lim sup n!1

 ı0 for some constant

are certain indices in the interval

.0/ X

naj 2

˛ 2 < "0 :

j 1

Lx

na

.0/

Now, to choose the Jj , pick for each of the ˛ j (which are finite in number) a frequency interval h a.0/ i .0/ a
1 .0/ .0/
1 nj Rj ; nj Rj

307

9.3 Step 2: Frequency localized approximations to the data .0/

with Rj

large enough such that

lim sup P n!1

.0/

naj

.0/ a j

Œlog.n

.0/ a j

.0/

1

/

log Rj ;log.n

/

1 Clog R.0/ c j

˛



L2x

 ı0 ;

na

(9.8)

.0/

which is possible due to the frequency localization of the atoms ˛ j . Here ı0 > 0 is a sufficiently small constant such that ı0 D ı0 .Ecrit ; "0 /, to be determined later. Picking n large enough, we may assume that the intervals h a.0/ i .0/ a
1 .0/ .0/
1 nj Rj ; nj Rj are disjoint. We can now exactly specify how to select the Jj : inductively, assume that J1 D Œa1 ; b1 ; : : : ; Jk 1 D Œak 1 ; bk 1  have been chosen. Then pick JQk D ŒaQ k ; bQk  such that aQ k D bk 1 and such that the integer bQk is maximal with the property that .0/ X nA0 2

P w

ŒaQ k ;bQk  ˛

2  "0 : Lx

˛D0;1;2 .0/  a
Then if bQk 2 log nj let

1

a

.0/

.0/

log Rj ; log nj a

.0/

bk D log nj




1

1




.0/ 

C log Rj

for some j , we

.0/

C log Rj :

Otherwise, we let bk D bQk . The point of this construction is that if the endpoint 0 for of JQk happens to fall on a “small atom” which may still be too large in BP 2;1 our later purposes, we simply absorb this atom into Jk . We can now state the approximate admissibility fact alluded to above. Recall nA

.0/

.0/

1nA

that Re wk 0 D wk 0 . Moreover, the constant ı0 controls the Besov norm of the tails and is kept fixed. We begin with a statement which does not involve the J` . Lemma 9.7. There is an admissible map R2 ! H2 with derivative components nA

.0/

˚˛ 0 such that

P

nA.0/

w˛ 0 e i kD1;2 4

1@

.0/ 1nA0 k wk

.0/

nA ˚˛ 0

e

i

P

kD1;2

4

.0/ 1nA0 1@ ˚ k k



L2x

.0/

nA0

as n ! 1. The same applies to the difference w˛

.0/

nA0

˚˛

.

!0

308

9 The Bahouri–Gérard concentration compactness method

.0/

1nA0

D .yn /

wj

X

1

1nA0

1

2nA0

C iwj

D wj

4

.0/

.0/

.0/

nA0

Proof. Recall the relation that defines wj

:

 1nA.0/  @k @j wQ k 0 yn ;

kD1;2 .0/ 2nA0

X

D

wj

4

1

.0/

2nA0

@k @j wQ j

:

kD1;2 .0/

1nA0

We now claim that the components wj

.0/

2nA0

, wj

are oL2 .1/- close to the

.0/

1;2nA0 ˚j

derivative components of a map, when n ! 1. Moreover, the error 1 satisfies r oL2 .1/ D oL1 .1/. First, observe that by Corollary 8.6, the compo.0/

1nA0

nent wj

is close in the above sense to .0/

1nA0

˚j

.0/

WD .ynA0 /

X

1

1

4

.0/

1nA0

@k @j ŒwQ k

.0/

ynA0 ;

kD1;2 .0/

ynA0 WD e

P

kD1;2

1@

4

.0/

.0/ 2nA Qk 0 kw

:

.0/

Next, introduce the auxiliary map .xnA0 ; ynA0 / W R2 ! H2 , with components defined by .0/

X

xnA0 WD

4

1

 1nA.0/ .0/  @k wQ k 0 ynA0 ;

.0/

ynA0 D e

P

kD1;2

4

1@

.0/ 2nA Qk 0 kw

:

kD1;2

Furthermore, as before we have .0/

1nA0

wj

D .yn /

X

1

1

4

 1nA.0/  @k @j wQ k 0 yn ;

kD1;2 .0/ 2nA0

wj

X

D

1

4

.0/

2nA0

@j @k wQ k

kD1;2 .0/

2nA0

and we set ˚j .0/

nA w˛ 0

e

as n ! 1.

i

P

.0/

2nA0

WD wj

kD1;2

4

.0/

.0/

1;2nA0

1;2nA0

D wQ 0

, w0

.0/ 1nA0 1@ w k k

.0/

D

nA ˚˛ 0

e

i

P

kD1;2

. In view of the preceding, 4

1@

.0/ 1nA0 k ˚k

C oL2 .1/ 

309

9.4 Step 3: Evolving the lowest-frequency nonatomic part

A similar result now applies to the frequency localized pieces. This time one has to use Lemma 8.5. Lemma 9.8. Given any ı1 > 0 one can choose ı0  ı1 as above such that for all large n X

.0/

nA PJj w˛ 0

e

i

P

kD1;2

4

1@

k

P

j `

.0/ 1nA0

PJj wk

;

`1

j `

may be approximated within ı1 in the energy topology by Coulomb components .0/

`nA0

˛

.0/

`nA0

WD ˚˛

e

i Re

P

kD1;2

4

.0/ `nA0 1@ ˚ k k

(9.9)

of actual maps from R2 ! H2 , uniformly in `. The same statement holds for the functions without any exponential phases. Proof. This follows exactly along the lines of the proof of Lemma 9.7: for the .0/ P nA components j ` PJj w˛ 0 we use the approximating maps .0/

.0/

x`nA0 WD .y`nA0 /

X X

1

4

1

.0/  .0/  1nA @k PJj wQ k 0 y`nA0 ;

kD1;2 j ` .0/

`nA0

y

WD e

P

kD1;2

4

1@

k

P

j `

.0/ 2nA0

w Qk

:

However, this time, the smallness of the error is contingent on the k
kBP 0 .0/

2nA wQ ˛ 0

2;1

-norm

of the non-atomic part of , while the contribution of the atomic part can be made small by choosing n large enough. More precisely, (8.25) holds for all large n due to the frequency separation properties which we have imposed on the various components, see (9.8) and (9.6). These separations become effective for large n due to the orthogonality of the scales involved.  As a general comment, we would like to remind the reader that all constructions here are not unique; moreover, they are subject to errors of the form oL2 .1/ as n ! 1.

9.4

Step 3: Evolving the lowest-frequency nonatomic part nA

.0/

As far as the evolution of w˛ 0 is concerned, we claim the following result. Note that we phrase it in terms of the derivative components that we just constructed.

310

9 The Bahouri–Gérard concentration compactness method

Once we have evolved all constituents of the decomposition from Step 1, the perturbative theory of Chapter 7 will then allow us to conclude that the representation that we obtain is accurate up to a small energy error globally in time. .0/

nA0

Proposition 9.9. Let ˚˛

.0/

nA0

˛

be as in Lemma 9.7 and set .0/

nA0

WD ˚˛

e

i

P

kD1;2

4

1@

.0/ 1nA0 k ˚k

:

Then provided "0  ı1  ı0 > 0 above are chosen sufficiently small, and pronA

.0/

vided n is large enough, the ˚˛ 0 exist globally in time as derivative components of an admissible wave map. Moreover, there is a constant C1 .Ecrit / such that the .0/

nA0

solution of the gauged counterparts of these components, i.e., ˛ bound

nA.0/ sup ˛ 0 2  C1 .Ecrit /: T0;1 >0 nA

satisfy the

S.Œ T0 ;T1 R /

.0/

Finally, ˛ 0 has essential Fourier support contained in .0; .1n / 1 /. More precisely, for some sequence fRn g1 nD1 going to 1 sufficiently slowly, one has .0/

Pk ˛nA0

S Œk

 Rn 1 e

 jk 1n j

(9.10)

for all k > 1n D log 1n and some absolute constant . As usual, all functions belong to the Schwartz class on fixed time slices. The proof of this result will occupy this entire section. The idea is to run an .0/

`nA

induction in ` on a sequence of approximating maps with data ˛ 0 , see (9.9). As we start from the low frequencies, it will turn out that the differences between two consecutive such approximating components is of small energy (provided ı1  ı0 are both sufficiently small). This allows us to pass from one approximation to the next better one by applying a perturbative argument, albeit with a linear operator involving a magnetic potential. Moreover, we need to divide the time-axis into a number of intervals which is controlled by the total energy. A key fact here which prevents energy build-up as we pass from one time interval to the next, is that the differences between these approximating components essentially preserve their energy, see Corollary 9.13. The approximate energy conservation, in turn, comes from the fact that the difference of consecutive approximating Coulomb components is essentially supported at much larger frequencies than the lower frequency approximating components.

311

9.4 Step 3: Evolving the lowest-frequency nonatomic part .0/

For the remainder of this section we drop the superscript A0 from our notation since we will limit ourselves entirely to the low frequency part. We begin by showing that (still at time t D 0) the step from ˛` 1;n to ˛`;n amounts to adding on a term of much larger frequency, up to small errors in energy. Lemma 9.10. One has ˛`;n

˛`

1;n

D ˛`;n D PJ` ˛`;n C Q˛`;n

with kQ˛`;n kL2x . ı1 . Furthermore, ˛` with k Q ˛`

1;n

1;n

D P[j `

1 Jj

˛`

1;n

C Q ˛`

1;n

kL2x . ı1 . Similar statements hold on the level of the ˚-components.

Proof. In view of Lemma 9.8 we may switch from `;n to the corresponding expressions involving w n . For simplicity, write P P X n 1 PJj w˛n e i Re kD1;2 4 @k j ` PJj wk DW f` e ig` j `

with g` real-valued. Since the Fourier support of f` is contained in [j ` Jj D . 1; b` , for any k  b` C 10 one has

Pk .f` e ig` / . kf` PkCO.1/ e ig` k2 . 2 k kf` k2 krPkCO.1/ e ig` k1 2 .2

k

. 2 b`

kf` k2 k
k

1

D 2 f` k1 . 2

k

kf` k2 k


1

D 3 f ` k2

kw n k22

. Ecrit 2b`

k

where Ecrit controls the total energy, and thus also the L2 -norm of w n . By construction of w˛n , one has for any L > 0

lim sup PŒb` L;b`  w˛n 2 . Lı0 : n!1

Together with the preceding bound this implies that

X P P n 1

PJj w˛n e i Re kD1;2 4 @k j ` PJj wk

j `

P[j ` Jj

X j `

PJj w˛n e

i Re

P

kD1;2

4

1@ k

P

j `

PJj wkn



2

. logŒ.Ecrit C 1/ı0 1 ı0  ı1

312

9 The Bahouri–Gérard concentration compactness method

for small ı0 . Next, observe that X

PJj w˛n e

i Re

P

D

X

kD1;2

4

1@ k

P

j `

PJj wkn

j `

PJj w˛n e

i Re

P

kD1;2

4

1@

k

P

j `

PJj wkn

j ` 1

C PJ` w˛n e

i Re

P

kD1;2

4

1@

k

P

j `

PJj wkn

:

The first assertion of the lemma therefore follows from the following claims: ı The function P P n 1 PJ` w˛n e i Re kD1;2 4 @k j ` PJj wk has frequency support in J` D Œa` ; b`  up to exponentially decaying errors, and we also have P P

 n  1 lim sup PJ`c PJ` w˛n e i Re kD1;2 4 @k j ` PJj wk L2 < ı1 : x

n!1

ı Furthermore, we have

X

PJj w˛n e

i Re

P

kD1;2

1@

4

k

P

j `

PJj wkn

j ` 1

X

PJj w˛n e

i Re

P

kD1;2

4

1@

k

P

j ` 1

PJj wkn



L2x

j ` 1

< ı1

for n large enough. As for the first claim, note that we have already dealt with the case of frequencies larger than b` . Thus, assume that j  a` 10 and estimate j

j

kPj .PJ` w n e ig` /k2 . 2 3 kPJ` w n e ig` k 3 . 2 3 kPJ` w n k2 kPJ` CO.1/ e ig` k6 2

j 3

. 2 kPJ` w n k2

X

2

k 3

kPk re ig` k2

k2J` CO.1/ j

. Ecrit 2

a` 3

:

Furthermore, as before one can “fudge at the edges” meaning

lim sup PŒa` ;a` CL w˛n 2 . Lı0 n!1

313

9.4 Step 3: Evolving the lowest-frequency nonatomic part

which concludes the first claim. For the second claim we need to show

X P P n 1

PJj w˛n e i Re kD1;2 4 @k j ` 1 PJj wk

P j ` 1 n
1 1 e i Re kD1;2 4 @k PJ` wk

L2x

. ı1

where the implied constant is absolute (not depending on any of the other parameters). However, this follows easily from the frequency localization up to exponentially decaying errors of X

PJj w˛n e

i Re

P

kD1;2

4

1@

k

P

j ` 1

PJj wkn

j ` 1

as well as the fact that

lim sup PŒa` ;a` CL[Œb` n!1

n L;b`  PJ` wk L2x

. Lı0

and we are done. The claim of the lemma about ˚ is easier since it does not involve any phases, cf. Lemma 9.8 and Lemma 9.7.  Our strategy now is to inductively control the nonlinear evolution of the ˛`;n , the Coulomb components of the approximation maps, starting with ` D 1 . At each induction step we add a term ˛`;n of energy less than "0 . The key then is the following perturbative result. Recall that "0 > 0 is a small constant which determines the perturbative energy-cutoff (it depends on Ecrit ). Proposition 9.11. Let ˛`;n , ˛`;n , be as before, with 1  `  C1 .Ecrit ; "0 /. Also, let X

 21

.` 1/ ` 1;n 2 c WD max 2  jr kj Pr P[ 2 J k

˛

j ` 1 j

˛

Lx

r2Z

for some small enough constant  > 0 (an a priori constant). We now make the following induction hypotheses, valid for all large n: there is a decomposition ˛` 1;n D Q ˛` 1;n C M ˛` 1;n so that

.` 1/ max Pk Q ˛` 1;n S Œk.Œ T0 ;T1 R2 / < C2 ck (9.11) ˛

` 1;n

M

< C2 ı1 (9.12) ˛ S for some positive number C2 .

314

9 The Bahouri–Gérard concentration compactness method

Then there exists a partition ˛`;n D Q ˛`;n C M ˛`;n so that

.`/ max Pk Q ˛`;n S Œk.Œ T0 ;T1 R2 / < C3 ck ˛

`;n

M < C3 ı1 ˛ S

(9.13) (9.14)

provided ı1 < ı10 D ı10 .C2 ; Ecrit /, ı0  ı1 with ı0 as in the discussion preceding Lemma 9.7, and provided n is sufficiently large. Here C3 D C3 .C2 ; Ecrit /. ;"0 / It is important to note that we iterate Proposition 9.11 O. C1 .E"crit / many 0 times, obtaining the induction start from the small data result of [23]. It is clear that there is some constant ı11 > 0 (depending only on Ecrit ) such that choosing ı1 < ı11 in each step, this proposition can be applied. This ı11 > 0 dictates our choice of A.0/ in the decomposition

w˛n D

X

.0/

naj

˛

C w˛nA

.0/

j

from before, see (9.7). Another essential feature of the construction is that

`;n

 K.Ecrit / (9.15) S where K is some rapidly growing function of the energy. This follows immediately from the inductive nature of the proof and the fact that the number of steps is controlled by the energy alone. However, it is crucial to the argument that we do not have to make "0 small depending on the function K.Ecrit / as we go through the inductive process. In other words, we have to make sure that one can fix "0 throughout. The idea of the proof of Proposition 9.11 is as follows: under the assumptions (9.11) and (9.12) we can find time intervals I1 ; I2 ; : : : ; IM1 , M1 D M1 .CQ 2 / as in Chapter 7, such that locally on Ij , `

1;n

D L`

1;n

` 1;n C NL :

Here L is a linear wave and NL is small in a suitable sense, see Lemma 7.6 and Corollary 7.27. In order to control the evolution of ˛`;n , we need to control the evolution of ˛`;n D ˛`;n ˛` 1;n : This we do inductively, over each interval Ij , starting with the one containing the initial time slice t D 0.

9.4 Step 3: Evolving the lowest-frequency nonatomic part

315

I4

I3

I2

I1

J1

J2

J3

J4

J5

Figure 9.2. The two directions of the induction

At this point one encounters the danger that the energy of ˛`;n keeps growing as we move to later (or earlier) intervals Ij , thereby effectively leaving the perturbative regime. The idea here is that we have a priori energy conservation for the components ` 1;n , `;n , while at the same time, due to our assumptions on the frequency distribution of energy for ` 1;n , ˛`;n , there cannot be much energy transfer between the latter two types of components; more precisely, we can enforce this by choosing ı1 small enough. This means that we have effectively approximate energy conservation for ˛`;n , whence the induction can be continued to all the Ij . We can now begin the proof in earnest. Proof. (Proposition 9.11) We inductively control the nonlinear evolution of ˛`;n . For ease of notation, we set ˛ WD ˛`;n and ˛ WD ˛` 1;n and for the most part we also ignore the ˛ subscript. Note that while exists globally in time,  exists only locally in time but we will of course need to prove global existence and bounds for . But for now, any statement we make for  will be locally in time on some interval I0 around t D 0. Applying the divisibility statements Lemma 7.6 and Corollary 7.27 to generates a decomposition of R into intervals fIj gjMD1 where M D M."0 ; k kS /. We may of course intersect these intervals with I0 which we will tacitly assume. Fix one of these intervals, say I1 , which contains t D 0.

316

9 The Bahouri–Gérard concentration compactness method

It will of course be necessary for us to pass to later intervals in the temporal sense until we have exhausted the entire existence interval I0 . In other words, our induction has two direction, namely a temporal one (referring to the interval Ij ), as well as a frequental one (referring to the interval J` ). These two directions are indicated as vertical and horizontal ones, respectively, in Figure 9.2. By construction, there is a decomposition D where k

L kS

L

C

1 4

2  "2 Ecrit and such that k

(9.16)

NL

2 NL kS

< "2 , 1

k

NL kS k L kS

< "24 :

(9.17)

Here "2 is small depending on Ecrit and with "0  "2  1. We note the following important improvement over (9.15): 1

2 max k `;n kS.Ij / . "2 4 Ecrit

(9.18)

j

Thus by restricting ourselves to one of the intervals Ij , we have essentially much reduced the nonlinear behavior of the . Proposition 9.11 will follow from a bootstrap argument, which is based on the following crucial result. Recall that J` is the Fourier support of .0/ up to errors which can be made arbitrarily small in energy. Proposition 9.12. Let satisfy the inductive assumptions (9.11) and (9.12) and let  be defined as above. Suppose there is a decomposition  D 1 C 2 which satisfies the bounds k2 kS.I1 R2 / < C2 C4 ı1 kPk 1 kS Œk.I1 R2 /  C4 dk

8k2Z

(9.19)

where we define dk WD

X

2

jr kj



2  21 Pr PJ` .0;
/ L2 x

r2Z

for some C4 D C4 .Ecrit / sufficiently large, and some small absolute constant  > 0. Then we can improve this to a similar decomposition with C4 C4 C2 ı1 ; kPk 1 kS Œk.I1 R2 /  dk (9.20) 2 2 for all k 2 Z, provided we satisfy the smallness condition ı1 < ı1 .C2 ; Ecrit / and ı0  ı1 with ı0 as in the discussion preceding Lemma 9.7. k2 kS.I1 R2 / 0, ı > 0, whence we conclude that



P0 Pk "1 r 1

1

r

.2

1

Πr

 jk1 j

1

.

2

/




kPk1 "1 kS Œk1 

L2t;x .Ij R2 /

X

Πr k2Z

1

.

2

2 / 2

L t HP

 21 1 2 .Ij R2 /

:

322

9 The Bahouri–Gérard concentration compactness method

Using Lemma 7.26, we can then arrange that the right hand side is  d0 , as desired. The corresponding estimate for 2 r 1 . 2 / is essentially the same, the only difference being that one square-sums over the frequencies at the end. We recall here how one infers the desired bound on " in the small-time case as in the proof of lemma 7.6 from the above considerations: letting .t / 2 C01 .R/ be a (potentially very sharp) cutoff localizing to a sufficiently close dilate of the interval I1 , and letting 1 .t / be a cutoff localizing to an interval of length  1 centered at t D 0, we write

 P0 ".t;
/ D 1 .t / P0 ".0;
/ C

Z 0

t

.s/rx P0 "Q.s;
/ ds Z t C .s/P0 ŒQ"r

1

2

.

0

 / ds C : : : ;

where "Q is a Schwartz extension of " satisfying the bootstrap estimate; more precisely, we can split "Q D "Q1 C "Q2 with each one satisfying suitable bootstrap estimates R t as in the proposition. The estimate for the first time-dependent term 1 .t/ 0 .s/rx P0 "Q.s;
/ ds is immediate: Z t



.s/rx P0 "Q.s;
/ ds

1 .t/ 0

S Œ0.I1 R2 /

. kkL2 krx P0 "QkL1 L2x  kP0 "QkS Œ0.R2C1 / t

t

Next, we can again crudely bound Z t



.s/P0 "Qr

1 .t/ 0

1

.

2



/ ds

S Œ0.I1 R2 /

Z t



.s/P0 "Qr . 1 .t /

1

.

0

2



/ ds

L2t;x

Z t



C @ t 1 .t / .s/P0 ŒQ"r 0

1

.

2



/ ds

L2t;x

:

The only difference of this compared to the estimates above is the inclusion of the cutoff .s/, which may, however, be very sharp. To deal with this, introduce a

323

9.4 Step 3: Evolving the lowest-frequency nonatomic part

C D C.Ecri t / sufficiently large, and split Z

t

1 .t/ 0

 .s/P0 "Qr

1

.

2

 / ds t

Z D 1 .t /

0

  Q 0 (absolute constant independent of the other smallness parameters), and furthermore k2 D k3 C O.1/ > Bj log ı1 j, since otherwise the desired smallness follows as in the preceding Case (b). We may thus essentially assume k1 D O.1/; k D O.1/, and reduce to the simplified expression X   P0 Pk1 r 1 Pk .Pk2 Pk3 / : k1 DO.1/Dk; k2 >Bj log ı1 j

Suppressing the frequency localizations for now, we use the schematic relation P0



r

D P0

1

h

 . 2 / r

1

.R  1 Rj  2

Rj  1 R  2 / C

r

C ::: C

1

r

1

.r

.r

1

1

. 2 //R /
i Œr 1 . 2 /2 C : : : .r

1

where we omit the remaining quintilinear and septilinear terms. More precisely, we shall use this provided both inputs  have relatively small modulation, i.e., are of hyperbolic type. In the immediately following we shall be a bit careless about the order in which we apply space-time frequency localizations and apply the Hodge decomposition. Due to the fact that the functions " are a priori only defined locally in time, this is a potential technical issue (which did not come up when we applied the Hodge decomposition to the ’s, as these are a priori defined globally in time). We shall explain how to del with this difficulty further below, when we explain how to construct the contribution to the actual Schwartz

9.4 Step 3: Evolving the lowest-frequency nonatomic part

327

extension of " from the present case. Thus for k2 D k3 C O.1/ > Bj log ı1 j, we write   P0 Pk1 r 1 .Pk2 Pk3 /     D P0 Pk1 r 1 .Pk2 Q>k2 Pk3 / C P0 Pk1 r 1 .Pk2 Qk3 / (9.27)  1 C P0 Pk1 r .R Pk2 Q 0, since we then have Z t



.s/P0 r

1 .t/

1

0

.2

2 k2



."2 / ds

sup 2

 jkj

k2Z

S Œ0.I1 R2 /



. kkL2 P0 r

1

2 L1 t Lx

t

2 B1


kPk kSŒk  ı1

sup 2

jkj



."2 /


kPk kS Œk ;

k2Z

which is more than enough for inclusion of this contribution into the "2 -part. Next, fixing some 1  0  and letting 1 be a smooth cutoff localizing to I1 and 0 which equals 1 for all t at distance  2 k2 from the endpoint of I1 , we write Z

t

  .s/P0 r 1 ."2 / ds 0 Z t Z t    1 2 D 1 .t/ 1 .s/P0 r ." / ds C 1 .t / 2 .s/P0 r 1 .t/

0

with 2 D 1 Z k1 .t/

1

 ."2 / ds;

0

1 . Then as before we obtain

t

2 .s/P0 0



r

1

 ."2 / dskS Œ0.I1 R2 /

0

. ı12

B1

sup 2

 jkj

kPk kS Œk




k2Z

which is again more than enough to include this term into "2 . Next, we decompose for some 1  00  0 Z

t

 r 1 ."2 / ds Z t   D 1 .t / Q< 00 k2 .1 /.s/P0 r 1 ."2 / ds 0 Z t  C 1 .t / Q 00 k2 .1 /.s/P0 r

1 .s/P0

1 .t/ 0



1

 ."2 / ds:

0

00

The second term on the right is again small since kQ 00 k2 .1 /kL2 . ı12 t For the first term on the right, we note that Q< 00 k2 .1 / D 1 Q< 00 k2 .1 / C O.2

N 00 k2

/;

k2

.

331

9.4 Step 3: Evolving the lowest-frequency nonatomic part

whence up to errors which can again be immediately absorbed into "2 , we can perform the Hodge-type decomposition for the factors in r 1 ."2 / and continue the calculations as after (9.27). Note that the localization due to the factor 1  also allows us to reduce the high-frequency inputs Pk2;3 " by their hyperbolic reductions Pk2;3 Q 0 is an absolute constant that depends only on the energy. Then recall from Section 9.3 that the atoms ˛na “split” the error term w˛nA0 into

359

9.6 Step 4: Adding the first large component nA

.i/

finitely many pieces w˛ 0 , 0  i  A0 , ordered by the size of jj in their Fourier support. Of course our eventual goal is to describe the evolution of the Coulomb components (with ˛n D ˛n1 C i ˛n2 ) n ˛

D ˛n e

i

P

kD1;2

1@

4

n1 k k

:

Our strategy then is to construct “intermediate wave maps” bootstrapping the bounds from one to the next, starting with the low frequency ones to the higher frequency ones. In the previous section, we have shown that we can derive an a priori bound .0/

nA0

k ˛

nA.0/ kS D ˚˛ 0 e

i

P

kD1;2

1@

4

.0/ nA0 1 k ˚k



< C10 .Ecrit /; S

provided we choose A0 above large enough and also pick n large enough. Moreover, we can then prove frequency localized bounds of the form

 .0/ nA

Pk ˚˛ 0 e

P

i

kD1;2

4

1@

.0/ nA0 1 k ˚k



S Œk.R2C1 /

 C11 .Ecrit /ck

P for a suitable frequency envelope ck with k2Z ck2  1, say, and ck rapidly decaying for k … . 1; log.1n / 1 /, where the frequency scales of the  na are given by .an / 1 . We now pass to the next approximating map, with data given by  nA.0/  w˛ 0 C ˛n1 e

i

P

kD1;2

4

.0/

nA0

D w˛

e

1@ k

i

.0/ nA0 1



P

wk

kD1;2

C ˛n1 e

i

Ckn1

4

P

1@

k





kD1;2

C oL2 .1/ .0/ nA0 1

wk

4

1@

k



Ckn1



.0/ nA0 1

wk

Ckn1



C oL2 .1/:

Here the first component satisfies .0/

nA w˛ 0

e

i

P

kD1;2

1@

4

k



.0/ nA0 1

wk

Ckn1

 .0/

D .0/

nA

nA w˛ 0

e

i

P

kD1;2

4

1@

.0/ nA0 1 k wk

nA

C oL2 .1/

.0/

1

as n ! 1 since w˛ 0 is singular with respect to the scale of k 0 . Technically speaking, this follows by means of the usual trichotomy considerations. We

360

9 The Bahouri–Gérard concentration compactness method

now need to understand the lack of compactness of the large added term Q ˛na D Q ˛n1 WD ˛n1 e

i

P

kD1;2

4

1@ k



.0/ nA0 1

wk

Ckn1



;

which is where the second phase of Bahouri–Gérard needs to come in. We now normalize via re-scaling to 1n D 1. This means now that the frequency support of Q ˛na with a D 1 is uniformly concentrated around frequency P

1

.0/ nA0 1

, jj  1. Observe that here we cannot remove the phase e i kD1;2 4 @k wk which may indeed “twist” the Coulomb components additionally. This will have a negligible effect, however, since the -system (1.12)–(1.14) is invariant with respect to the modulation symmetry 7! e i . For technical reasons4 , we now apply a Hodge type decomposition to the comna ponents 1;2 (here 1; 2 refer to the derivatives on R2 with respect to the two cona ordinate directions), as well as for Q 1;2 . Thus write 1na D @1 Q na C @2 V na

(9.58)

2na Q 1na Q 2na

V na

(9.59)

na

(9.60)

na

(9.61)

Q na

D @2  D @1 

na

D @2 

na

@1  C @2  @1 

More precisely, we define the components Q na ; V na ,  na ; na using the preceding relations, imposing a vanishing condition at spatial infinity. All of this is at time t D 0, of course. Now following the procedure of the preceding section, using the bound

nA.0/

˛ 0 < C10 .Ecrit /; S we can select finitely many intervals Ij (whose number depends on C10 .Ecrit /) such that .0/

nA

.0/

nA

.0/

nA0 jIj D jL 0 C jNL0

(9.62)

for each interval j , see Corollary 7.27. Moreover, it is straightforward to verify that our normalization 1n D 1 implies that jIj j ! 1 as n ! 1; indeed, this follows from L1 -bounds. 4

This has to do with the fact that the energy of the free wave equation involves a derivative.

361

9.6 Step 4: Adding the first large component

Next, pick the interval I1 containing the initial time slice t D 0. Consider the magnetic potential (note that we do not use the Hodge decomposition here) An WD

X

1

4

h 1nA.0/ 2nA.0/ @j  0 j 0

.0/

1nA0

j

.0/

2nA0



i :

j D1;2

Here we restrict everything to a non-resonant situation, i.e., we shall replace the above by An D

X

X

1;2 2K n jk k1 j 0 sufficiently small, just as in the proof of Proposition 9.12. In order to ensure the asymptotic (in n) energy conservation, one writes PŒ

C 2i @ PŒ R;R Q 0 is small and will be determined, kP0 u Pk1

L kL2t;x n˝k

1

 "2

k1 2

kuk2 kPk1

1 /C

and so

L k2 ;

(9.71)

S where ˝k WD  2Tk  . In our case u is of course not a free wave; however, by 1 Remark 5.12 as well as Remark 6.6 in conjunction with Lemma 2.22, we conclude that we can write u D u1 C u2 ; where ku2 kSQ < "2 kukSQ while

Z u1 D

fa ua .da/

is a superposition of free waves ua with the same frequency support properties as u and Z kfa ua kL2x .da/  C."2 /kukSQ ; fa 2 L1 kfa kL1  C: t;x ; t;x Thus for u in the original sense, choosing " in (9.71) of the form C.Ecrit ; "2 / we obtain kP0 u Pk1

L kL2t;x n˝k

 "2 2

1

k1 2

kukS Œ0 kPk1

1"

L k2 :

Inserting this bound in (9.70) yields k@˛ P0 u A˛ krx 1 N  X  C.Ecrit ; "1 / 2

k1

kP0 u Pk1

k1 < C

 X C C.Ecrit ; "1 / 2 k1 < C

X

k1

2 L kL2 n˝ k1 t;x

 12

k i P0 u Pk1 k

k

L kS

2 L kL2

t;x

ki 2Tk

 C.Ecrit ; "1 /"2 kukS Œ0 k L k2S  X X C C.Ecrit ; "1 / 2 k1 kP0 uk2L1 L2 k i Pk1 k1 < C

ki 2Tk1

t

x

 12

k

k

L kS

2 L kL2 L1 t

x

1

k

L kS :

 21

2,

367

9.6 Step 4: Adding the first large component

By picking "2 small enough in relation to "1 , we can achieve the desired smallness gain for the first expression on the right. Next, by a standard T T  estimate, and for all k1 2 Z, k i Pk1 k

L kL2t L1 x

.2

k1 2

kPk1

L k2 ;

whence X

2

k1

X

k i Pk1 k

k1 2Z

ki 2Tk1 1

2 L kL2 L1 t

x

 12

. .Ecrit "

1 C

/ k

L k2 :

Therefore, the exist intervals fIj gjMD1 as claimed. Since the constants C."2 / and C.Ecrit ; "1 / depend polynomially on the parameters, we are done.  We can now prove Proposition 9.20. We will assume that the energy of the data .f; g/ is also controlled by Ecrit although this is only a notational convenience. Proof of Proposition 9.20. With fIj g1j M as in the lemma, we relabel them as follows: with initial time 0 2 Ij0 , we set J0 WD Ij0 . At the next step, we define J1 D Ij1 and J 1 WD Ij2 where Ij1 is the successor of Ij0 (with respect to positive orientation of time), whereas Ij2 is the predecessor. In this fashion one obtains a sequence Ji with 0  i  M 0 and M 0  .Ecrit "1 1 /C as in Lemma 9.21 where "1 is small depending only on Ecrit . Next, let u be the solution of u C 2i @˛ u A˛ D 0;

uŒ0 D .f; g/:

ˇ We claim that u.0/ WD uˇJ0 can be written as an infinite Duhamel expansion in the form u

.0/

WD

1 X

u.J0 ;`/ ;

u.J0 ;0/ .t / WD S.t /uŒ0;

`D0

u

.J0 ;`/

Z WD

2i

t

U.t

s/@˛ u.J0 ;`

1/

A˛ .s/ ds

0

, V .t / D where S.t/ D .U; V /.t / is the free wave evolution, and U.t / D sin.tjrj/ jrj cos.t jrj/. Of course, t 2 J0 in this equation. Due to the energy estimate of Section 2.3 and Lemma 9.21, this series converges with respect to the S -norm. In

368

9 The Bahouri–Gérard concentration compactness method

ˇ a similar fashion, we can pass to later times: u.i / WD uˇJ satisfies i

u

.i /

WD

1 X

u.Ji ;`/ ;

u.Ji ;0/ .t / WD S.t

ti /u.i

1/

Œti ;

`D0

u.Ji ;`/ D

t

Z 2i

2i

1/

A˛ .s/ ds

(9.72)

ti

where t 2 Ji and ti WD max Ji u.Ji ;0/ .t/ WD S.t

s/@˛ u.Ji ;`

U.t

ti

D min Ji for i  1 and t0 WD 0. Observe that

1

.i 1/ Œti 1  1 /u

1 Z X `D1

ti

U.t ti

s/Ji

1

.s/@˛ u.Ji

1 ;`/

.s/ A˛ .s/ ds (9.73)

1

for all t 2 Ji . If i  2, we expand further to obtain S.t

ti

.i 1/ Œti 1  1 /u

2i

WD S.t 1 Z ti X `D1

ti

ti

.i 2/ Œti 2  2 /u

1

U.t

s/Ji

2

.s/@˛ u.Ji

2 ;`/

.s/ A˛ .s/ ds:

2

This procedure can be continued all the way back to t0 D 0 and yields u

.Ji ;0/

.t/ WD S.t /.f; g/

i 1 X kD0

2i

1 Z X `D1

tkC1 tk

U.t s/Jk .s/@˛ u.Jk ;`/ .s/ A˛ .s/ ds

(9.74) for all t 2 Ji . Inductively, one passes from this term to u.Ji ;`/ for all `  0 by means of (9.72). We next claim that for each i , the functions u.Ji ;`/ become small with respect to k
kSQ provided ` is large enough. This is a direct consequence of applying Lemma 9.21 to the above iterative definition of u.Ji ;`/ as well as the basic energy estimate. Now fix a number > 0. We will show that there exist t0 D t0 . / and n0 . / with the property that if jtj > t0 . / and n > n0 . /, then we can write u D u1 C u2 where ku2 kSQ < and ju1 .t; x/j <

369

9.6 Step 4: Adding the first large component

for jtj > t0 , uniformly in n > n0 . /. We start by reducing ourselves to a double light cone. Indeed, pick a large enough disc D in the time slice f0g  R2 with the property that kD c uŒ0kL2x  : Here D c is a smooth cutoff localizing to a large dilate of D . If we denote the covariant propagation of D c uŒ0 by uQ 2 , then we can achieve that kuQ 2 kSQ  by means of Lemma 9.15. We are thus reduced to estimating uQ 1 D u uQ 2 , which by construction is supported in a (large) double cone whose base depends only on

. We can then expand uQ 1 in terms of Volterra iterates just as before, and there exists ` with the property that X X J ;`

uQ i Q  : (9.75) 1 S i

`>`

Furthermore, note that all the iterates uQ 1Ji ;` are supported in the same double light Ž Ž cone with base D . We now show that uQ 1 D u1 C u2 where ku2 kSQ  and Ž u1 has the desired dispersive property. Setting u2 WD uQ 2 C u2 then concludes the argument. First, in view of (9.75) and the fact that the total number of Ji is Ž controlled by the energy, we may include the contributions of ` > ` in u2 . By Huygens principle, uQ 1 D .t; x/uQ 1 where for the remainder of the proof .t; x/ is a smooth cut-off to the region jxj  jt j C  with  being the radius of D . Then we can write Z t   .Ji ;`/ .J ;` 1/ uQ 1 .t/ D 2i.t; x/ U.t s/PŒ k0 0 extremely small depending on the profiles ab V1;2 Œ0, 1  b  B1 , but here B1 itself was defined based on ı2 . This is clarified ab by noting that all the profiles V1;2 Œ0 are small (more precisely, the square sum of their energies is small) for b sufficiently large, and this implies that enlarging B1 past a certain cutoff will not affect the condition on ı2 ; for more clarification see the “important technical observation” below. Proof. (Proposition 9.30) We will prove the inequality for Pk  using a bootstrap argument. The challenge consists in careful book-keeping of all the possible interactions. The idea is to essentially replicate the proof of Proposition 9.12 with  D 2 . The main novel feature here is that we now have to deal with a large number of additional source terms stemming from the nonlinear interactions of

406

9 The Bahouri–Gérard concentration compactness method n.B0 / C W naB 

bD1

 C P0 Œ

.0/

nA0

C

B0 X

Q nab C Q a.>B0 / C W naB r

1

 . 2 /

bD1

C P0 Œr

1

2

. / C interaction terms

“Interactions terms” here refers to all possible expressions which do not involve the radiation term  such as   .0/ P0 nA0 r 1 Œ. Q nab /2  : Indeed, the complete list of the error terms included under this heading is complicated, due to our construction of the evolutions Q nab in (I)-(III) above. Recall that for the temporally bounded type components, we use the nonlinear wave maps flow on a large time interval T abı2 , but we then use the covariant linear evolution past that time. This means that on Œ0; T abı2 , we generate error interaction terms like the preceding one coming from the interactions with the low frequency part .0/ nA0 , while on the interval ŒT abı2 ; t nab2 C  generate errors due to the nonnab Q linear self-interactions of . On the other hand, for the temporally unbounded type components, we use the linear covariant evolution on Œ0; t nab2 C , which means that we generate errors due to the nonlinear self-interactions. In addition to all these, we generate errors due to different concentration profiles interacting with each other, as well with the small frequency component .0 nA0 , or the weakly small error, and the latter also generates nonlinear errors due to interactions with itself. We will deal with this rather large collection of errors later, showing that we can make its N Œ0-norm arbitrarily small by choosing B1 large enough, and then n large enough. We also use the notation Q a.>B0 / for the evolution of X nab V3;4 .0 t nab ; x x nab / C oL2 .1/; b21 [2

X


@ t SAQnab V1ab Œ0 .0

t nab ; x

x nab / C oL2 .1/;

b21 [2

as explained in the “important technical observation” above. We first deal with the terms involving . Our task is to gain a smallness constant that allows us to improve the a priori bound we are assuming about .

411

9.7 Step 5: Invoking the induction hypothesis

(i.1) Terms involving . These can be handled exactly as Case 1 in the proof of Proposition 9.12, in light of the bound



.0/

nA0

C

B0 X

 Q nab C Q a.>B0 / C W naB  C. S

.0/

nA0

0 ; f Q nab gB ; Ecrit /: bD1

bD1

Thus for example paralleling Case 1 (a) in the proof of Proposition 9.12, one obtains a bound X 

I Pk r j

1

.Œ

.0/

nA0

C

k2Z

B0 X

Q nab C Q a.>B0 /2 / 2 2

L t HP

bD1

1 2

 kk2S

provided we choose the time cutoffs suitably (such that the number M1 of such time intervals is as above). .0/

(i.2) Errors due to nonlinear (self)interactions of the Q nab , nA0 , W naB . Note that these errors serve as source terms for , and hence we need to show that they are extremely small (of order controlled by ı2 ). The mechanism for this is first choosing B1 sufficiently large (for the contributions involving W naB ), and then choosing n large enough. As these estimates are analogous to those in Case 1 of Proposition 9.12, we explain here only the mechanism for generating arbitrary smallness (as n ! 1). (i.2.a) Errors generated by the temporally bounded type Q nab . If Q nab is the evolution of a temporally bounded concentration profile, then recall that we let Q nab be the wave maps evolution on the interval Œ0; T abı2 , provided Q nab is supported at frequency scale  1. Now we want to track the evolution of an arbitrary frequency mode Pk , which we have scaled to k D 0. But then we have also re-scaled all the source terms. Now the source terms generated by Q nab itself come from a number of sources: first, the “gluing definition” of (9.85) implies that we generate errors of the form (before frequency localization) 0.

1;T abı2 C10/

.t /

ab

0.

1;T abı2 C10/

.t /SAn

ab

ŒT abı2 :

The only way for this term to contribute in the Case (i) for a fixed frequency (which we assume equals one after scaling) is when the original frequency (which is scaled to one) is extremely large. But this contribution is then easily seen to be 2 Q ab . Next, the selfvery small in L1 t Lx , say, due to the frequency localization of interaction errors generated from the usual div-curl system are (schematically)   Q nab r 1 Œ. Q nab /2  ; P0 @ t Q nab rx Q nab

412

9 The Bahouri–Gérard concentration compactness method

which vanishes provided the I1 fits into the re-scaled interval Œ0; T abı2 . Otherwise, one obtains a contribution of the above form on the complement of the re-scaled interval Œ0; T abı2  inside I1 , and which is of the above form. We need to show that picking n large enough, this can be made arbitrarily small. For this purpose we use the following observation. Lemma 9.33. Let Q nab be the evolved Coulomb components of a temporally bounded type concentration profile, concentrated at frequency  1. Then letting T abı2 be the time indicating transition from nonlinear to linear evolution (as explained in the preceding discussion), we have Q ˛nab .T abı2 ;
/ D @˛ Q nab .T abı2 ;
/ C error where kerrorkL2x ! 0 as ı2 ! 0, and furthermore Q nab D

X

4

1

@k Q knab :

kD1;2

The proof of this lemma follows exactly as in the proof of Case (B) of Proposition 9.24. It then follows that in case we are on the complement of the re-scaled interval Œ0; T abı2  inside I1 , we generate errors of the form   P0 Q nab r 1 Œ. Q nab /2  C error; with error as in the preceding lemma, in addition to errors stemming from inter.0/ actions of Q nab with the other components nA0 etc. to be considered later. But Q nab .t;
/ as jt j ! 1, it is seen that then, using the L1 t;x -dispersion for the

 nab 1 nab 2 

P0 Q r Œ. Q /  L1 L2 .I1 \Œ0;T abı2 c R2 /  ı2 t

x

if we choose T abı2 large enough in relation to ı2 . Next, we need to analyze the .0/ errors generated by Q nab through interaction with the other ingredients nA0 , Q a.>B0 / , and W naB . We begin with the interactions between two distinct terms Q nab , b D 1; 2; : : : ; B1 . Thus we are considering   P0 Q nab1 r 1 . Q nab2 Q nab3 / where bi ¤ bj for some i; j . By the frequency localization of all these factors, we may assume that, up to negligible errors, each of them satisfies Q nabj D

413

9.7 Step 5: Invoking the induction hypothesis

PŒ C6 ;C6  Q nabj where C6 is a potentially extremely large constant depending on the frequency localizations (i.e., how well-localized  a 1  the factors are in frequency space), as well as ı2 and B1 , and that log .n / 2 Œ C6 ; C6 . Now assume first that I1  Œ0; T abj ı2  for all j , i.e., our time interval is such that we are in the “nonlinear regime” for each of these factors. But then choosing n large enough, we can force

 nab 1

P0 Q r

1

 1 . Q nab2 Q nab3 / L1 L2  ı2 100 x t C6

by the essential disjointness of the supports of the factors, and this suffices to handle Case 1, see the proof of Proposition 9.12. Indeed, the expression  nab  1 Q r 1 . Q nab2 Q nab3 / is essentially supported in a frequency interval Œ 10C6 ; 10C6 , and repeating the above estimate for each of these frequencies and square summing easily yields the bound X  

Œ T ;T  .2k t /Pk Q nab1 r 1 . Q nab2 Q nab3 / 1  ı2 : 1 1 2 P k

Lt H

2

If, on the other hand, at least one of the factors Q nabj is in the “linear regime” (i.e., satisfies the covariant wave equation), then smallness follows from the L1 decay. Next we consider the term   .0/ P0 Q nab r 1 Œ. nA0 /2  : Here of course it is essential that we are in Case (i) and so it suffices to estimate the 2 2 L1 t Lx or also L t;x norm of this term, see Case 1 of the proof of Proposition 9.12. .0/ Due to the essential disjointness of the Fourier supports of Q nab and nA0 , see Proposition 9.9, we may assume that the first input Q nab has frequency of size one, while the second input has extremely small frequency (controlled by picking n large enough). But then we may estimate this contribution by placing .0/ r 1 Œ. nA0 /2  into L1 t;x , and re-scaling and square-summing over the output frequencies results in the desired small bound. Finally, the interactions of temporally bounded Q nab with the remaining weakly small errors W naB1 are handled similarly by exploiting the smallness of the latter with respect to L1 t;x . Here the “Important Technical Observation” from before becomes important again.

414

9 The Bahouri–Gérard concentration compactness method

(i.2.b) Errors generated by temporally unbounded Q nab . Again the errors generated are of the form   Q nab r 1 Œ. Q nab /2  ; P0 @ t Q nab rx Q nab .0/

as well as terms involving interactions of Q nab with nA0 , Q a.>B0 / , as well as W naB . From Part (B) of the proof of Proposition 9.24, we know that Q nab is of gradient form up to an error which can be made arbitrarily small. Hence the above simplifies, up to a negligible error, to the nonlinear term   P0 Q nab r 1 Œ. Q nab /2  : To estimate this, we can first reduce this to  P0 Q nab r 1 PŒ C6 ;C6  Œ.PŒ

C6 ;C6 

 Q nab /2  ;

arguing as in Case 1 of the proof of Proposition 9.12, and then by using the L1 t;x dispersion, i.e., Lemma 9.20, to write PŒ

C6 ;C6 

Q nab .0;
/ D oL1 .1/

from which the desired smallness follows easily. The interaction terms of tempo.0/ rally unbounded Q nab with the remaining components nA0 , Q a.>B0 / , W naB , are handled as before and are omitted. (i.2.c) Errors generated by the weakly small remainder W naB . Again recalling Part (B) of the proof of Proposition 9.24, and Lemma 9.26, we know that W˛naB is of pure gradient form up to a negligible error (provided B and n are large enough). The conclusion is that the error of the form   P0 @ t W naB rx W naB W naB r 1 .ŒW naB 2 / reduces up to a negligible error to the nonlinear self-interaction term P0 Œ W naB r

1

.ŒW naB 2 /;

which can be estimated as in the preceding case, using the smallness of kW naB kL1 after reducing to frequencies of size O.1/. t;x (ii) jI1 j > T1 , T1 as in Case 1. Proceeding as in Case 2 of the proof of Proposition 9.12, we decompose P0  into P0  D P0 Q 0 have the property that Z

2 X ˇ 1 ˇ ˇ .0; x/ˇ2 dx < ": ˛

jxj> M 2 ˛D0

(10.2)

430

10 The proof of the main theorem

Then 2 X ˇ 1 ˇ ˇ .t; x/ˇ2 dx < C " ˛

Z

(10.3)

jxj>2M Ct ˛D0

for all t 2 I C . Here C is an absolute constant. Proof. By definition, there exist un D .xn ; yn / W I C ! H2 which are admissible wave maps such that (9.90) holds. Now define .xn2 ; yn2 /.0;
/

 xn .0;
/ WD Œjxj>M  yn0

xn0

yn

;e

Œjxj> M  logŒ yn .0;
/  2

0

where Œjxj>M  is a smooth cutoff to the set fjxj > M g which equals one on fjxj > 45 M g, say, and xn0 WD yn0 WD exp

− ŒM <jxj< 54 M 

− 5 ŒM 2 <jxj< 8 M 

The construction here is such that yn2 D be the wave map evolution of the data 

xn .x/ dx1 dx2 ;

.xn2 ; yn2 /.0;
/;

yn yn 0


log yn .x/ dx1 dx2 : on the set frŒjxj>M  ¤ 0g. Let uQ n

@ t xn .0;
/ @ t yn .0;
/
 ; : yn0 yn0

By construction, the energy of uQ n does not exceed C ". This requires the use of Poincaré’s inequality as in the proof of Lemma 7.22. One now concludes by means of finite propagation speed for classical wave maps, and by passing to the limit n ! 1.  Next, one has the following lower bound on .t / in Corollary 9.36. Lemma 10.4. Assume I C is finite. After rescaling, we may assume that I C D Œ0; 1/. There exists a constant C0 .K/ depending on the compact set K in Corollary 9.36, such that C0 .K/ 0<  .t / (10.4) 1 t for all 0  t < 1.

10.1 Some preliminary properties of the limiting profiles

431

1 2 g˛D0 with Proof. Take any sequence tj ! 1. Consider the limiting profile f Q ˛;j 2 1 1 1 data .tj / ˛ .tj ; .
x.t N j //.tj / /g˛D0 . By the well-posedness theory of 1 2 g˛D0 have a fixed the limiting profiles in Section 7.2, one infers that the f Q ˛;j life-span independent of j which depends only on the compact set K. By the uniqueness property of the solutions and rescaling, .1 tj /.tj /  C0 .K/ as claimed. 

Next, combining this with Lemma 10.3 one concludes the following support property of the ˛1 with finite life-span. Lemma 10.5. Let ˛1 be as in the previous lemma. Then there exists x0 2 R2 such that
supp ˛1 .t;
/  B.x0 ; 1 t / for all 0  t < 1, ˛ D 0; 1; 2. Proof. This follows the exact same reasoning as in Lemma 4.8 of [14]. One uses Lemma 10.3 instead of their Lemma 2.17 and Lemma 10.4 instead of their Lemma 4.7.  Next, we turn to the vanishing moment condition of Propositions 4.10 and 4.11 in [14]. Proposition 10.6. Let ˛1 be as above and assume that I C is finite. Then for i D 1; 2, Z Z i1 N 01 dx D 0 h@i U; @ t U i dx D Re R2

R2

for all times in I C . Proof. Assume that Z Re R2

11 N 01 dx > > 0:

This implies that the approximating sequence un satisfies Z h@1 un ; @ t un i dx > > 0 R2

for large n. Following [14] we apply a Lorentz transformation  t dx x1 1 Ld .t; x/ WD p ;p 1 d2 1

dt d2

; x2



432

10 The proof of the main theorem

to the un . Note that for any " > 0 one has from Lemma 10.5 that 2 Z X j@˛ un .t; x/j2 dx < " ˛D0 jxj1 t

for all t 2 I C D Œ0; 1/ and sufficiently large n. Then the argument in [14] implies that there exists d small with the property that lim sup E.un ı Ld / < Ecrit : n!1

By our induction hypothesis, k n;d kS.I C R2 / < M < 1 for all sufficiently large n. Here n;d are the Coulomb components of the admissible wave maps un ı Ld . Note that the Coulomb components n;d do not obey a simple transformation law relative to the Coulomb components n of un . Nonetheless, it is possible to conclude from this that lim sup k n!1

n

kS.I C R2 / < M1 < 1

via Remark 7.8 which gives us the desired contradiction. Thus, we need to prove that for each k1 > k2 X 2 k2 Pk1 ;1 n Pk2 ;2 n D fk1 ;k2 C gk1 ;k2 (10.5) 1;2 2Cm0 dist.1 ; 2 /&2

m0

where m0 is a large depending on Ecrit , where we have the bounds (7.20) for fk1 ;k2 and gk1 ;k2 . Furthermore, we need to show that Pk Q>k

n

D hk C ik

with the bounds stated in Remark 7.8. We establish this for the bilinear expression, the corresponding computations for Pk Q>k n being similar. First, we claim the following bound for n;d : X X 2 2 k2 kPk1 ;1 n;d Pk2 ;2 n;d kL < 0 (10.6) 2 k1 >k2

1;2 2Cm0 dist.1 ; 2 /&2

t;x

m0

This, however, is immediate from the angular separation and (2.30) with a constant 0 which depends on M and Ecrit . In fact, we need something slightly stronger due to the usual tail issues: X X 2 sup < 0 (10.7) 2 k2 kPk1 ;1 n;d y Pk2 ;2 n;d kL 2 y

k1 >k2

1;2 2Cm0 dist.1 ; 2 /&2

t;x

m0

433

10.1 Some preliminary properties of the limiting profiles

where y is a translation by y 2 R2 . Next, we claim the following estimate: X k1 >k2

X

2

1;2 2Cm0 dist.1 ; 2 /&2

k2

2 kPk1 ;1  n;d Pk2 ;2  n;d kL < 0 2

(10.8)

t;x

m0

where  n;d are the derivative components of the un ı Ld . This is the same as X

X

k1 >k2

2

1;2 2Cm0 dist.1 ; 2 /&2

Pk1 ;1

n;d

k2

e

i@

1  n;d




m0

n;d


Pk2 ;2

e

1  n;d

i@

L t;x

P where we wrote the phase i @ 1  n;d D i Re j2D1 .
/ cally. This follows from (10.6) and the Strichartz estimate X

2

3 2k

sup

X

j 10 c2D

k2Z

2

.1 2"/j

2 kPc  n;d kL 4 1 L t




2

2 < 0

 21

x

1 @  n;d j

schemati-

. M:

(10.9)

k;j

To prove (10.9), one uses the corresponding bound on n;d (which is part of the S -norm), energy conservation, and a simple Littlewood–Paley trichotomy. To prove (10.8), one argues as follows. Split X k1 >k2

.

X 1;2 2Cm0 dist.1 ; 2 /&2

X

2

k2

Pk1 ;1

n;d

e

1  n;d

i@




m0

X

k1 >k2 1;2 2Cm0 dist.1 ; 2 /&2

n;d


Pk2 ;2

2 k2 Pk1 ;1

n;d

Pk2 1;2 2Cm0 dist.1 ; 2 /&2

n;d

Pk1 ;1

2

P>k1

m0

m0 e

n;d


Pk2 ;2

1  n;d

i@

Pk2 1;2 2Cm0 dist.1 ; 2 /&2

n;d

P>k1

m0

m0 e

n;d


Pk2 ;2

1  n;d

i@

P>k2




i@

m0 e

1  n;d


2

2

L t;x

(10.13) 1 In (10.10) one reduces matters to (10.7) by placing the exponential in L1 t Lx . Next, to bound (10.11) one notes that

P>k 2

m0 e

i@

1  n;d



L4t L1 x

k2 4

.2

M

where the implicit constant depends on Ecrit . Therefore, X

X

k1 >k2 1;2 2Cm0 dist.1 ; 2 /&2

X

.

k2

2

n;d

Pk2 1;2 2Cm0 dist.1 ; 2 /&2

X

Pk1 ;1

kPk1

m0 e

i@

n;d

n;d 2 kL1 L2 x t

1  n;d

P>k2 X




m0 e

i@

1  n;d


2

2

L t;x

`>k2

m0

n;d

kPc P`

Pc 0 P`CO.m / .e kL4 L1 0 x t

c;c 0 2D`;k2 ` dist.c;c 0 /.2k2

i@

1  n;d

2

1/ L4 L1 t

x

. M6 using the Strichartz estimate from above. The remaining terms are the same. This concludes the proof of (10.8). By the same logic, one also obtains X k1 >k2

X 1;2 2Cm0 dist.1 ; 2 /&2

2

2 Pk1 ;1 Qk1 CC2  n;d Pk2 ;2 Qk2 CC2  n;d L2

k2

t;x

m0

< 0

435

10.1 Some preliminary properties of the limiting profiles

where C2 is a large constant depending only on the energy which will be determined later. This then implies the following version without the Lorentz transforms X X

2 2 k2 Pk1 ;1 Qk1 CC2  n Pk2 ;2 Qk2 CC2  n L2 < 0 ; t;x

k1 >k2

1;2 2Cm0

0

dist.1 ; 2 /&2

m00

provided d is chosen small enough, but depending only on Ecrit (so that m00 is close to m0 ). Finally we claim that Pk1 Q>k1 CC2  n Pk2  n n

Pk1  Pk2 Q>k2 CC2 

(10.14)

n

(10.15)

can both be included in gk1 ;k2 . To see this, one first expands  1 n Pk1 Q>k1 CC2  n D Pk1 Q>k1 CC2 n e i @  X  D Pk1 Q>k1 CC2 P` n P` e i @

1n



i@

1n

(10.16)

`>k1 CC2 10

C

X

 Pk1 Q>k1 CC2 P`

n

P` e



(10.17)

k1 k1 CC2 Pk1 CC2 Pk1 n Pk1 CC2  n into L4t L2x followed by an application of (10.9) with caps of size 2k1 ; more pre1 n n gets placed into L1 L2 cisely, P` e i @  goes into L4t L1 x as before, and P` t x 2 -norm of n over (see Lemma 2.18 for the issue of square-summing the L1 L t x C2 caps of size 2k1 ). Note that one gains a smallness factor of the form 2 10 due to the improved Strichartz bounds. Next, we consider (10.19) and the remaining terms (10.17) and (10.18) will follow similar arguments. Now we decompose further:  1 n Pk1 Q>k1 CC2 Pk1 n Pk1 CC2 Q>k1 CC2 10 Pk1 n Pk1 CC2 Qk1 CC2 10 Pk1 n Pk1 CC2 10 e i @  (10.21)

436

10 The proof of the main theorem

For the contribution of (10.20) to (10.14) one estimates



Pk Q>k CC Q>k CC 10 Pk n Pk1 CC2 . 2k2 2

k1 CC2 2

10 Pk1

kPk1

n

n

1n



Pk2  n L2

t;x

n

kL2 kPk2  kL1 L2x t;x

t

kS Œk1  kPk2  n kL1 L2x t

C2

which is sufficient since it gains the smallness 2 2 . Finally, we use (1.6) for the case when we substitute (10.21) for Pk1 Q>k1 CC2  n ; one can then write Pk1 Q>k1 CC2  n Pk2  n  D Pk1 Q>k1 CC2 Qk1 CC2

n

10 Pk1

@ t 1 Pk1 CC2 10

. n  n //e

i@

1n




Pk2  n

where we have written (1.6) schematically in the form @t @

1 n

 D n C r

1

. n  n /:

The contribution of  n is easy, it is placed again in L4t L1 x (of course after apply1n n i @ ing the usual trichotomy to  e ). On the other hand, due to the determi1 n n nant structure of r .  / we have r

1

. n  n / D r

1

.

n

n

/:

By using a further Hodge decomposition of the inputs on the right, we have for each k 2 Z kPk r 1 . n n /k 2 P 1 . k n k2S ; Lt H 2

and from here we obtain

1

@ Pk1 CC2 10 .r

1

.

n

n

/e

i@

1n

k1 2

/ L2 L1  2 t

x

k k2S ; (10.22)

and from here we obtain @ t 1 Pk1 CC2

1 n  .r 1 . n  n /e i @  / Pk2  n L2



Pk Q>k CC Qk CC 1 1 2 1 2

10 Pk1

n

10

t;x

k2

2

k1 2

kPk2  n kL1 L2x k k2S : t

437

10.1 Some preliminary properties of the limiting profiles

This concludes the proof that (10.14) may be included into gk1 ;k2 . For (10.15) one argues similarly. By following the same Littlewood–Paley trichotomies, one is eventually lead to the most difficult case Pk1  n Pk2 Q>k2 CC2  n  D Pk1  n Pk2 Q>k2 CC2 Qk2 CC2

n

10 Pk2

@ t 1 Pk2 CC2 10

. n  n //e

i@

1n




where we again used the curl equation (1.6). The  n term is again easier, whereas for the nonlinear term we again use r

1

. n  n / D r

1

n

.

n

/:

Then as before we use (10.22), in order to infer that



Pk  n Pk Q>k CC Qk CC 10 Pk n @ 1 Pk2 CC2 10

1

x

t

2

k2 2

10

k2S kPk1  n kL1 L2x t

k

which justifies us in including it into gk1 ;k2 . In conclusion, we have now shown that we can write X Pk1 ;1  n Pk2 ;2  n D fQk1 ;k2 C gQ k1 ;k2 (10.23) 1;2 2Cm0

0

dist.1 ; 2 /&2

m00

with bounds as in (7.20). The goal is now to deduce (10.5) from this estimate. For this purpose, fix k1 > k2 C C1 and caps 1 ; 2 2 Cm00 as above. We now describe how to break up Pk1 ;1

n


Pk2 ;2

n

D Pk1 ;1 . n e

i@

1n

/
Pk2 ;2 . n e

i@

1n

/

into various pieces which then constitute fk1 ;k2 and gk1 ;k2 , respectively when summed over the caps. First, write Pk1 ;1 . n e D

i@

1n

3 X i1 ;i2 D1

/
Pk2 ;2 . n e Pk1 ;1 . n Ai1 e

i@

i@

1n

/

1n

/
Pk2 ;2 . n Bi2 e

i@

1n

/ (10.24)

438

10 The proof of the main theorem

where A1 D Pk1

C2 

n

Ai1 e

i@

1n

/ D Pk2 ;2 .P>k2

C2 

n

Bi2 e

i@

1n

/

and /

for otherwise one obtains smallness from Bernstein’s inequality. For example, consider now i1 D 1, i2 D 2 which is Pk1 ;1 .P>k1

C2 

n

Pk2 D Pk1 ;1 .P>k1

C2 

n

m00 10 e C2 

Pk2

n

i@

Pk2

m00 10 e C2 

n

1n

/

m00 10
k1

C2 


Pk2 ;2 .P>k2

C2 

m00 10 e n

Pk2

Pk1

n

P 0 so that for all t  0 one has Z ˇ 1 ˇ ˇ .t; x/ˇ2 dx  ": (10.28) ˇ ˇ ˛ N ˇ ˇxC x.t/ .t/ R0 ."/ ˇ ˇ ˇ x.t N /ˇ ı There exists M > 0 so that for all t  0, one has ˇ .t / ˇ  t C M . These are a consequence of the compactness in Corollary 9.36 and Lemma 10.3. Recall from the proof of Proposition 10.1 that upon passing to a suitable subsequence of the approximating maps un , we may extract an L2 -limit for the standard derivative components ˛n ; denote this by ˚˛1 (which, in contrast to ˛1 , we do not claim to be canonical). Now define for each d > 0, R > 0,  t dx  x1 dt 1 Z˛d;R .t; x/ WD ˚˛1;R p ;p ; x2 1 d2 1 d2 where ˚˛1;R .s; y/ WD R˚˛1 .Rs; Ry/: These rescaled limiting profiles again have energy Ecrit . Now define  to be a smooth cutoff function supported on jxj  2 and  D 1 on jxj  1. The main calculation in the proof of Proposition 4.11 of [14] now reveals that, see (4.20) there, uniformly in t0 2 Œ1; 2, 2 Z X ˛D0

R2

ˇ ˇ2  2 .x/ˇZ˛d;R .t0 ; x/ˇ dx D Ecrit

d C d.R; d / C .R; Q d / C O.d 2 /

(10.29) with .R; d / and .R; Q d / ! 0 as R ! 1, uniformly in 0 < d < d0 and with O.d 2 / uniform in R. Furthermore, the argument in [14] yields that for fixed " > 0, R > 0, d > 0 as above, one may find t0 2 Œ1; 2 such that Z ˇ d;R ˇ ˇZ .t0 ; x/ˇ2 dx  ": ˛ 1 2 jxj2

We shall later pick , R depending on ; d and d depending on ; Ecrit . Now for fixed choices of these parameters, pick n large enough such that for un D .xn ; yn / an element of the approximating sequence of wave maps from R2C1 ! H2 , denoting by ˛n;d;R the Coulomb components of un ı Ld dilated by factor R as above, and similarly by ˛n;d;R the standard derivative components, an averaging argument over different time-like foliations yields that we may also assume Z ˇ n;d;R ˇ2 ˇ .t0 ; x/ Z˛d;R .t0 ; x/ˇ dx < ": ˛ R2

443

10.1 Some preliminary properties of the limiting profiles

Note that now t0 may depend on n, but this does not affect the argument. The idea now is to truncate the data
un ı Ld .Rt0 ; Rx/; R@ t un ı Ld .Rt0 ; Rx/ ; solve the Cauchy problem backwards, and undo the Lorentz transform. We thereby obtain a good approximation to the original essentially singular sequence n ˛ , but which satisfies good S -estimates, which gives us the desired contradiction. Thus, write un ı Ld .Rt; Rx/ D .xn;d;R ; yn;d;R /. To do this, we consider data n;d;R

h



.t0 ;
/ WD Œjxj< 1 

xn;d;R 0

xn;d;R .t0 ;
/ yn;d;R 0

2

;e

ŒjxjM  is a smooth cutoff to the set fjxj > M g which equals one on fjxj > 45 M g, say, and Œjxj<M  WD 1 Œjxj>M  . Moreover, − n;d;R x0 WD xn;d;R .x/ dx1 dx2 ; Œ 41 <jxj< 12 

yn;d;R WD exp 0

− Œ 21 <jxj 0 there exists R0 ."/ > 0 such that for all t  0 one has Z ˇ ˇ ˇ@˛ U.t; x/ˇ2 dx  " ˇ ˇ N ˇxC x.t/ ˇR0 ."/ .t/

since .t/  0 > 0 for all t  0. Lemma 10.9. There exists "1 > 0, C > 0 such that if " 2 .0; "1 / there exists R0 ."/ so that if R > 2R0 ."/ then there exists t0 D t0 .R; "/, 0  t0  CR with the property that for all 0 < t < t0 one has ˇ x.t ˇ ˇ x.t ˇ ˇ N /ˇ ˇ N 0/ ˇ ˇ ˇ < R R0 ."/; ˇ ˇ D R R0 ."/: .t / .t0 / Proof. As a preliminary argument, we show that there exists ˛ 2 R with Z Z ˇ 1 ˇ2 ˇ ˇ .t; x/ dxdt  ˛ > 0 (10.30) 0 I

R2

for all intervals I of length one. If not, there exists a sequence of intervals Jn WD Œtn ; tn C 1 with the property that tn ! 1 and Z Z ˇ 1 ˇ2 ˇ ˇ .t; x/ dxdt  1 : (10.31) 0 n Jn R2 Then there exist times sn 2 Jn with the property that k 01 .sn ;
/k2 ! 0 as n ! 1. By Corollary 9.36 one has that n

o1 .sn / 1 ˛1 sn ;
x.s N n /0 .sn / 1 nD0

forms a compact set for ˛ D 0; 1; 2. Passing to a subsequence, we may assume that strongly in L2

.sn / 1 ˛1 sn ;
x.s N n / .sn / 1 ! ˛ .
/:

446

10 The proof of the main theorem

By Lemma 7.10 there exists some nonempty time interval I  around zero such that

.sn / 1 ˛1 sn C t .sn / 1 ;
x.s N n / .sn / 1 ! ˛ .t;
/  2 2 in L1 loc .I I L .R //. Distinguish two cases: f.sn /g is bounded or not. In the former case, note that .t /  0 > 0 implies that there exists a nonempty I Ž  I  such that sn C .sn / 1 I Ž  Jn for each n. Therefore, (10.31) implies that Z Z ˇ  ˇ2 ˇ ˇ .t; x/ dxdt D 0: 0 IŽ

R2

This implies that 0 .t;
/ D 0 for all t 2 I Ž . On the other hand, if f.sn /g is unbounded for every sequence fsn g with sn 2 Jn , we invoke the covering argument from [50]. Thus write for each n [   s  1 .s/; s C  1 .s/ : Jn D s2Jn

By the Vitali covering lemma, we may pick a disjoint subcollection of intervals fIs gs2An , Is WD Œs  1 .s/; s C  1 .s/ for some subset An  Jn with the property that [ 1 jIs j  : 5 n s2A

But then the defining property of the Jn implies that for each Jn , we may pick times sn 2 Jn with the property that Z

1




.t;
/ 2 2 dt D o  1 .sn / : Isn \Jn

0

Lx

Alternatively, this implies that as n ! 1 Z 1




J 1 .sn C t  1 .sn /;
/ 2 2 dt D o.1/: n 0 L x

1

Now pick a converging subsequence of .sn /

1

01 sn C t 

1


.sn /;
x.s N n / .sn /

to again obtain a limiting object ˛ with the property that 0 .t;
/ D 0 provided t 2 I  , the latter its life-span interval.

1




10.2 Rigidity I: Harmonic maps and reduction to the self-similar case

447

We now deduce the desired contradiction from this situation: as in Proposition 10.1, we can associate a weak wave map U  from R2C1 ! S with the limiting object ˛ , and this wave map has the property that @ t U  D 0;

t 2 I :

Moreover, we have 2 X X 2

@˛ U  2 2 D

 2 ¤ 0: ˛ L L x

˛D1

x

˛D1;2

We have thus obtained a nonvanishing finite energy harmonic map U  W R2 ! S, which is impossible, see [39]. We therefore conclude that (10.30) holds. The remainder of the argument is essentially the same as that in Lemma 5.4 of [14]: by Corollary 10.2, 2 Z ˝ ˛ d X xi .x=R/ @ t U.t; x/; @i U.t; x/ dx 2 dt iD1 R Z ˇ ˇ
ˇ@ t U.t; x/ˇ2 dx C O r.R/ (10.32) D R2

where Z r.R/ WD

2 X ˇ ˇ ˇ@˛ U.t; x/ˇ2 dx:

ŒjxjR ˛D0

Furthermore, by definition of R0 ."/ > 0, for all t  0 one has Z ˇ ˇ ˇ@˛ U.t; x/ˇ2 dx  ": ˇ ˇ x.t/ N ˇxC ˇR0 ."/ .t/

Therefore, if the lemma were to fail, then (assuming x.0/ N D 0 as we may) one would have ˇ x.t ˇ ˇ N /ˇ ˇ  R R0 ."/ ˇ .t / for all 0  t < CR. In view of the preceding, one concludes that r.R/  C5 " for some absolute constant C5 . Now choose " > 0 so small that Z  Z ˇ ˇ
 ˇ@ t U.t; x/ˇ2 dx C O r.R/ dt  ˛ 2 I R2

448

10 The proof of the main theorem

for all I of unit length. In view of the a priori bound ˇZ ˝ ˛ ˇˇ ˇ sup ˇ xi .x=R/ @ t U.t; x/; @i U.t; x/ dx ˇ  C6 REcrit t

R2

one obtains a contradiction by integrating (10.32) over a sufficiently large time interval.  Next, we obtain a contradiction to Lemma 10.9 by means of Proposition 10.7. This is completely analogous to Lemma 5.5 in [14]. Lemma 10.10. There exists "2 > 0, R1 ."/ > 0, C0 > 0 such that if R > R1 ."/, t0 D t0 .R; "/ are as in Lemma 10.9, then for 0 < " < "2 one has t0 .R; "/ >

C0 R : "

Proof. This follows from Proposition 10.7 by the same argument as in [14].



Proof of Proposition 10.8 for T1 D 1. Choosing " small in Lemma 10.9 and Lemma 10.10 leads to a contradiction.  It remains to prove Proposition 10.8 in case T1 < 1. This will be lead to a contradiction as in [14], by a reduction to the case of a self-similar blow-up scenario. More precisely, recall from Lemma 10.4 above that .t / 

C0 .K/ ; 1 t

0 C.k kS ; "0 / may be bounded by  "0 k 0 kS . In fact, similar reasoning allows us to reduce to the case when r1 < k0 C O.1/, k D k0 C O.1/, where the implied constant O.1/ may of course be quite large depending on k kS and "0 , and furthermore we may assume that ki D rj C O.1/, i D 1; 2; 3; 4, j D 1; 2; 3. The proof of Lemma 6.1 also implies that we may assume all inputs other than the ones of the null-form Qk .Pk3 3 ; Pk4 4 / to be essentially in the hyperbolic regime, i.e., we may replace Pkj j by Pkj Qk3 CC

3 ; Pk4


i

4/

where the implied constant C is large enough, depending on k kS ; "0 . Then if we write Pk3 Q>k3 CC

3

D Pk3 QŒk3 CC;k0 C10

3

C Pk3 Q>k0 C10

3;

the contribution of the first term on the right is seen to be very small, by placing 1; 1 ;1 the output into either XP k0 2 or L1t HP 1 . On the other hand, consider now the contribution of the second term on the right. Here one places the output into 1 C"; 1 ";2 provided the output is in the elliptic regime, or else into L11 HP 1 . XP k02 In either case, one verifies that provided r1 < C is sufficiently negative, the contribution is small in the above sense. Hence assume now that r1 D O.1/ (which again means an interval depending on k kS as well as "0 ), and as before Pk3 3 D Pk3 Q>k0 C10 3 . Then we may replace Pk4 4 by Pk4 Qk3 CC

3 ; Pk4 Qk3 CC 3 , provided we dyadically localize the latter. But then a straightforward argument using the “divisibility” of L2t;x 1 reveals that we may pick intervals fIj gjMD1 with M1 D M1 .k kS ; "0 / such that  h X

rx;t Ij Pk Pk0

0r

1

Pr1 Pk1

1r

1

Pr2

k0 2Z

.Pk2

2r

1

Pr3 Qk .Pk3 Q>k3 CC

3 ; Pk4 QkC10 jrj 1 .jrj 1 Œ jrj 1 . 2 / /
C Pk PrCO.1/

jrj

1

Pr jrj

1

Pr1 Πjrj

1

Pr .

2

/Pr2


:

465

11.3 Completion of a proof

Now observe that for the first factor on the right we have the estimate

X


jrj 1 Pr jrj 1 Pr1 Πjrj 1 PrCO.1/

D

X

X

r1 Dr2 CO.1/>rCO.1/ c1;2 2Dr1 ;r 1

jrj .2

r .1 "/.r r1 /

2

. 2.1

"/.r r1 /

1

rj

r1 dist.c1 ;

Pc1 Πjrj

c2 1

Pr

/.2r 2

P r1 C O.1/. Furthermore, all inputs may be assumed to be in the hyperbolic regime (up to large constants only depending on Ecrit ). But then the smallness can be forced by shrinking Ij suitably and forcing that X

I Qk Pr .PrCO.1/ ; PrCO.1/ / 2 1  1; j 2 P r2Z

Lt H

2

see the proof of Proposition 6.1. For the higher order errors of long type (recall the discussion in Chapter 6), the smallness is achieved by exploiting the “divisibility” of the norms L8t;x . (3) Only 2 factors present in addition to factors . All of these terms contribute to 2 . If at least two factors 2 are present, we clearly obtain the desired smallness from Proposition 6.1. Hence now assume that only one such factor is present. If this factor is in the position of 0 , then we obtain smallness via “divisibility” or L2t;x as in Case (2). If this factor is in the position of some j with j D 1; 2; 3; 4, one obtains smallness via a slightly different divisibility argument: first, reduce to the case when 0 and one of the j which represents a have angular separation between their Fourier supports: to do this, consider for example 
 rx;t Pk Pk0 r 1 Pr1 Pk1 r 1 Pr2 .Pk2 2 r 1 Pr3 Qk .Pk3 ; Pk4 / : Again we may assume that kj D ri C O.1/ for j D 1; 2; 3; 4, i D 1; 2; 3, and k0 D k C O.1/ > r1 C O.1/. Here we can use the divisibility of L2t;x by placing 1 Pr3 Qk .Pk3 ; Pk4 / into L2t HP 2 , see the proof of Proposition 6.1. On the other hand, for the expression 
 rx;t Pk Pk0 r 1 Pr1 Pk1 r 1 Pr2 .Pk2 r 1 Pr3 Qk .Pk3 2 ; Pk4 / ;

469

11.5 Competion of proofs

one obtains smallness from the divisibility of L4t L1 x , more precisely, that of X 4 kPk kL 4 1: L k2Z

t

x

11.5 Completion of the proof of Proposition 9.12, Part II Here we show how to obtain the bootstrap for the elliptic part of , i.e., QD . Recall that we solve for QD  via the equation QD  D QD

5 hX i D1

F˛2iC1 .

C /

i

QD

5 hX

i F˛2i C1 . /

i D1

where the F˛2iC1 are obtained as described in Chapter 3. In particular, F˛3 . / constitutes the trilinear null-forms. Of course the proper interpretation of the right-hand side is that we substitute suitable Schwartz extensions for and  but which agree with the actual dynamic variables on the time interval that we work on. We start by considering the trilinear null-forms, which with the appropriate localizations we schematically write as     rx;t P0 QD . C/r 1 Qj . C; C/ rx;t P0 QD r 1 Qj . ; / : We need to show that we can write the above expression as the sum of two terms, which, when evaluated with respect to k
kN Œ0 , improve the bootstrap assumption (9.19). Now we distinguish between various cases: (1) Here we consider the trilinear terms which are schematically of the form   rx;t P0 QD r 1 Qj . ; / : We decompose this into two further terms according to the type of : (1a): This is the expression rx;t P0 QD Œ1 r 1 Qj . ; /. Recalling the fine structure of the trilinear terms described in Chapter 3, we see that this can be decomposed into two types of terms   rx;t P0 QD 1 r 1 Qj . ; /   D rx;t P0 QD 1 r 1 Qj I c . ; / (11.2)   1 C rx;t P0 QD .R /1 r Qj I. ; / (11.3)

470

11 Appendix

where in the last term an operator R may be present or not. Start with the first term on the right, which we write as  Qj I c . ; / X  D rx;t P0 QD Pk1 1 r

 rx;t P0 QD 1 r

1

1

 Pr Qj I c .Pk2 ; Pk3 / :

k1;2;3 ;r

Now the fundamental trilinear estimates in Chapter 5, see in particular (5.39), imply that under the bootstrap assumption kPk 1 kS Œk  C4 dk with some C4 D C4 .Ecrit /, we have

X 

rx;t P0 QD Pk1 1 r

1



Pr Qj I c .Pk2 ; Pk3 /

N Œ0

jk1 j1;k2;3 ;r

 C4 d0 ;

which is as desired. In fact, the proof of (5.39) cited above implies that one also obtains

X  

1 c r P Q P  r P Q I .P ; P /  C4 d0 ;

x;t 0 D r j k1 1 k2 k3 N Œ0

k1;2;3 ;jrj1

and finally, again the trilinear estimates from Chapter 5 imply that we may also assume k2;3 D O.1/ (implied constant depending on Ecrit ). Hence we may assume for the present term that all frequencies are O.1/. Thus we may now reduce to considering X   rx;t P0 QD Pk1 1 r 1 Pr Qj I c .Pk2 ; Pk3 / : k1;2;3 CO.1/DrDO.1/

Now if one of the inputs of the null-form Qj I c .Pk2 ; Pk3 / is of elliptic type, either at least one of 1 and the other input has at least comparable modulation, or else the output inherits the modulation from the large modulation input. In the former case, it is straightforward to obtain smallness: Indeed, consider for example X

X

rx;t P0 QD

k1;2;3 CO.1/DrDO.1/ l1

 Pk1 Q