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 . .
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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
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. 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
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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<` 10 .e i j D1 @j j /PŒ` 10;`C10 ˛ H ı2 P2
1 1 C P` ŒPŒ` 10;`C10 .e i j D1 @j j /P<`C15 ˛ H ı2
8
1 Introduction and overview
C
P2 X
P` ŒPk .e i j D1
1@ 1 j j
/PkCO.1/
˛ H ı2
k>`C10
. kPŒ`
10;`C10 ˛ kH ı2 P2
1 1 C PŒ` 10;`C10 .e i j D1 @j j / H ı2 kP<`C15 ˛ k1 P2 X
Pk .e i j D1 1 @j j1 / ı kPkCO.1/ ˛ k1 C H 2 k>`C10
Next, one has the bounds
rx e i
1
P2
1 j D1 @j j
kP<`C15
2 L1 t Lx
. 2.1 ˛ kL1 x
. k j1 kL1 L2x ; t
ı1 /`
k
˛ kH ı1
where the first one is admissible due to energy conservation for the derived wave map, see (1.5). In conclusion, kP` ˛ kH ı2 . kP`CO.1/
˛ kH ı2
C 2.ı2
ı1 /`
k kL2x k kH ı1 :
Summing over ` 0 yields k kL1 t
. T0 ;T1 /IH ı2 .R2 /
< 1:
(1.18)
By the subcritical existence theory of Klainerman and Machedon, see [18]– [16] as well as [20], [21], the solution can now be extended smoothly beyond this time interval. More precisely, the device of frequency envelopes allows one to place the Schwartz data in H s .R2 / for all s > 0 initially, and as it turns out, also for all times provided s > 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˙ >0b . ; /:
1
s;p;q Then the relevant XP k spaces here are defined as X q k kXP 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 k kL1 L2x . k k t
1 ;1 0; 2
XP k
as well as k kL1 . k k 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 Π>0 FO .; / _ C a . / O Π<0 FO .; / _ : P F D a ./ We will also encounter other rectangles R which are obtained by dividing Rk; in the radial direction into 2 m many subrectangles of comparable size where m < 0 is some integer parameter (it will suffice for us to consider ` m 0 where 2 C` ). The collection of these Prectangles will be denoted by Rk;;m , and we introduce projections PR so that R2Rk;;m PR D Pk; . Figure 2.1 exhibits
27
2.1 Preliminaries
such a collection of rectangles. The sector ABCD is of length 2k and width 2`Ck , whereas the shorter segments AP1 , P1 P2 etc. are of length 2kCm . O ˝, where We shall frequently use Bernstein’s inequality: if supp. / 1 1 ˝ R2 is measurable, then k kq . j˝j p q k kp for any choice of 1 p q 1. We shall also require the following variant of Bernstein’s L2 ! L1 bound, which is obtained by combining the standard form of this bound with the 1C2 /-Strichartz estimate for the wave equation. This type of estimate L4t L1 x .R appears in [59], but the following formulation is from [23], which involves one further localization on the Fourier side. We present the proof for the sake of completeness. ˚ Lemma 2.1. Let Dk;` be a cover of jj 2k by disks of radius 2kC` . Then for all j k, X c2Dk;`
2 kPc Qj kL 2 1 t Lx
12
`
. 22 2
3k 4
2
j
k 4
k kL2 L2x ; t
(2.2)
for any which is adapted to k. Proof. We follow the argument in [58], but use the small-scale Strichartz estimate of Klainerman–Tataru at a crucial place, see Lemma 2.17 below. First, set j D 0, whence k 1. Construct a Schwartz function a.t / whose Fourier transform is supported in j j 1, and which satisfies 1D
X
a3 .t
s/
s2Z
for all t 2 R. Then
X 3
kPc Q0 kL2 L1 a .t x t
s/Pc Q0 L2 L1 t
s
X
a2 .t . s
.
X
a.t s
s/Pc Q0
x
21
2
2
L t L1 x
21
2 s/Pc Q0 L4 L1 : t
x
Now one notes that the function a.t s/Pc Q0 satisfies almost the same assumptions about modulation ( 1) and frequency localization as Pc Q0 . Therefore,
28
2 The spaces S Œk and N Œk
we can apply the improved Strichartz estimate of Klainerman–Tataru [22] to estimate
`
a.t s/Pc Q0 4 1 . 2 3k 4 2 2 kP c k 0; 1 ;1 : L L XP k
x
t
2
Thus kPc Q0 kL2 L1 .2 x
3k 4
t
`
22
21
2 s/Q0 Pc 0; 1 ;1
X
a.t
XP k
s2Z
.2
3k 4
2
`
2 2 kPc Q0 kL2 L2x : t
The lemma follows via Plancherel’s theorem.
The previous proof was based on the following small-scale version of the usual L4t L1 x -Strichartz estimate. It was obtained by Klainerman and Tataru [22]. Lemma 2.2. With Dk;` as above, one has X
2 kPc e i t jrj f kL 4 1 L t
c2Dk;`
x
12
`
. 22 2
3k 4
kf kL2x
(2.3)
for any k-adapted f . In particular, X
2 kPc kL 4 1 L
c2Dk;`
t
x
12
`
. 2 2 k k
3 ; 1 ;1 2
XP k4
(2.4)
for any Schwartz function which is adapted to k.
2.2
The null-frame spaces
In contrast to sub-critical HP s .R2 / data with s > 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 kGkNFAΠWD inf dist.!; / !622
k kPWAΠWD inf k kL2
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 ; kPWΠWD k k;! kL2
Z
1 t! Lx!
d! D
k
k;! kPWAΠd!:
In other words, PWΠis 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 kPWAΠ1 (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 k F kNFAŒ dist.; 0 /
1
k kPWAΠ0 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 k kL1 L2x dist.!; /. The previous t! ! estimate then implies the bound k F kNFŒ dist.; 0 /
1
k kPWΠ0 kF kL2 L2x :
(2.11)
t
The dual space NFΠis characterized by the norm k kNFΠD sup dist.!; /k kL1 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 k kS Œk; WD k kL1 L2x C jj
1 2
t
2
k 2
k kPWŒ C k kNF Œ
(2.12)
k kS Œk WD k kL1 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 k kS Œk; : kf kS Œk; kf kL1 tx
(2.16)
One has the following scaling property: k kS 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 k kS 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 k kS 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 k k 2 2 k kL1 L2x . k k L L L Lx t
t
k kL2
1 t! L x !
1 2
1
O 2 . .2j jj/ k k L
2 ! L !
D .2j jj/ 2 k kL2 L2x t
j 2
k kL1 L2x . t!
!
2 k kL2 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
. k k
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 k kL1 L2 . k k 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
j j/
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
k kS Œk ; "/
; 1/2
(2.23) j 2
k kS 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 . k kS 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
k F kNFŒ
j 0 j 2 2 2 . k kS Œk 0 ; 0 kF kL2 L2x ; t dist.; 0 /
k kL2 L2x
jj 2 2 2 . k kS 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 j jj 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 k kS 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<`C`0 Ck F k2NFΠ0
12
C 2k (2.33)
with some absolute constant C . Proof. By scaling invariance, we can set k D 0. Moreover, fix `0 ` and `0 j 0. Choose ! 0 D !. 0 / 2 S 1 n .2 0 / for each 0 which attain the respective
38
2 The spaces S Œk and N Œk
NFΠ0 norm. Then one has X X X kPR P 0 Q<`C`0 F k2NFΠ0 0 2C`0 R2R0; 0 ;j 0
.
X X
X
d.! 0 ; 0 /
0 2C`0 R2R0; 0 ;j 0
.
X
inf
1 !2S n.2/
d.!; /
2
2 kPR P 0 Q<`C`0 F kL 1
2 t! Lx!
X
2
X
2 kPR P 0 Q<`C`0 F kL 2
0 2C`0 R2R0; 0 ;j 0
x!
21 2
1
L t!
(2.34) .
X
inf
1 !2S n.2/
d.!; /
2
2 kF kL 1
2 t! L x !
:
(2.35)
To pass to (2.34) we used the inclusion `2 .L1t! / L1t! .`2 /, whereas orthogonality implies (2.35). Indeed, first note that [ supp ŒPR P 0 Q<`C`0 F /.t! ; /^ ˘! supp F ŒPR P 0 Q<`C`0 F t! 2R
where the Fourier transform on the left-hand side is in x! and on the right-hand side in .t! ; x! /. Second, the sets on the right-hand side enjoy a finite overlap property by Lemma 2.6. In what follows, we will often need to split a wave into C C where O / _; O / _: C WD Œ 0 .; WD Œ <0 .; The question arises whether the spaces S Œk and N Œk are preserved under these operations. Lemma 2.12. For any Schwartz function which is adapted to k, k ˙ kS Œk C k kS Œk ;
kF ˙ kN Œk C kF kN Œk ;
with some absolute constant C . Proof. We set k D 0 and assume that is adapted to k D 0. Let 0 be a bump function on the line with 0 . / D 1 on C 1 and 0 . / D 0 if 2C 1 where C > 1 is some large constant. Then C c . ; / D 0 jj Π0b . ; / C .1 0 / jj Π0b . ; /:
39
2.2 The null-frame spaces
Denote the two functions on the right-hand side by .C;1/ and .C;2/ , respectively. Then .C;1/ D (2.36) where is a measure of bounded mass. Therefore, k .C;1/ kS Œ0 C k kS Œ0 : Next, k .C;2/ kS Œ0 C k .C;2/ k
1 ;1 0; 2
XP 0
C k kS Œ0
where we used the Plancherel theorem in the final step. For N Œk it will suffice to check the case of L1t L2x -atoms. For these, we write F C D F .C;1/ C F .C;2/ as above. The first term here is fine from (2.36), whereas the second is placed in L2t L2x and bounded by means of (2.32). Another piece of terminology used by Tao is the following: Definition 2.13. We shall say that a family fm˛ g˛ is disposable, if T˛ f WD .m˛ fO/_ D f ˛ where ˛ are measures with uniformly bounded mass: sup k˛ k C < 1 ˛
with some universal constant C . Clearly, disposable multipliers give rise to bounded operators on any translation invariant Banach space. Thus, if X is a Banach space of functions on RnC1 with the property that for all f 2 X one has kf .
y/kX D kf kX ;
8 y 2 RnC1 ;
then sup˛ kT˛ f kX C kf kX . The following observation will be a useful device for removing frequency cut-offs. Lemma 2.14. The families ˚ Pk; k; ;
˚ Pk Qj j k ;
˚ Pk Q<j j k ;
˚ ˙ Pk Q
C k
40
2 The spaces S Œk and N Œk
are disposable. In the first family is any cap, whereas in the last family C > 0 has to be chosen such that the support of the multiplier associated with Pk Q
100
Proof. Without loss of generality one may take k D 0. Then these statements reduce to simple exercises in harmonic analysis. The following fact will serve as a substitute for the previous problem in a non-disposable context. p
Lemma 2.15. Qj , Q<j are bounded on L t L2x for every 1 p 1 with a constant independent of j 2 Z. Proof. The inverse Fourier transform of Q<j with respect to time alone is Z b . ; / d e i t m0 2 j . jj/ F Z O ds D 2j m c0 2j .t s/ e i jj.t s/ F .s; / O in the second line denotes the Fourier transform with respect to the where F .s; / second variable. Consequently, kQ<j F kL1 L2x kc m0 k1 kF kL1 L2x t
as claimed.
t
The previous result, combined with Lemma 2.7, implies the following square function bound. Corollary 2.16. For all j; k 2 Z and all k-adapted Schwartz functions one has kQ<j kS Œk C k kS Œk with some absolute constant C . 2 Proof. We may again take k D 0. The L1 t Lx -component of the S Œ0-norm is covered by Lemma 2.15. The XP s;b;q -components are obvious, the Strichartz norms as well by construction, and the square function is a consequence of Lemma 2.7.
41
2.2 The null-frame spaces
We remark that the analogous statement for N Œk holds as well, see Corollary 2.23 below. Next, for the sake of completeness we state the full range of Strichartz estimates that follow from (2.14). Lemma 2.17. For any 4 p 1 and 2 q 1 which satisfy X
2 kPc kL p q L t
12
C 2`.1
x
2 p
2 q
4" p /
1 p
2k.1
2 q/
1 p
C
1 2q
k kS Œk
14 , (2.37)
c2Dk;`
for any k 2 Z, ` 0, and with an absolute constant C . Proof. Assume first that X
2 kPc kL p q L t
x
12
1 p
C
1 2q
D 14 . By interpolation, and with D
X
X
x
t
c2D0;`
c2D0;`
2.1 / 2 L1 t Lx
2 kPc kL 4 1 kPc k L
2 kPc kL 4 1 L x
t
c2D0;`
2 X
4 p,
12
2 kPc kL 1 2 L t
c2D0;`
1 2
x
(2.38) .2
. 12
"/`
k kS Œ0 :
To pass from (2.38) to the last line, one uses (2.14) as well as the energy compo1 2 1 nent of (2.15). For larger q, one gains a factor 22`. 2 p q / by Bernstein’s inequal1 2 ity, and rescaling to frequency 2k yields a factor of 2k.1 p q / as claimed. Finally, we conclude this section with the following useful fact. Lemma 2.18. Let be adapted to 0. Then for any m0 X
kP0; k2L1 L2 t
2Cm0
12
x
10,
. jm0 jk kS Œ0
Proof. First, X 2Cm0
kP0; Q2m0 k2L1 L2 x t
12
.
X
kP0; Q2m0 k2S Œ0;
21
2Cm0
. k kS Œ0
42
2 The spaces S Œk and N Œk
by (2.15). Second, X
X 2m0 `0
kP0; Q` k2L1 L2 t
2Cm0
12
X
X
.
kP0; Q` k2 0; 1 ;1
x
2m0 `0
XP 0
2Cm0
21
2
. jm0 jk kS Œ0 : And third, kP0; Q0 kL1 L2x . kP0; Q0 k
1 ;1 0; 2
XP 0
t
. kP0; Q0 kXP 0;1
";2
;
0
whence X
kP0; Q0 k2L1 L2 t
2Cm0
12
.
X
x
kP0; Q0 k2P 0;1 X0
2Cm0
. kP0 Q0 kXP 0;1 0
";2
12 ";2
. k kS Œ0
as claimed.
The central problems concerning the S Œk and N Œk spaces are how to obtain an energy estimate and how to control the trilinear nonlinearities appearing in the gauged wave map system. We begin with the energy estimate, and then develop bilinear bounds which are preliminary to the central trilinear bounds.
2.3
The energy estimate
The purpose of this section is to prove the energy estimate in the context of the SŒk and N Œk spaces, see Proposition 2.26 below. First, we require some technical lemmas. The first two of these lemmas will arise in the Duhamel integral. Lemma 2.19. For any F which is k-adapted and satisfies F D QkCC F , k RC F kN Œk . kF kN Œk
(2.39)
where RC acts only in time. Proof. We may assume that k D 0. This is clear if F is an energy atom. Next, we 1 consider the XP 0; 2 ;1 -atoms. First, let FO be supported on jj 1; jjj j jj 2j
43
2.3 The energy estimate
with kF kL2 L2x . 2j=2 . Then t
k RC F kN Œ0 . kP<j . RC /F kN Œ0 C kPj . RC /F kN Œ0 .2
j=2
kP<j . RC /F kL2 L2x C kPj . RC /F kL1 L2
.2
j=2
kP<j . RC /k
.2
j=2
t
L1 t
kF kL2 L2x C kPj . RC /kL2 kF kL2 L2 t
t
kF kL2 L2x . 1: t
Now let F be a wave-packet atom, i.e., for some ` 100, X ˚ c / > 0; jj 1; jjj j 22` ; 2 F ; supp.F F D 2C`
and
P
kF k2NFΠ1. We write, with j D 2`, RC D P<j . RC / C Pj . RC /
as before. Then P<j . RC / does not significantly change the support properties of F . Moreover, since kP<j . RC /k1 . 1, we see that P<j . RC /F is essentially a wave-packet atom. On the other hand, since kF kL2 L2x . 2j=2 from (2.32) t we conclude that kPj . RC /F kL1 L2 . 2
j=2
kF kL2 L2x . 1 t
(2.40)
which proves (2.39).
It is important to note that the previous lemma fails for functions in N Œ0 1 C"; 1 ";2 -norm is finite on functions which which are “elliptic” since the XP k 2 are too singular. But in the elliptic regime, there will be no need for the Duhamel formula and thus for Lemma 2.19. A technical variant of the preceding lemma will be needed in the proof of Proposition 9.14: Lemma 2.20. Let F be as in the preceding lemma, S 1 , and ! … ˙2. Then for any c 2 R we have ˙ kP Q
where the implied constant only depends on C and jj.
44
2 The spaces S Œk and N Œk
Proof. It is essentially identical to the preceding one: one replaces RC by Œt! c and L1t L2x by L1t! L2x! . The Duhamel formula (in other words, 1 ) introduces a Hilbert transform in the normal direction to the light-cone. The following lemma is of this type. Lemma 2.21. Let be a smooth function on R such that 0 1, .u/ D 1 on 1 u 1, supp./ Œ 2; 2, and 0 .u/ 0 on u 0, and 0 .u/ 0 on u 0 0. Define C T .t / WD Œ0;1/ .t =T / for each T 1. Then, with WD Œ0;1/ , 1 b .T / C 1 : i
C c T . / D
(2.41)
c C C In particular, c T . / D a T .a / for all 0 < a < 1 and a
ˇc ˇ ˇC . /ˇ . j j T
1
ˇd c ˇ ˇ C . /ˇ . j j T d
;
2
:
Moreover, let D . / be a smooth function on Œ 1; 1 with .0/ D 1 and 1 on Œ 1; 1. Then ˇ ˇˇ ˇc c C C sup ˇT . / T . / ˇ C k0 k1 j j1
with an absolute constant C . Finally, if T 0 2 ŒT =2; 2T , then
b
ˇc ˇC . /
ˇ ˇ C T 0 . / C T min 1; .T j j/
T
100
with a constant C that only depends on . Proof. Integrating by parts in C c T . / D
Z e
i u C T .u/ du
yields (2.41). In particular, ˇ ˇc ˇC . /ˇ . j j T
1
min.T j j; .T j j/
100
/;
and similarly for the derivatives. Next, write C c T . /
C c T . / D
1 1 b T. / C 1 : b .T / C 1 C i i. /
45
2.3 The energy estimate
In view of our assumptions on , ˇ 1 ˇ .1 . /
1
ˇ /ˇ . k0 k1 ;
and similarly for the terms involving b .T /. The final statement is an immediate consequence of (2.41). The following representation of waves 1 F with F a null-frame atom will be useful in several instances. Hence, we state it as a separate fact. Lemma 2.22. Assume that F 2 N Œ0 is a wave-packet atom, i.e., F D F C D P 2C` F with X kF k2NFŒ 1 2C`
for some `
100, see Definition 2.9. Then Z t sin .t 1 .t / WD F .t / D 0
s/jrj F .s/ ds jrj
admits a decomposition of the form D
1
F1 C
XZ 2C`
1 2 ;a C B;a ;a da
(2.42)
R
where kF1 kL1 L2x . 1 and t
sup kB;a kL1 C; t;x ;a
2 Z X
j k k ;a
j D1
with an absolute constant C , whence XZ j sup k ;a k j D1;2
Finally,5 for j D 1; 2
b
j c ; supp ;a C supp F
0; 1 2 ;1
XP 0
0; 1 ;1 XP 0 2
da
3
da C kF kNFŒ
2
. 1:
2 c supp B;a ;a C supp F
(2.43)
for all a and and some absolute constant C . 5
Here cE denotes the dilation of the convex set E about its center of mass by the constant c.
46
2 The spaces S Œk and N Œk
Proof. As in the proof of Lemma 2.19, we first write RC D P2` . RC / C P<2` . RC / DW 1 C 2 : Then F1 WD 1 F satisfies kF1 kL1 L2x . 1, see (2.40). On the other hand, F2 WD t F is again a wave-packet atom at essentially the same scale as F , i.e., F2 D 2 P e 2C` F with X e k2 kF 1: NFŒ
2C`
Define ˚ WD
1F . 2
Then ˚ D Z
1
˚T; .t / WD
1
P
limT !1 ˚T; with sin .t s/jrj C e .s/ ds: T .t s/F jrj
It suffices to prove that Z ˚T; D
1 2 T;;a C BT;;a T;;a da
where sup sup kBT;;a k1 . 1
T 1 a
and
Z
j
k T;;a k
sup sup
0; 1 2 ;1
XP 0
j D1;2 T 1
da . kFQ kNFŒ
(2.44)
both uniformly in . Fix 2 C` and ! D !./ 2 S 1 n .2/ so that d.!; /
1
e k 1 kF L
2 t! Lx!
e kNFΠ: 2kF
As usual, we foliate relative to t! . More precisely, define e .t; x/.a; x! / fa .x! / D F where t! D a means that .t; x/.a; x! / WD a!C C x! : By Lemma 2.6 supp.fOa / ˘!
˚ . ; / W jj 1; b 2 ; j
jjj . 22`
DW R;! : (2.45)
47
2.3 The energy estimate
Let .t! ; x! / denote the null-frame coordinates. Then
jj D
2!1 ! C jj
h.! /
(2.46)
where !1 WD ! ! and, with j! j2 D .!1 /2 C .!2 /2 , one has h.! / WD 2 /2 .! 1 . 2!
Moreover, j!1 j d.!; /2 and j!2 j . d.!; / by elementary geometry (cf. Lemma 2.6). We define P;! f WD F 1 R;! .! /f . ! ; ! / ; where R;! is a smooth cut-off adapted to the rectangle R;! in the ! -plane. Furthermore, we set h i C Qj;! f WD F 1 m0 2 j C d 2 .; !/. ! h.! // f . ! ; ! / : C By construction, P;! Q2`;! is essentially the same as P0; Q2` , see Lemma 2.6. In fact, one has
e D P;! QC e F F 2`;! C and P;! Q2`;! is disposable. Clearly, Z ˚T; D ˚T;;a da
where ˚T;;a .t/ WD
C P;! Q2`;!
Z
1
sin .t
1
s/jrj C T .t jrj
s/ı.s!
a/fa ds:
C Then ˚T;;a D P0;C Q2`CC ˚T;;a and C ˚T;;a D P;! Q2`;! F
1
c .C T .
jj/
C c T .jj C //e
i ! a
fOa .! / : (2.47)
C We claim that the contribution of jc T .jj C /j . 1 to (2.47) can be added 1 to T;;a . In fact,
C
1 i ! a O Q F O.1/e . / f . /
2`CC R;! ! a ! 0; 1 ;1 XP 2
0
(2.48) . 2` m0 2 2` C . jj/ R;! .! /fOa .! / 2 2 L L
. 22` d.!; /
1
kfa kL2 ;
48
2 The spaces S Œk and N Œk
which is better than needed. To pass to the final estimate here we used Lemma 2.6, especially (2.27); the latter estimate can be applied for fixed , since then ! D C .; /. Next, we split the contribution of c .jj / to (2.47) into several !
T
h.! / D 0 implies that p 2jj D C jj D 2 h.! /= 2 C ! e1 DW g.! /
pieces. Since !
where e1 D .1; 0; 0/, one has by Lemma 2.21 C c T .
jj/
1 C 2! c T g. / . ! !
1 h.! / D O 1 D O d.!; / !
2
:
(2.49)
In view of (2.48) (which gains a factor of 22` . d.!; /2 ), the contribution 2 1 1 of (2.49) to (2.47) can again be added to T;;a . Set b D b.! / D g.!! / . Fur.0/
.0/
thermore, set b0 WD b.! / where ! Lemma 2.21, C c T
2 1 ! . ! g.! / Db
b
h.! / D b
1 C b0 T
!
2 R;! is fixed, cf. (2.45). In view of
b
1 C bT
!
h.! /
h.! / C O T min.1; .T j
jjj/
100
/
(2.50)
where we used that b b0 on R;! . The computation from (2.48) above now 1 shows that the O./ term in (2.50) can be added to T;;a . It therefore remains to analyze the contribution of the first term in (2.50) to (2.47). Define Z BT;a; .t; x/ WD C .t s! /e i a.t! s! / c m0 .s! / ds! b0 T ! where 2j CC d 2 .; !/ DW (recall that m0 is even). On the one hand, kBT;a; k kc m0 k1 and on the other hand, C P;! Q2`;! F
D BT;a;
b
1 C b b0 T . ! h.! //e i ! a fOa .! / h i g.! / ih.! /a O F 1 ı ! h.! / R;! .! / e f . / a ! 2!1 1
2 DW BT;a; T;;a : 2 2 By inspection, the Fourier support of T;;a as well as that of BT;a; T;;a are no larger than that of the original wave-packet F (up to a dilation by a constant).
49
2.3 The energy estimate
Finally, by a calculation similar to (2.48), 2 k T;;a k
. F
1 ;1 0; 2
XP 0
1
g.! / h.! / e 2!1
ı !
. d.!; /2
lim sup F
1
ŒM M.
M !1
. d.!; /
1
ih.! /a
R;! .! /fOa .! /
g.! / jj/ e 2!1
ih.! /a
0; 1 2 ;1
XP 0
R;! .! /fOa .! /
0; 1 2 ;1
XP 0
kfa kL2 :
This concludes the proof of the lemma.
In passing, we now prove the analogue of Lemma 2.15 for null-frame coordinates, which then gives Corollary 2.16 for the N Œk spaces. Corollary 2.23. For all F 2 N Œk and all j 2 Z one has kQj F kN Œk C kF kN Œk with some absolute constant C . P s;b Proof. This Pis clear if F is either an energy or a X -atom. Therefore, suppose that F D F is a wave-packet atom with k D 0. It suffices to prove that kQj F kNFŒ C kF kNFŒ : This in turn follows from kQj F kL1
2 t! Lx!
C kF kL1
2 t! Lx!
(2.51)
which holds uniformly in ! 2 S 1 n .2/. Fix such an ! and apply Plancherel’s theorem in x! . By (2.46), F2 Q<j F .t! ; ! / Z 2!1 D m0 2 j ! C jj
h.! / e i ! .t!
s! /
d ! F2 F .s! ; ! / ds!
where for our purposes here F2 refers to a partial Fourier transform relative to the
50
2 The spaces S Œk and N Œk
second variable x! . In view of j!1 j d.!; /2 , ˇZ 1 ˇ ˇ ˇˇ ˇF2 Q<j F .t! ; ! /ˇ ˇ m0 2 j 2! ! ˇ C jj .N 2j d.!; /
2
Z
˝
2j d.!; /
2
.t!
ˇ ˇ ˇ s! / d ! ˇ ˇ ˇ ˇ ˇF2 F .s! ; ! /ˇ ds! ˇ ˛ Nˇ ˇF2 F .s! ; ! /ˇ ds! : s! / h.! / e i ! .t!
Performing an L2! estimate followed by an L1t! bound yields (2.51).
Finally, there is the following simple fact that will play a role in the proof of the Strichartz component of k kS Œk . Lemma 2.24. Let akm 0 for all 1 m M and 1 k K. Suppose PK a kD1 km for all m where 0 is arbitrary. Then K X M X kD1
2 akm
12
M
1C 2
K
1 2
;
(2.52)
mD1
for all 0 1. Proof. Denote the sum in (2.52) by S. On the one hand, S
K X M X
akm M:
kD1 mD1
On the other hand, K X M 12 p X p 2 S K akm KM ; kD1 mD1
and the lemma is proved.
Now we can state the main energy bound. We begin with the easier elliptic regime. Lemma 2.25. Let F be a space-time Schwartz function which is adapted to k 2 Z. Assume furthermore that F D I c F and set WD 1 F , which is defined via division by 2 jj2 on the Fourier side. Then k kS Œk . kF kN Œk with an absolute implicit constant.
51
2.3 The energy estimate
Proof. We may again assume that k D 0. We then need to prove that k kXP 0;1 ";2 . min kF kL1 L2x ; kF kXP 0; 1 ";2 ; k
t
(2.53)
0
since, as we observed after Definition 2.3, the norm on the left-hand side dominates the other norms which make up k kS Œk . If we select kF kXP 0; 1 ";2 on the 0 right-hand side of (2.53), then this inequality is obvious. On the other hand, if we select kF kL1 L2x , then one concludes via Bernstein’s inequality in time. t
Next, we deal with the hyperbolic regime. Proposition 2.26. Let k 2 Z and suppose 0 ; 1 are Schwartz functions in R2 which are adapted to k. Further, suppose F is a space-time Schwartz function which is adapted to k, and which is moreover hyperbolic, i.e., F D IF . Then the unique smooth solution of D F; .0/; @ t .0/ D . 0 ; 1 / satisfies k kS Œk . k. 0 ; 1 /kL2 HP
1
C kF kN Œk
(2.54)
with an absolute implicit constant. Proof. By scaling we may assume that k D 0. We first assume that F D 0. Then
b
.t /./ D cos.t jj/ b0 ./ C
sin.tjj/ b 1 ./: jj
Consequently, (2.54) follows from (2.21) upon sending T ! 1. Next, we assume that 0 D 1 D 0. By the Duhamel formula, Z t sin .t s/jj F .s/./ ds: .t / D jj 0
b
b
In other words, we need to show that
Z t sin .t s/jrj
Œ0;1/ .s/F .s/ ds . kF kN Œ0 :
S Œ0 jrj 1 In view of Lemma 2.19, we may remove the indicator function Œ0;1/ .t / D RC .t/ on the left-hand side. This is where we use that F D IF , but after this point we may no longer assume that F D IF since RC F loses this property.
52
2 The spaces S Œk and N Œk
The goal is now to prove uniformly in T 1
Z 1 C
sin .t s/jrj T .t s/F .s/ ds
S Œ0
1
. kF kN Œ0
(2.55)
where C T .u/ WD .u=T / RC .u/ is a bump function as specified in Lemma 2.21. Denote Z 1 .t / D sin .t s/jrj C s/F .s/ ds: (2.56) T .t 1
Then the space-time Fourier transform of equals (up to a multiplicative constant) C C b b . ; / D c jj/ c (2.57) T . T . C jj/ F . ; /; whence, by Lemma 2.21, ˇ ˇ jb . ; /j . ˇj j jjˇ
1
2
Œj j<10 C
b . ; /j Œj j10 jF
(2.58)
. kF kN Œ0
(2.59)
and thus also kQ0 k
1 ;1 0; 2
XP 0
C kQ>0 kXP 0;1 0
";2
from (2.32) and Lemma 2.25. By Lemma 2.4 it suffices to assume that F is either 1 an energy or a wave-packet atom. Moreover, in each of these cases the XP 0; 2 ;1 and XP 0;1 ";2 -components of the S Œk norm of can be ignored due to (2.59). Moreover, since the XP 0;1 ";2 -norm controls the entire S Œ0-norm in the elliptic regime, it suffices to consider only Q0 . In case F is an energy atom, i.e., kF kL1 L2 1 standard X s;b and Strichartz norms for the wave equation bound the norms in (2.13) and (2.14), see Lemma 2.2. We are therefore reduced to bounding (2.15), for which it suffices to verify that
C
sup sup sup P0; Q<2` sin.t jrj/C T .t /f S Œ0; . kf kL2x ` 100 T 1 2C`
for any f which is 0-adapted (the case of Q being analogous). We can ignore the further localization to the rectangle R due to orthogonality, cf. (2.15). The Fourier transform of the function inside the norms on the left-hand side is C C O 0 2 2` .jj / c O ./m jj/ c T . T .jj C / f ./ C where is a cut-off adapted to the cap . The contribution by c T .jj C / is c controlled by (2.19). As for C /, one needs to show that T .jj
C
22` m c0 .22` / P0; e i t jrj f S Œ0; . kf kL2x : T
53
2.3 The energy estimate
However, since the term in brackets is a bounded function uniformly in `, one can again apply Lemma 2.4. P Now assume that F is a wave-packet atom, i.e., F D 2C` F with X kF k2NFΠ1 (2.60) 2C`
where the F have the wave-packet form as specified in Definition 2.9. We need to show that X 21 X ˙ 2 . 1: (2.61) sup sup sup kPR Q2` 0 C kS Œ0; 0 ˙ `0 100 `0 m0
0 2C`0 R2R0;˙ 0 ;m
We first consider the case `0 `. Lemma 2.11 implies that it suffices to assume that `0 D ` and to show that, uniformly in 2 C` , k kS Œ0; C kF kNFŒ with an absolute constant C , where Z 1 WD sin .t 1
s/jrj C T .t
s/F .s/ ds:
However, this follows immediately from Lemma 2.22 applied to , the stability property (2.16), and the imbedding (2.19); note that the term 1 F1 in (2.42) can be ignored as it was dealt with in the beginning of this proof. Finally, the case `0 ` is reduced the to `0 D ` by means of Lemma 2.7 (note that the Fouriersupport of equals that of F ). It remains to control the Strichartz norms (2.14). Due to Corollary 2.23, we may ignore the projection Q<j . We split the argument into two parts: First, we will prove the estimate
2 12 X Z 1 `
e ˙i.t s/jrj F .s/ ds 4 1 (2.62) . 2 2 kF kN Œ0
Pc c2D0;`
L t Lx
1
for any F as in (2.60), cf. Lemma 2.1. Second, we take the C T cut-off as in (2.56) into account which then yields the full result. This second step is done by an adaptation of the Christ–Kiselev argument and will result in the loss of a power 2`ı where ı > 0 can be made arbitrarily small. Lemma 2.2 reduces the proof of (2.62) to the bound
Z 1
i sjrj e F .s/ ds 2 . kF kN Œ0 :
1
Lx
54
2 The spaces S Œk and N Œk
By orthogonality, it suffices to show that uniformly in
Z 1
i sjrj e F .s/ ds 2 . inf d.!; / 1 kF kL1
Lx
1
2 t! L x !
!622
with F as in (2.60). By Plancherel, this is the same as
b F .˙jj; / L2 . d.!; / 1 kF kL1
2 t! Lx!
where we choose an arbitrary ! 62 2. As above, we may set F D ı.t! .0/ .0/ t! /f .x! / where t! 2 R is an arbitrary number and f 2 L2x! is an arbitrary function whose Fourier support is contained in the projection of the Fourier support of F onto the ! -plane. This reduces us further to the bound kfO .! /kL2 . d.!; /
1
kf kL2
!
(2.63)
where on the left-hand ˇ ˇ side we regard ! as a function of . By Lemma 2.6 the ˇ @ ˇ Jacobian obeys ˇ @! ˇ d.!; / 2 which implies (2.63). This concludes the first step, i.e., the proof of (2.62). Note Pthat our proof of (2.62) applies to any F which can be written in the form F D F provided F satisfy b / f 2 R2 W jj 1; b supp.F 2 g: In other words, one does not need any condition on the modulations of F . This fact will be most important for the remainder of the proof (since we will need to multiply F by cutoff functions in time). Our next goal is to establish the estimate, with ı > 0 arbitrarily small,
2 2 X Z 1 1
. 2. 2 ı/` kF kN Œ0 e ˙i.t s/jrj C .t s/F .s/ ds
Pc
T 4 1 c2D0;`
1
L t Lx
(2.64) for any F as in (2.60) but without any restriction on the modulations of each F . For the remainder of the proof we will fix such a Schwartz function F . Moreover, k k2 without any subscripts will mean the sum in (2.60). As mentioned before, we prove (2.64) by an adaptation of the Christ–Kiselev lemma. The latter does not apply directly since the null-frame norm in (2.60) is not of pure Lebesgue type. We make the following preliminary observation. Let E D E .t / act only in the time variable and define the map .E/ WD k E F k2 as a set function on the Borel sets of R. Then one has the following -subadditivity property with fEj g R an
55
2.3 The energy estimate
arbitrary collection of pairwise disjoint Borel sets: X X XX .Ej / D k Ej F k2 D inf d.!; / j
j
X
inf d.!; /
j
2
X
!…2
X
D
X
2
!…2
inf d.!; /
2
X
2 t! Lx!
2 t! Lx!
X
2 k Ej F kL 2
x!
j
21
2 kL 1
t!
2 kF kL 1
2 t! Lx!
!…2
k
2 k Ej F kL 1
2 k Ej F kL 1
j
inf d.!; /
D
2
!…2
kF k2NFΠ1
In view of this property it suffices to prove (2.64) for F which are supported on intervals6 of size T in time and we may also replace C T by the indicator Œs
jIm;n j D jJm;n j D T 2 dist Im;n Jm;n ; fs D tg/ 2 .T 2
n
n
;
=10; 10T 2
n
8 1 m Mn ; n 0 ; 8 1 m Mn ; n 0:
We call any two intervals I; J of length T 2 n related provided I J 2 Q. Note that any I can be related to at most 20 of the J intervals. To each n 0 we now also associate 2n pairwise disjoint intervals fe J m;n g1m2n which partition Œ0; T and with the property that .e J m;n / D .e J m0 ;n / 6
8 1 m; m0 2n :
Strictly speaking, one would need to choose something like 10T here to accommodate the support of C T , but we ignore this issue.
56
2 The spaces S Œk and N Œk
The subadditivity of implies that .e J m;n / 2 n . Finally, we introduce an auxiliary function ˚ which is piece-wise linear, strictly increasing on Œ0; T and which has the property that ˚.Jm;n / D e J m;n . In view of all these properties Z X
Pc
t
c2D0;`
e ˙i.t
2
F .s/ ds 4
s/jrj
L t L1 x
1
Mn 1 X X X
˚.Im;n / .t /Pc .
nD0
c2D0;`
mD1
Z
t
e ˙i.t
s/jrj
2
˚.Jm;n / .s/F .s/ ds
L4t L1 x
1
:
Applying Cauchy–Schwarz to the sum over n allows one to bound this further as .
1 X X
Mn
X
˚.Im;n / .t/Pc .1 C n/2 mD1
c2D0;` nD0
Z
t
e ˙i.t
s/jrj
1 X
2n
Z X X
P
c
.1 C n/2
nD0
c2D0;`
1
2
˚.Jm;n / .s/F .s/ ds 4
L t L1 x
1
.
(2.65)
e ˙i.t
s/jrj
1
mD1
4 21
e .s/F .s/ ds
4 1 : J m;n L L t
x
(2.66) Label the disks c 2 D0;` by fck gK ,K 2 kD1 Z
WD Pck
akm;n
1
e ˙i.t
s/jrj
1
2` ,
and denote for fixed n,
2
e .s/F .s/ ds
4 1: J m;n L L x
t
The previous bound now takes the form Z X
Pc c2D0;`
t
e 1
˙i.t s/jrj
2
F .s/ ds 4
L t L1 x
.
1 X
n
2
.1 C n/
nD0
K X 2 X kD1
2 akm;n
12
:
mD1
In view of (2.62) (and the remark at the end of its proof concerning time cutoffs) K X kD1
2
D 2` .e akm;n . 2` e F J m;n / 2` 2 J m;n
n
:
57
2.4 A stronger S Œk-norm, and time localizations
By Lemma 2.24 with D 2`
n,
M D 2n , K D 2
n
K X 2 X kD1
2 akm;n
21
. 2.1
2` ,
2ı/`
2
ın
mD1
for any 0 ı 1. In view of (2.66), one obtains (2.64).
As a simple corollary, we now obtain the following continuity result. Recall that the norm of S Œk can also be defined for non-integer k, cf. (2.18). The continuity in k is not obvious due to the various Fourier multipliers in (2.14) and (2.15) over infinitely many scales. Corollary 2.27. Let be a Schwartz function in R1C2 which is adapted to k 2 R. Then lim k kS ŒkCh D k kS Œk
h!0
Proof. By (2.18), 1
k .
1
/kS Œk D k kS ŒkClog2
It therefore suffices to note that by the energy estimate ˇ ˇ
1
k .
1
k
ˇ k kS Œk ˇ
/kS Œk
1
.
k.
1
. k.
1
1
/
.
1
.
1
/ /
kS Œk /Œ0kL2 HP /Œ0kL2 HP
1 1
C k.
1
C k.
1
.
1
/
/kN Œk
.
1
/
/kL1 HP
as ! 1.
2.4
t
1
!0
A stronger SŒk-norm, and time localizations
The energy estimate of Proposition 2.26 and Lemma 2.25 can be summarized as the statement that k kS Œk . jjj jjjS Œk where jjj jjjS Œk WD k kL1 L2x C k kN Œk t
(2.67)
58
2 The spaces S Œk and N Œk
for any space-time Schwartz function which is adapted to k 2 Z. To see this, one estimates k kS Œk . kI kS Œk C kI c kS Œk . k .I /.0/; .@ t I /.0/ kL2 HP
1
C kI kN Œk C kI c kN Œk
. k kL1 L2x C k kN Œk D jjj jjjS Œk : t
To remove I from the right-hand side here one uses Corollary 2.23. We shall henceforth use this stronger norm and the resulting smaller S Œkspace. We introduce this norm because it leads to an improvement over the bilinear bound (2.30) in the case of high-high interactions, see Lemma 4.5 below. This improvement reflects a smoothing effect of convolutions of measures supported on the light cone. It thus cannot be obtained using the S Œk; norms alone, since (2.30) is based on Hölder’s inequality 1 2 2 2 L2t! L1 x! L t! Lx! ,! L t Lx
which does not allow for any gain in regularity. It will be essential to note that Corollary 2.16 still applies to the stronger norm jjj jjj: Lemma 2.28. For all which are adapted to k 2 Z and all j 2 Z one has jjjQj jjjS Œk C jjj jjjS Œk with some absolute constant C . Proof. This follows immediately from Lemma 2.15 and Corollary 2.23.
Another property which the stronger norm inherits is that it is finite on free wave, cf. Lemma 2.4. More precisely, for any which is adapted to k and satisfies D Qk , jjj jjjS Œk D k kL1 L2x C k kN Œk t
k kL1 L2x C k k t
0;
XP k
1 ;1 2
. k k
0; 1 2 ;1
XP k
:
As in [23], one needs to allow for time-localized versions of S Œk, both relative to the original k kS Œk , as well as the stronger jjj jjj-norm. This has to do with the fact that the we need to derive a priori bounds in these spaces for Schwartz functions ˛ which satisfy (1.12)–(1.14) on some time interval Œ T; T . Since the norms of the S Œk and N Œk spaces are defined in phase space, one cannot simply define these norms by time truncations. Rather, one proceeds as in [59] and [23]
59
2.4 A stronger S Œk-norm, and time localizations
by means of Schwartz extensions: with T 0, k kS Œk.Œ T;T R2 / WD Q jŒ
jjj jjjS Œk.Œ
T;T R2 /
WD
Q jŒ
and Q both Schwarz functions, and kPk Q kS Œk ;
inf T;T D
jŒ
T;T
T;T D
(2.68)
jjjPk Q jjjS Œk :
inf jŒ
T;T
It is easy to see that the triangle inequality holds for these expressions and that they are actually norms. Moreover, it is clear that these norms are nondecreasing in T . Following [23], we now verify that these norms are continuous in T . Lemma 2.29. Let be the restriction of some Schwartz function the time interval Œ T0 ; T0 where T0 > 0. Then k kS Œk.Œ
and jjj jjjS Œk.Œ
T;T R2 /
0
in R1C2 to
T;T R2 /
are nondecreasing and continuous in 0 T < T0 . Proof. The definition of S Œk with respect to either norm can be extended to noninteger k. Given T > 0, let j"j be very small and set WD TTC" . Then kPk kS Œk.Œ
T ";T C"R2 /
D kPkC
kS ŒkC.Œ T;T R2 /
where WD log2 and .t; x/ WD .t; x/, and similarly for jjj jjj. Clearly, for " > 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 /Œ0kL2 HP
C kPkC . . k.
/kN ŒkC
/Œ0kL2 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) jjj jjjS Œk.Œ
T;T R2 /
WD k kL2
I IL2 .R2 /
C k k
N Œk.Œ T;T R2 / :
Furthermore, later we will also need localized norms on asymmetric time intervals ΠT 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 f j g subordinate to the cover fIj g, such that supp j Ij . We shall select the j in such fashion that jsupp j0 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 kkS 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 k 1 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 Œ 1 Q 1 C 2 Q 2 for some j 1. We decompose this by applying a frequency trichotmoy P0 Qj Œ 1 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 supp 01 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 Q
T0 ;T0 kS
C kF kN
(2.74)
where the implied constant is absolute (the T0 1 here comes from the timederivative in the initial data). Indeed, this follows from .I /Œt0 D I. Œ T0 ;T0 / Œt0 C I.Œ1 Œ T0 ;T0 / Œt0 and min
t0 2ΠT0 ;T0
as well as
I. Œ
T0 ;T0
.I.Œ1
/ Œt0
L2x
Œ
T0 ;T0
D r t;x I. Œ
// Œt0 L2 HP x
T0 ;T0
/
L2x
1
k kS
. T0 1 jŒ
T0 ;T0
S
due to our choice of I . The above energy inequality then follows immediately. It is apparent that in order to use this energy inequality, one needs to establish an a priori bound for on a small time interval ΠT0 ; T0 . In fact, in later applications we will always split the estimates for Pk into the small-time case jt t0 j "1 2 k and the large time case jt t0 j "1 2 k (with a small "1 that is determined by the specific context - this then requires the constant C in the definition of I to be large). In the small time case, the necessary a priori bound is derived from the div-curl system (1.12), (1.13) for the gauged components. This information is then fed into the large-time case as described above.
7
In the sequel, we shall understand the operator otherwise stated.
1
to be division by the symbol unless
3
Hodge decomposition and null-structures
Here we introduce the actual system of wave equations for which our S and N spaces allow us to deduce a priori estimates. From the discussion at the very beginning, we recall that the Coulomb components ˛ satisfy the system (1.14), which has the schematic form
˛
D i @ˇ Œ
i @ˇ Œ
˛ Aˇ
ˇ A˛
ˇ
C i @˛ Œ
Aˇ
(3.1)
where Aˇ denotes the Coulomb gauge potential Aˇ D
X
4
1
@j
1 2 ˇ j
2 1 ˇ j
:
j D1;2
This system in and of itself does not appear to lend itself to good estimates, and to overcome this we have to use a key additional feature, namely the fact that the flow of (1.14) preserves the div-curl system (1.12), (1.13) in the obvious sense: if the ˛ at time t D 0 are the Coulomb derivative components of an actual map, whence (1.12), (1.13) holds at time t D 0, then the corresponding solution of (1.14) satisfies this system on its entire time interval of existence. The div-curl system allows us to decompose the components ˛ as the sum of a gradient term and an error term solving an elliptic equation, see (1.15). Thus we have schematic identities of the form ˛
D R˛
C ˛ :
Substituting the gradient terms introduces the desired null-structure. The present section serves to make this decomposition of the nonlinear source terms precise. We now describe this procedure for each of the three terms on the right-hand side of (3.1). First, define @j 1 WD 1 @j and Qˇj . ; / D Rˇ
1
1 2 j 1 ˇ Rj 2
Qˇj . ; / D Rˇ Qˇj . ; / D
Rj
Qˇj . ; / D 1ˇ j2
2
Rj
1
1 2 ˇ 1 j Rˇ 2
Rj
j1 2ˇ :
Rˇ
2
66
3 Hodge decomposition and null-structures
Then, adopting the Einstein summation convention, i@ˇ Œ ˛ Aˇ D i @ˇ ˛ I c @j 1 Qˇj . ; / C i @ˇ ˛ I @j 1 Qˇj . ; / C i @ˇ ˛ @j 1 Qˇj . ; / C i @ˇ ˛ @j 1 Qˇj . ; / C i @ˇ ˛ @j 1 Qˇj . ; / :
(3.2)
The two main terms here are the trilinear ones in . We introduced the modulation cutoff I in front of Qˇj since the two resulting expressions are estimated differently: for the second, one uses a trilinear null-form structure, see (5.44) below, whereas for the first the bilinear null-form Qˇj suffices. Note that the other three terms involving are quintilinear and septilinear in , respectively, due to (1.16). These are discussed in greater detail below, under the heading “higher order errors”. Next, i @ˇ ˇ @j 1 Q˛j . ; / i@ˇ Œ ˇ A˛ D i @ˇ ˇ @j 1 Q˛j . ; / i @ˇ ˇ @j 1 Q˛j . ; / i @ˇ ˇ @j 1 Q˛j . ; / : The -terms need to be decomposed further, whereas the main term here is again the trilinear one in , which we now rewrite as follows: i@ˇ ˇ @j 1 Q˛j . ; / D i @ˇ ˇ @j 1 I c Q˛j . ; / i @ˇ ˇ @j 1 I Q˛j . ; / (3.3) The first term on the right-hand side will be estimated as is, whereas the second term now needs to be rewritten according to the Littlewood–Paley trichotomy, in order to make it amenable to our estimates: X i@ˇ ˇ @j 1 I Q˛j . ; / D i Pk @ˇ ˇ @j 1 IP
i
X
@j 1 IP
Pk Rˇ
5@
ˇ
Q˛j . ; /
(3.4)
k
i
X
Pk ˇ @j 1 IP
5@
ˇ
Q˛j . ; /
k
i
X
@ˇ Pk P>k Rˇ
@j 1 IP>kC5 Q˛j . ; /
k
i
X
@ˇ Pk P>k ˇ @j 1 IP>kC5 Q˛j . ; /
k
(3.5)
67
3 Hodge decomposition and null-structures
i
X
@ˇ P
@j 1 IPŒk
5;kC5 Q˛j .
; /
k
i
X
@ˇ P
5;kC5 Q˛j .
; / : (3.6)
k
The terms involving are expanded further as explained below. For the first term on the right-hand side of (3.4) one replaces @ˇ ˇ by the right-hand side of (1.13) which leads to a quintilinear term. The second term can be estimated since the @ˇ -term falls on the small frequencies. Finally, the third term in (3.1) is treated as follows:
i @˛ Œ
ˇ
Aˇ D i @˛ Œ
ˇ
I c A ˇ C i @˛ Œ
ˇ
IAˇ
(3.7)
The first term on the right-hand side of (3.7) is estimated as is; in fact, it is essential that one does not perform the Hodge decomposition in the first slot since otherwise ˇ D 0 would create problems if has large modulation. For the second term, one needs to distinguish frequency interactions as before:
i @˛ Œ
ˇ
IAˇ D i
X
P k @˛
ˇ
@j 1 IP
5 Qˇj .
; /
k
Ci
X
Pk
ˇ
@j 1 IP
5 @˛ Qˇj .
; /
(3.8)
k
Ci
X
@˛ Pk P>kC5 Rˇ
@j 1 IP>k Qˇj . ; /
k
Ci
X
@˛ Pk P>kC5 ˇ @j 1 IP>k Qˇj . ; /
(3.9)
k
Ci
X
@˛ P
@j 1 IPŒk
5;kC5 Qˇj .
; /
k
Ci
X
@˛ P
5;kC5 Qˇj .
; / (3.10)
k
The -terms need to be expanded further, see below, whereas the -terms in (3.9) and (3.10) are estimated as they are. The second term on right-hand side of (3.8)
68
3 Hodge decomposition and null-structures
is expanded by means of the Hodge decomposition: X i Pk ˇ @j 1 IP
Di
X
Pk R ˇ
@j 1 IP
5 @˛ Qˇj .
; /
(3.11)
k
Ci
X
Pk ˇ @j 1 IP
5 @˛ Qˇj .
; /
(3.12)
k
The trilinear estimates of Chapter 5 cover (3.11), and (3.12) is handled below, under “higher order errors”. Finally, the first term on the right-hand side of (3.8) is rewritten by means of (1.12): i
X
Pk @˛
ˇ
@j 1 IP
5 Qˇj .
; /
k
Di
X
Pk @ˇ
˛ @j
1
IP
5 Qˇj .
; / C quintilinear terms (3.13)
k
where the quintilinear terms arise by using the curl identity for @˛ ˇ @ˇ ˛ into this expression. Note that we have switched the derivatives @˛ and @ˇ . We still have to explain how to deal with the higher order terms involving at least one factor of . Higher order errors. Note that these arise in two ways: first, we generate errors by replacing the Gauge potential Aˇ in i @ˇ Œ
˛ Aˇ
by a Qˇj . ; / null-form, and similarly for the remaining types of terms i @ˇ Œ
ˇ A˛ ;
i @˛ Œ
ˇ
Aˇ :
We shall call the higher order terms generated by this process (and later further Hodge decompositions applied to them) of the first type or kind. Second, we generate errors of the schematic form r
1
IQˇj . ; /;
and we call these together with all the terms generated by them upon applying further Hodge decompositions of the second type or kind. For simplicity, we omit frequency localizations in the ensuing discussion.
3 Hodge decomposition and null-structures
69
Figure 3.1. An example of an expansion graph
Considering the errors of the first kind, these are of the schematic form rx;t r 1 Œ ; rx;t r 1 Œ where we recall from the very beginning, Section 1, that D r 1 r 1. 2/ ; whence the above terms may be thought of as quintilinear and septilinear. Now as they are written, we cannot yet quite estimate these expressions, and we need to introduce more null-structure, by expanding the r 1 . 2 / in D r 1 r 1. 2/ ; into a Qj -null-form as well as even higher order error terms. To keep track of things we associate an expansion graph, i.e., a simple binary tree with the expressions generated: Represent the original terms rx;t Œ Aˇ by a simple node, and whenever we replace one of the factors in the (schematically written) Aˇ D r 1 . 2 / by the corresponding , we draw a downward edge pointing left or right corresponding to which factor we replace. We can now exactly specify the full expansion of the higher order errors of first type:
70
3 Hodge decomposition and null-structures
Precise description of expansion for errors of first type: Keep applying Hodge decompositions to the inner r 1 . 2 / in all factors 1
Dr
r
1
.
2
/;
generated until the associated expansion graph has a directed subgraph of length four. Then the process stops. Note that formally, the terms with a directed subgraph of length four thereby generated are up to at least the 11th degree in . Next, we apply a similar process to the errors of the second type. We represent the first such error, schematically given by rx;t r
1
IQj . ; /
by a simple node, and whenever we apply a Hodge decomposition to one of the factors of r 1 . 2 / in D r 1 r 1. 2/ we draw a downward edge pointing left or right, thereby generating an associated expansion graph. Then we have: Precise description of expansion for errors of second type: Keep applying Hodge decompositions as above until the associated expansion graph has a directed subgraph of length three. Then the process stops. Again we generated a list of errors of degree of multilinearity up to order 11 and more in . To summarize this discussion, we have now recast our system of equations in the form
˛
D
5 X
F˛2i C1
i D1
where the superscript indicates the minimum degree of multilinearity of the corresponding terms in (i 1 indicates the length of a directed subgraph in the corresponding graph representation), and the leading cubic terms F˛3 can be ex-
71
3 Hodge decomposition and null-structures
pressed as F˛3 D i@ˇ
I c @j 1 Qˇj . ; / C i @ˇ ˛ I @j 1 Qˇj . ; / i @ˇ ˇ @j 1 I c Q˛j . ; / X i Pk Rˇ @j 1 IP
˛
k
i
X
i
X
@ˇ Pk P>k Rˇ
@j 1 IP>kC5 Q˛j . ; /
k
@ˇ Pk P
@j 1 IPŒk
5;kC5 Q˛j .
; /
k
C i @˛ ˇ I c @j 1 Qˇj . ; / X Ci Pk @ˇ ˛ @j 1 IP
Ci
X
Ci
X
Ci
X
Pk Rˇ
@j 1 IP
5 @˛ Qˇj .
; /
k
@˛ Pk P>k Rˇ
@j 1 IP>kC5 Qˇj . ; /
k
@˛ P
@j 1 IPŒk
5;kC5 Qˇj .
; /
(3.14)
k
Here it is very important to note that the second as well as the eighth term on the right contribute a magnetic potential interaction term of the form X 2i Pk @ˇ ˛ @j 1 IP
the idea being that we interpret the low-frequency term @j 1 IP
k Rˇ @j 1 IP>kC5 Q˛j . ; / k
72
3 Hodge decomposition and null-structures
i
X
i
X
Pk Rˇ
@j 1 IP
5@
ˇ
Q˛j . ; /
k
@ˇ P
@j 1 IPŒk
5;kC5 Q˛j .
; / : (3.16)
k
Furthermore, we denote by F˛3k .
1;
2;
3/
the corresponding multilinear expressions. We also introduce frequency localized versions F˛3k . 1 I P<` I 2 ; 3 / in which one includes a cutoff P<` in front of all instances of Q˛j . similarly for other multipliers P` etc.
2;
3 /,
and
4
Bilinear estimates involving S and N spaces
In this section we develop some of the required bilinear bounds. First, we present some bounds from S S into L2tx , in particular one which involves a gain in the high-high case and which does not appear in [59] or [23], see Lemma 4.7 below. This result allows for better control on products 1 2 of S-waves and will be most useful in the trilinear case. In addition, as in the aforementioned references we consider the case of 1 2 S and 2 2 N . This section concludes with bilinear estimates for null-forms.
4.1
Basic L2 -bounds
To begin with, we present the following geometric lemma for cones, see [58] for a similar result. It will be used repeatedly. Lemma 4.1. Suppose 1 ; 2 are such that ˇ ˚ supp. bj / .; / j jj 2kj ; ˇjj
ˇ j jˇ 2`j
for j D 1; 2. Let `0 ; k0 2 Z and assume that there exists j0 2 f0; 1; 2g so that `j0 > `j C C
8 j 2 f0; 1; 2g n fj0 g:
(4.1)
Then there is the following dichotomy: (A) If k0 D kmax C O.1/, then X Pk0 Q`0 1 2 D Pk0 Q`0 Pk1 ;1 1 Pk2 ;2 2 1 ;2
where 1 ; 2 are caps of size C
1r
and separation dist.1 ; 2 / r with
r WD 2.`max In particular, `max kmin C O.1/.
kmin /=2
:
(4.2)
74
4 Bilinear estimates involving S and N spaces
(B) If k0 < kmax Pk0 Q`0
C , then X X ."/ Pk1 ; 1 Pk2 ; 1 2 D Pk0 Q`0 X
(4.3)
"D˙
C
."/
2
Pk0 Q`0
X
."/
Pk1 ;1 1 Pk2 ;
. "/
2 2
(4.4)
1 ;2
"D˙
1r
the sum in (4.4) runs over caps of size C r WD 2k0
with
kmax .`max kmin /=2
2
and with separation dist.1 ; 2 / r, whereas the sum in (4.3) runs over caps of size r 0 where 2k0 kmax r 0 1 is arbitrary but fixed. The sum (4.3) is empty if `max < kmax C and (4.4) is nonzero only if `max kmin C O.1/. Finally, if (4.1) fails, then the same representations hold provided r . 1 and one replaces dist.1 ; 2 / r with dist.1 ; 2 / . r. Proof. We consider first the .CC/ and . / cases, i.e., when 1 ; 2 have the same sign. Then j1 j C j2 j j1 C 2 j 2`max ; (4.5) whence .j1 j C j2 j/2
j1 C 2 j2 2`max Ckmax
and thus ^.1 ; 2 / 2.`max Ckmax
k1 k2 /=2
:
Now assume further that k0 D kmax C O.1/. Then it follows that ^.1 ; 2 / 2.`max
kmin /=2
:
If on the other hand k0 < kmax C , then k1 D k2 C O.1/ D kmax C O.1/ and from (4.5), `max D kmax C O.1/. Furthermore, 2 D 1 C O.2k0 / implies that ˇ ˇ ˇ^.1 ; 2 /ˇ j1 ^ 2 j D O.2k0 j1 jj2 j
kmax
/:
Next, consider the .C / or . C/ cases. Then j1 C 2 j
jj1 j
j2 jj 2`max
(4.6)
which implies that j1 C 2 j2
jj1 j
j2 jj2 2`max j1 C 2 j C jj1 j
j2 jj
4.1 Basic L2 -bounds
75
Figure 4.1. Opposing .CC/ waves
or equivalently, 2k1 Ck2 ^2 .1 ; 2 / 2`max Ck0 :
(4.7)
If k0 D kmax C O.1/, then ^.1 ; 2 / 2.`max If, on the other hand, k0 kmax
kmin /=2
:
C , then
^.1 ; 2 / 2k0
kmax .`max kmin /=2
2
and we are done. While it is clear that `max kmin C O.1/ if k0 D kmax C O.1/, some proof is needed in case k0 < kmax C . Thus, suppose j1 j j2 j whence j1 C 2 j
j1 j C j2 j 2`max
which implies that 2k0 Ck1 ^2 .1 C 2 ; 2 / 2`max Ckmax since 2k0 Ck1 2kmin Ckmax , the claim follows.
76
4 Bilinear estimates involving S and N spaces
Finally, if (4.1) fails, then (4.5) turns into j1 j C j2 j
j1 C 2 j . 2`max
which then leads to the claimed loss of separation between the sectors. However, their maximal distances are controlled by the same quantities as before. The special appearance of (4.3) derives from the contributions of waves which lie on opposing sides of the light-cone. In fact, Figure 4.1 shows two vectors on the same half (i.e., > 0) but opposing sides of the light cone. They add up to produce a wave of small frequency but large modulation, as described by (4.3). This is the mechanism by which nonlinearities can turn free waves into “elliptic objects”. This phrase refers to functions whose Fourier support has large separation from the characteristic variety of . Also, following Tao we refer to (4.1) as the modulation imbalanced case, whereas its opposite is the modulation balanced case. Remark 4.2. Lemma 4.1 is optimal in the following sense: ı Given `0 k0 10 there exist 1 ; 2 2 Rn with 1 j1 j; j2 j 2, ^.1 ; 2 / 2.`0 Ck0 /=2 and such that j1 C 2 j
jj1 j
j2 jj 2`0
j1 C 2 j 2k0 : ı Given `0 k1 10 there exist 1 ; 2 2 Rn with 2k1 1 j1 j; j2 j 2 and ^.1 ; 2 / 2.`0 k1 /=2 and so that j1 j C j2 j
1
j1 j 2k1 ,
j1 C 2 j 2`0 :
Our immediate goal now is the proof of Lemma 4.5. It is important to note that k the improvement of 2 2 over (2.30) which is obtained in Lemma 4.5 coincides with the gain for the case of free waves. In order to accomplish this, we require three preparatory lemmas, all of which are well-known. The first is Mockenhaupt’s “square function estimate” (more precisely, its geometric content), see [34], [35]. O Recall that D sign. /. Lemma 4.3. Let ; Q 2 C` with dist.; / Q jj 1 and suppose that Fi C`i for i D 1; 2 are partitions of and , Q respectively, by pairwise disjoint caps. Further, let r 2 .0; 1/, 2 .1; 2/, and define for any cap 0 S 1 T 0 ;;r WD f. ; / W jjj
j r; 2 0 ; jj j
jjj j 0 j2 g:
4.1 Basic L2 -bounds
Set Mi WD #Fi . Then
X X
sup T1 ;1 ;r CT2 ;2 ;r 1 ;2 1
L1 .R3 /
1 2F1 2 2F2
77
C max 1; r.M1 C M2 / (4.8)
where C is some absolute constant. Proof. Fix r 2 .0; 1/, and 1 ; 2 1. Applying a Lorentz transform, one may assume that ` D 10, say. Also, suppose without loss of generality that `1 `2 whence M1 M2 . We first consider the case where rM1 1. Fix . ; / 2 R3 such that1 X X 1 .T ; ;r CT ; ;r / . ; / 1: 2
1
1
2
2
1 2F1 2 2F2
Suppose T1 ;1 ;r with 1 2 F1 contributes to the sum on the left-hand side. Define a mirror-image T1 ;1 ;r of T1 ;1 ;r by reflecting T1 ;1 ;r about the point . ; /. Due to ` D 10 and the dimensions of the tubes T , the mirror images of all fT1 ;1 ;r g1 2F1 have uniformly bounded overlap. The same applies with the role of F1 and F2 reversed. In conclusion, each T1 ;1 ;r can pair up with at most O.1/-many T2 ;2 ;r so as to give a contribution to (4.8), whence the bound of M1 for (4.8). To obtain the factor r improvement, we further note that due to fixed 1 and 2 , only those contributions to (4.8) need to be counted which derive from pairs .T1 ;1 ;r ; T2 ;2 ;r / which lie in fixed cylinders jji j i j < r, i D 1; 2. In terms of equations, we are given .; / 2 R3 and we need to consider the sets of . i ; ri !i /, i D 1; 2 with !i 2 S 1 satisfying the transversality condition 1 1 ^.!1 ; !2 / 2 Π100 ; 50 , say, and such that 1 C 2 D ; jr1 jj 1 j
1 j < r;
r1 !1 C r2 !2 D jr2
r1 j < 22`1 ;
2 j < r jj 2 j
r2 j < 22`2 :
It follows from the second, third, and fourth conditions that 1 !1 C 2 !2 D C O.r/ and since the circular arcs containing !1 and !2 are transverse to each other, they must be of lengths . r. Consequently, we can only count tubes which correspond to an r r disk on the light-cone and of those there are at most rM1 -many. In case 1
The 12 -factor is a convenient modification that can be made due to scaling.
78
4 Bilinear estimates involving S and N spaces
rM1 1, then the number of the allowed pairs is . 1 in light of this construction and we are done. Next, we present a standard bilinear L2 bound for free waves. Lemma 4.4. Let ; Q 2 C` with dist.; / Q jj WD ˇ and suppose 1 , 2 Q are arbitrary caps. Let r 2 .0; 1/ and 1 ; 2 1. Then q ke i t jrj f1 e ˙i t jrj f2 kL2 L2x . ˇ 1 min rˇ; j1 j; j2 j kf1 k2 kf2 k2 ; (4.9) t
provided ˚ b1 / 2 R2 W O 2 1 ; jjj 1 j . r supp.f ˚ b2 / 2 R2 W O 2 ˙2 ; jjj 2 j . r supp.f
and the sign in the last sign is chosen to be the same as in (4.9). Proof. The proof reduces to the following well-known property of convolutions: Suppose ˚ 3 O 1 j . r ; 1 WD .jj; / 2 R W 2 1 ; jjj ˚ 3 O 2 j . r : 2 WD .˙jj; / 2 R W 2 ˙2 ; jjj Note that ^.; ˙/ & ˇ for any .jj; / 2 1 and .˙jj; / 2 2 . Then q 1 kf 1 g 2 kL2 .R3 / . ˇ min rˇ; j1 j; j2 j kf kL2 .d / kgkL2 .d 1
2
/
(4.10) where 1 and 2 are the lifts of the measure in to the sectors 1 ; 2 on the light-cones. To prove (4.10), interpolate between L1 and L1 . On L1 we have the standard fact that k k kkkk for measures and their total variation norms. This fact does not use the angular separation of the supports nor their sizes. On L1 , however, this separation and size are crucial and yield R2
kf
1
g 2 kL1 .R3 / . ˇ
1
min.r; j1 jˇ
1
/kf kL1 .d
1
/ kgkL1 .d
2
/
assuming as we may that j1 j j2 j. To obtain this bound, consider ıneighborhoods of 1 and 2 , respectively. In other words, replace d1 by .ı/
de j
WD ı
1
Œdist..; /;
j /<ı
d d
4.1 Basic L2 -bounds
79
for small ı > 0 and observe that .ı/
.ı/
lim sup kde 1 de 2 kL1 .ˇ ;
1
min.r; j1 jˇ
1
/
(4.11)
ı!0C
by elementary geometry. To pass from (4.10) to estimates for the wave equation use Plancherel’s theorem. We can now state the aforementioned improved bilinear L2 bound. The norm jjj jjj is the one from (2.67). Lemma 4.5. Let i be adapted to ki for i D 1; 2. Assume further that we are in the high-high case k1 D k2 C O.1/ and that i D Qj Ck 2k1 C i for i D 1; 2. Then k1 k j (4.12) kPk Qj . 1 2 /kL2 L2x . 2 2 2 4 jjj 1 jjjS Œk1 jjj 2 jjjS Œk2 t
for any j k k1 C O.1/. Moreover, in the same range of j , kPk Qj .R˛ 1 Rˇ 2
Rˇ 1 R˛ 2 /kL2 L2x . 2 t
k1 2
2
3kCj 4
jjj 1 jjjS Œk1 jjj 2 jjjS Œk2 (4.13)
for any ˛; ˇ D 0; 1; 2. Proof. We assume that k1 D k2 C O.1/ D 0. At first, we also assume that k C so as to exclude the opposing .CC/ and . / waves in Lemma 4.1. We need to prove that kPk . 1 2 /kL2 L2x . 2 t
k j 4
k.f1 ; g1 /kL2 HP
1
C kF1 kN Œk1
k.f2 ; g2 /kL2 HP
1
C kF2 kN Œk2
for any ki -adapted Schwartz functions fi ; gi ; Fi , i D 1; 2 and Z t sin t jrj sin .t s/jrj i .t/ D cos.t jrj/fi C gi C Fi .s/ ds jrj jrj 0 We reduce this to three cases:
k j
Pk Qj . 1 2 / 2 2 . 2 4 .f1 ; g1 / 2 P 1 .f2 ; g2 / 2 P L t Lx L H L H
k j
Pk Qj . 1 2 / 2 2 . 2 4 .f1 ; g1 / 2 P 1 F2 L t Lx L H N Œk2
k j
Pk Qj . 1 2 / 2 2 . 2 4 kF1 kN Œk kF2 kN Œk 1 2 L L t
x
1
(4.14)
(4.15)
(4.16) (4.17) (4.18)
80
4 Bilinear estimates involving S and N spaces
where the absence of terms on the right-hand side implies that the corresponding functions are zero (thus, F1 D F2 D 0 in (4.16) etc.) We begin with (4.16) which follows easily from Lemma 4.4. To see this, we decompose i into caps of size ` D .j C k/=2 as in Lemma 4.1. Adopting the convention that 1 2 means that dist.1 ; 2 / 2` , and setting g1 D g2 D 0 for simplicity, one has2 X
Pk Qj . 1 2 / 2 2 .
Pk Qj .Pk ; 1 Pk ; 2 / 2 2 1 1 2 2 L L L L t
x
t
x
1 2 2C`
X
.
X
kPk1 ;1 Pc 1 Pk2 ;2 P
c 2 kL2 L2x t
c2D0;k 1 2 2C`
X
.
X
2
k ` 2
kPk1 ;1 Pc f1 k2 kPk2 ;2 P
c f2 k2
(4.19)
c2D0;k 1 2 2C`
.2
k ` 2
kf1 k2 kf2 k2
as needed. The estimate in (4.19) follows from (4.8) since k `. To prove (4.17) and (4.18) it will suffice as usual to assume that Fi are N Œki atoms for i D 1; 2. In fact, if F2 in (4.17) is either and energy or an XP s;b -atom, then one again reduces matters to the free case. Consequently, we may restrict ourselves to (4.18) when both F1 and F2 are null-frame atoms. Using Lemma 2.11 to refine these null-frame atoms one can thus assume that X X e 00 F (4.20) F1 D F 0 ; F1 D 0 2C`0
00 2C`00
where `0 ; `00 `. Again by Lemma 2.11, we can further assume that there exists a fixed c 2 D0;k so that Pc F1 D F1 and P c F2 D F2 . Applying the same decomposition as in (4.19), fix 1 2 . In view of Lemma 2.22, X Z 1 Pk1 ;1 1 D G1 C 10 ;a C B 0 ;a 20 ;a da (4.21) 0 2C`0 0 1
Pk2 ;2 2 D
1e
G 2 C
R
X Z 00 2C`00 00 2
e 200 da e 100 C B e 00 ;a ;a ;a
(4.22)
R
where the functions on the right-hand side satisfy the bounds specified in that lemma. Moreover, the Fourier supports of the functions appearing inside the integral in (4.21) and (4.22) satisfy (2.43), and they also retain the Pc and P c 2
Recall our convention about Pki ; which takes the sign of into account.
4.1 Basic L2 -bounds
81
localization property, respectively, due to the fact that k `. We can ignore the e 2 as they are reducible to free waves. For simplicity, terms involving G1 and G 1 1 e we also set 0 ;a D 00 ;a D 0. By Plancherel’s theorem and Lemma 4.3, kPk ŒPk1 ;1 1 Pk2 ;2 2 kL2 L2x t q . 1 C 2k .M1 C M2 / ! 12 X c2D0;k
X
X
0 2C
00 2C
`0
2 c 2 kL2 L2 t x
kPk1 ; 0 Pc 1 Pk1 ; 00 P
`00
0 1 00 2
0
00
where M1 D 2` ` ; M2 D 2` ` . On the other hand, applying Lemma 2.22 to Pc 1 , P c 2 and using Lemma 4.4 implies that kPk1 ; 0 Pc 1 Pk1 ; 00 P c 2 kL2 L2x t Z e c e2 c kBc0 ;a 2c . 0 ;a B 00 ;b 00 ;b kL2 L2 dadb t x 2 ZR e2 c . k 2c 0 ;a 00 ;b kL2 L2 dadb t x 2 R Z q 0 00 e2 c . 2 ` min.2kC` ; 2` ; 2` / k 2c 0 ;a k 0; 1 ;1 k 00 ;b k XP 0
R2
.2
`
q
2
0 00 e min.2kC` ; 2` ; 2` /kPc F 0 kNFΠ0 kP
1 ;1 0; 2
XP 0
dadb
c F 00 kNFΠ00 :
One checks that q
1C
2k .M1
C M2 / 2
`
q
0
00
min.2kC` ; 2` ; 2` / . 2
k ` 2
;
whence X
kPk1 ;1 Pc 1 Pk2 ;2 P
c 2 kL2 L2x t
.2
k ` 2
c2D0;k
! 12 X c2D0;k
X
X
0 2C
00 2C
`0
`00
0 1 00 2
e kPc F 0 k2NFΠ0 kP
2 c F 00 kNFΠ00
:
82
4 Bilinear estimates involving S and N spaces
In conclusion, kPk Qj . 1 2 /kL2 L2x t X X kPk1 ;1 Pc 1 Pk2 ;2 P .
c 2 kL2 L2x t
c2D0;k 1 2 2C`
.2
k ` 2
! 21 X
X
c2D0;k 1 2 2C` k `
. 2 2C
X
X
X
e c F 00 k2 00 kPc F 0 k2NFŒ 0 kP NFŒ
0 2C`0 00 2C`00 0 1 00 2
kF 0 k2NFΠ0
12 X
0 2C`0
e 00 k2 00 kF NFŒ
21
00 2C`00
as desired; in the last step we have also used Lemma 2.11. This concludes the proof of (4.18) for the case of null-frame atoms F1 ; F2 . As indicated, the other cases are easier since they can be reduced to free waves. Finally, if k D O.1/, then the proof is easier. In fact, it follows via a capdecomposition from the basic bilinear bound (2.30). We leave those details to the reader. The second bound (4.13) follows by the same argument. The only difference from (4.12) lies with an additional gain of 2` which is precisely the size of the angle in the above decompositions into caps. Later, we shall require the following technical variant of the previous bound. Corollary 4.6. Under the assumptions of Lemma 4.5, for any j k k1 CO.1/ and any m0 10, X
kPk Qj .Pk1 ;1 1 Pk2 ;2 2 /kL2 L2x . 2
k1 2
t
2
k j 4
jjj 1 jjjS Œk1 jjj 2 jjjS Œk2 ;
1 ;2 2Cm0 dist.1 ;2 /2m0
(4.23) X 2Cm0
kPk Qj .Pk1 ; 1 2 /k2L2 L2 t
x
12
. jm0 j 2
k1 2
2
k j 4
jjj 1 jjjS Œk1 jjj 2 jjjS Œk2 : (4.24)
Moreover, analogous bounds hold for the null form in (4.13) with an extra gain j Ck of 2 2 . Finally, the left-hand side in (4.23) vanishes unless j C k 2m0 100.
4.1 Basic L2 -bounds
83
Proof. The final statement here is due to Lemma 4.1. Note that one cannot simply square sumP the bounds of LemmaP 4.5 applied to Pk1 ;1 1 and Pk2 ;2 2 due 2 to the fact that jjjPk; jjjS Œk (or kPk; k2S Œk for that matter) cannot be controlled. However, since we may assume that j Ck 2 m0 , the angular decomposition induced by the frequency and modulation cutoffs Pk Qj is finer than the one superimposed by 1 and 2 . Inspection of the proof now reveals that either by orthogonality or by organizing the finer caps into subsets of the 1 ; 2 2 Cm0 , and applying the Cauchy–Schwarz inequality yields the stated bound. For (4.24) one needs to distinguish two cases: either m0 j Ck 2 or not. In the former case, the decomposition into caps in Cm0 is coarser than the one coming from Lemma 4.1 and one can again argue by means of Cauchy–Schwarz as before. In the latter case, however, we split the modulation of the first input as follows: Q<j Ck
C
D Q<2m0
C
C Q2m0
C <j Ck C
The contribution of Q<2m0 C 1 is handled exactly as in the Lemma 4.5 since one may always refine the null-frame representation, cf. (4.20). On the other hand, Q2m0 C <j Ck C 1 is controlled by means of Lemma 2.4. More precisely, for any 2m0 C ` < j C k C one has Q` 1 D Q` 1 F1 ; see (4.15). Since (2.32) implies that kQ` 1 k
1 ;1 0; 2
XP 0
1
D kQ`
F1 k
1 ;1 0; 2
XP 0
. kQ` F1 k
0;
XP 0
1 2 ;1
. kF1 kN Œ0
one can reduce the contribution of Q` 1 to the case of free waves as in the proof of Lemma 4.5. Summing over all ` in this range loses a factor of at most jm0 j, as claimed. Finally, the claim concerning the null-forms is immediate. Removing the modulation restrictions on the inputs in Lemma 4.5 results in the following estimates. Lemma 4.7. If 1 and 2 are adapted to k1 and k2 , respectively, then for j k k1 C O.1/ D k2 C O.1/, kPk Qj . 1 2 /kL2 L2x . 2
k j 4
t
2
k1 2
jjj 1 jjjS Œk1 jjj 2 jjjS Œk2 ;
(4.25)
whereas for j k2 k D k1 C O.1/, kPk Qj . 1 2 /kL2 L2x . 2 t
3k2 4
2
j 4
k 1 kS Œk1 k 2 kS Œk2 :
(4.26)
84
4 Bilinear estimates involving S and N spaces
Proof. Consider the high-high case j k k1 C O.1/ D k2 C O.1/ D 0. On the one hand, there is the bound
k j
Pk Qj .Qj Ck C 1 Qj Ck C 2 / 2 2 . 2 4 jjj 1 jjjS Œk jjj 2 jjjS Œk (4.27) 1 2 L L x
t
which is given by Lemma 4.5. On the other hand, by the improved Bernstein bound of Lemma 2.1,
j k
Pk Qj .Q>j Ck C 1 2 / 2 2 . 2 4 ^0 2k kQ>j Ck C 1 2 k 2 1 L Lx L L t
x
t
.2 .2
j
k 4
k j 4
2k kQ>j Ck
2 C 1 kL2t L2x k 2 kL1 t Lx
k 1 kS Œk1 k 2 kS Œk2 :
(4.28)
In the high-low case j k2 k D k1 C O.1/ D 0 consider the following three subcases. First,
j k k
Pk Qj .Q<j C 1 Q<j C 2 / 2 2 . 2 4 2 2 22 k 1 kS Œk k 2 kS Œk 1 2 L L t
x j
k2
j
k2 4
by a decomposition into caps of size 2 2 and the L2 -bilinear bound (2.30). Next, by the improved Bernstein estimate Lemma 2.1,
Pk Qj . 1 Qj C 2 / 2 2 . k 1 kL1 L2 kQj C 2 k 2 1 L Lx L L t
x
t
.2
k2
2 2
j 2
k 1 kS Œk1 k 2 kS Œk2 :
And third, kPk Qj .Qj C 1 Q<j C 2 /kL2 L2x t X
. Pk Qj .Qm 1 Q<j
C 2 / L2 L2x t
mj CO.1/
X
.
X
kPk1 ;1 Qm 1 Pk2 ;2 Q<j
mj CO.1/ 1 ;2
X
.
X
kPk1 ;1 Qm 1 kL2 L2x kPk2 ;2 Q<j t
mj CO.1/ 1 ;2
X
.
2
m 2
C 2 kL2t L2x
2k2 2
m k2 4
C 2 kL1 t;x
k 1 kS Œk1 k 2 kS Œk2
mj CO.1/
.2
3k2 4
2
j 4
k 1 kS Œk1 k 2 kS Œk2
as claimed. The inner sums run over 1 ; 2 2 C m
k2 2
with dist.1 ; 2 / . 2
m k2 2
.
4.1 Basic L2 -bounds
85
Later we shall also need the following technical variants, both of which are in the same spirit as Corollary 4.6. Corollary 4.8. Let be adapted to k1 and suppose for every 2 Cm0 with m0 100 there is a Schwarz function which is adapted to k2 . Then, provided j k2 k D k1 C O.1/, X
kPk Qj .Pk1 ;
/kL2 L2x . jm0 j2
3k2 4
t
2
j 4
k kS Œk1
X
2Cm0
k
2 kS Œk2
12
2Cm0
(4.29) Proof. One uses the argument for the high-low case of Lemma 4.7. In particular, k D k1 C O.1/ D 0. First, with m D j 2k2 , X
Pk Qj .Q<j
C Pk1 ;
Q<j
C
/
2Cm0
D
X
X
Pk Qj .Q<j
C Pk1 ;1 Pk1 ;
Q<j
C Pk2 ;2
/:
2Cm0 1 ;2 2Cm
If m m0 , then by the L2 -bilinear bound (2.30) X
Pk Qj .Q<j
C Pk1 ;
Q<j
C
/ L2 L2 t x
2Cm0
X
.
X
Pk Qj .Q<j
C Pk1 ;1
Q<j
C Pk2 ;2
/ L2 L2 t x
2Cm0 1 ;2 2Cm 1
X
.
X
j
2
k2 4
2
k2 2
2Cm0 1 ;2 2Cm 1
j
.2
k2 4
2
k2 2
k kS Œk1
kQ<j X
C Pk1 ;1 kS Œk1 ;1 kQ<j C Pk2 ;2
k
2 kS Œk2
kS Œk2 ;2
12
2Cm0
where we applied Cauchy–Schwarz twice to pass to the last line. If, on the other hand, m > m0 , then we first consider smaller modulations of . In fact, dropping
:
86
4 Bilinear estimates involving S and N spaces
the Q<j C on as we may one has X
Pk Qj .Q<2m C Pk ; Q<j 0 1
/ L2 L2 t x
C
2Cm0
X
.
X
Pk Qj .Q<2m 0
C Pk1 ;1
Q<j
C Pk2 ;2
/ L2 L2 t x
1 ;2 2Cm 2Cm0 1
X
.
j
X
2
k2 4
2
k2 2
1 ;2 2Cm 2Cm0 1
j
.2
kQ<2m0 X k2 k2 4 2 2 k kS Œk1 k
C Pk1 ; kS Œk1 ; kQ<j C Pk2 ;2 2 kS Œk2
kS Œk2 ;2
12
2Cm0
where we again applied Cauchy–Schwarz twice to pass to the last line. Finally, we need to account for Q2m0 C <j C . Fix ` with 2m0 C ` < j C and repeat the previous estimate. This yields X kPk Qj .Q` Pk1 ; Q<j C /kL2 L2x t
2Cm0 j
.2
k2 4
2
k2 2
k kS Œk1
X
k
2 kS Œk2
21
2Cm0
which, upon summing in ` yields the same bound with the loss of a factor of .j 2m0 /C . Replacing this by the larger jm0 j then implies the bound of the corollary. Next, by the improved Bernstein estimate of Lemma 2.1, and Lemma 2.18, X kPk Qj .Pk1 ; Qj C /kL2 L2x t
2Cm0
.
X
kPk1 ; kL1 L2 kQj
C
kL2 L1 x t
2Cm0 j
. jm0 j 2
k2 4
2 k2 2
j 2
k kS Œk1
X 2Cm0
And third, X kPk Qj .Qj 2Cm0
C Pk1 ;
Q<j
C
/kL2 L2x t
k
2 kS Œk2
12
:
4.1 Basic L2 -bounds
X
.
X
87
kPk Qj .Qm Pk1 ; Q<j
C
/kL2 L2x t
mj CO.1/ 2Cm0
X
.
X
X
kPk1 ;1 Qm Pk1 ; Pk2 ;2 Q<j
C
kL2 L2x t
mj CO.1/ 2Cm0 1 2
X
.
X
X
kPk1 ;1 Qm Pk1 ; kL2 L2x t
mj CO.1/ 2Cm0 1 2
2k2 2
. 2k2 2
j
k2 4
X
X
1 ;
mDj CO.1/
m k2 4
kPk2 ;2 Q<j 12 2 kPk1 ;1 Qm Pk1 ; kL2 L2 kPk2 ;2 Q<j
C
0
3k2 4
2
j 4
k kS Œk1
X
k
2 kS Œk2
t
k2L1 L2
2 ; 0
.2
kL1 L2x
x
t
X
C
t
21
x
12
2Cm0
as claimed. The inner sums run over 1 ; 2 2 C m dist.1 ; 2 / . 2
m k2 2
k2 2
and 1 2 denotes
.
Remark 4.9. We note here that one may also gain in terms of m0 at the expense of some losses in terms of the frequencies/modulations; specifically, using similar reasoning and under the same assumptions as above, one obtains X
kPk Qj .Pk1 ;
/kL2 L2x t
2Cm0
. 2ı1 m0 2ı2 .k1
j/
2
3k2 4
2
j 4
k kS Œk1
X
k
2 kS Œk2
21
(4.30)
2Cm0
for suitable ı1;2 > 0. We shall also require the following estimates which gain something in terms of the small angle. Corollary 4.10. Given ı > 0 small and L 1, there exists m0 .ı; L/ 1 with the following property: let k; k1 ; k2 2 Z so that maxi D1;2 jk ki j L. For any
88
4 Bilinear estimates involving S and N spaces
1 and 2 which are adapted to k1 ; k2 , respectively, and j k C C , X
kPk Qj .Pk1 ;1 1 Pk2 ;2 2 /kL2 L2x ı2
k j 3
2
t
k1 2
jjj 1 jjjS Œk1 jjj 2 jjjS Œk2 :
1 ;2 2Cm0 dist.1 ;2 /2m0
(4.31) In the high-low case k D k1 C O.1/, k2 k1 X
C,
kPk Qj .Pk1 ;1 1 Pk2 ;2 2 /kL2 L2x ı2
k2 j 3
t
2
k2 2
k 1 kS Œk1 k 2 kS Œk2
1 ;2 2Cm0 dist.1 ;2 /2m0
(4.32) as well as X 1
k j k
Pk Qj . 1 Pk ; 2 / 2 2 2 2 ı2 23 2 22 k 1 kS Œk k 2 kS Œk : 1 2 2 L L t
x
2 2Cm0
(4.33) Proof. Let k1 D 0 whence jkj L and jk2 j 2L. Implicit constants will be allowed to depend on L. By Corollary 4.6 and (4.27), X
Pk Qj .Qj Ck
C Pk1 ;1 1 Qj Ck C Pk2 ;2 2 / L2 L2x t
1 ;2 2Cm0 dist.1 ;2 /2m0
.2
kCj 4
2
j 2
jjj 1 jjjS Œk1 jjj 2 jjjS Œk2 ı2
j 4
jjj 1 jjjS Œk1 jjj 2 jjjS Œk2 m0
kCj
which is sufficient. Note that we used interpolation and 2 4 2 2 which gives the desired gain of ı provided m0 is small enough relative to ı and L. For the remaining cases we use a variant of (4.28): with 2 > r > 1, D 2r 1, and 1 1 1 p D r 2,
Pk Qj .Q>j Ck .2 .2
j 4 j 4 j
C Pk1 ;1 1
Pk2 ;2 2 / L2 L2 t
kQ>j Ck
C 1
kQ>j Ck
p C Pk1 ;1 1 kL2t L2x kPk2 ;2 2 kL1 t Lx
. 2 4 .
2/ m0 . 21
2
1 p/
(4.34)
x
Pk2 ;2 2 kL2 Lrx
kPk1 ;1 1 k
t
1 ;1 0; 2
XP 0
kPk2 ;2 2 kL1 L2x : t
(4.35)
4.1 Basic L2 -bounds
89
j
Taking close to 1, one can make this ı2 4 as desired. This bound can be summed over 1 ; 2 by Cauchy–Schwarz and the definition of the S Œk-norm; see also Lemma 2.18. In the high-low case j k2 k D k1 C O.1/ D 0 we proceed as follows. First,
Pk Qj .Q<j C Pk ; 1 Q<j C Pk ; 2 / 2 2 1
j
.2
k2 2
min 2
m0 2
1
2
j
;2
k2 4
2
kPk1 ;1 Q<j j
. ı2
k2 3
2
k2 2
kPk1 ;1 Q<j
L t Lx
2
k2 2
C 1 kS Œk1 kPk2 ;2 Q<j C 2 kS Œk2
C 1 kS Œk1 kPk2 ;2 Q<j C 2 kS Œk2 j
k2
by a decomposition into caps of size 2 2 and the L2 -bilinear bound (2.30). The summation over 1 and 2 can be carried out since it leads to the square function (2.15). Next, by the improved Bernstein estimate, see Lemma 2.1, kPk Qj .Pk1 ;1 1 Qj
C Pk2 ;2 2 /kL2t L2x
. kPk1 ;1 1 kL1 L2 kQj j
. min.2 j
ı2
k2 4
k2 3
2
;2
k2 2
m0 2
/2k2 2
C Pk2 ;2 2 kL2t L1 x j 2
kPk1 ;1 1 kL1 L2x kPk2 ;2 2 k
kPk1 ;1 1 kL1 L2x kPk2 ;2 2 k t
1 ;1 0; 2
XP k
t
2
0; 1 2 ;1
XP k
2
and summation over 1 ; 2 is again admissible. Finally,
Pk Qj .Qj C Pk ; 1 Q<j C Pk ; 2 / 2 2 1 1 2 2 L t Lx X
. Pk Qj .Qm Pk1 ;1 1 Q<j C Pk2 ;2 2 / L2 L2 t
x
mj CO.1/
X
X
mj CO.1/
10 ;20
X
X
mj CO.1/
10 ;20
. .
kPk1 ;1 Qm Pk1 ;10 1 Pk2 ;2 Q<j kQm Pk1 ;1 Pk1 ;10 1 kL2 L2x kQ<j t
X
.
2
m 2
2k2 min.2
m k2 4
;2
m0 2
C Pk2 ;20 2 kL2t L2x C Pk2 ;2 Pk2 ;20 2 kL1 t;x
/kPk1 ;1 1 kS Œk1 kPk2 ;2 2 kS Œk2
mj CO.1/
. ı2
k2 2
j
2
k2 3
kQj CO.1/ Pk1 ;1 1 k
0; 1 2 ;1
XP k
1
kPk2 ;2 2 kL1 L2x t
(4.36)
90
4 Bilinear estimates involving S and N spaces
as claimed. The inner sums run over 10 ; 20 2 C j
with dist.10 ; 20 / . 2
k2 2
j
k2 2
.
The bound in (4.36) can be summed over the caps 1 ; 2 by definition of the S Œk norm. Finally, (4.33) follows from the preceding since the gain of ı was obtained only from the low-frequency function 2 . We can therefore square-sum the final estimate to obtain the desired conclusion.
4.2
An algebra estimate for SŒk
The following bilinear bound expresses something close to an algebra property of the SŒk spaces. It is obtained by removing the restriction on the modulation of the output in Lemma 4.7. Lemma 4.11. For any j; k 2 Z,
Pk Qj . / . 2k1 ^k2 2 provided ; tively.
1
XP 0; 2 ;1
k k1 _k2 4
j
2
k1 ^k2 ^0 4
2.k1 _k2
j /. 21 "/^0
jjj jjjS Œk1 jjj jjjS Œk2 ; (4.37)
are Schwartz functions which are adapted to k1 and k2 , respec-
Proof. We commence with the high-high case k1 D k2 C O.1/ D 0 and k O.1/. We need to prove that
Pk Qj . /
j
k
. 2 4 min 2 4 ; 2
1
XP 0; 2 ;1
To begin with, one has
2j=2 Pk Qj .Q>j 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 "/
jjj jjjS 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
/k kS 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 k kS 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 "/
/k kS 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
/k kS 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
k kS 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 Œk1 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 . Q
1
XP 0; 2 ;1
. 2k1 1 C .k2 ^ a
k1 /C k kS Œk1 k kS Œk2
(4.39)
where k D O.1/ C k2 . Proof. Summing (4.37) over j yields (4.39) with a k2 . It thus suffices to consider a k2 . If a k1 we use Q
2j=2 Pk Qj . Q
x
t
j
X
2j=2 kPk Qj .Q
j aC10
X
C
2j=2 kPk Qj .Qj CO.1/ Q
j aC10
X
. 2k1 k kS Œk1 k kS Œk2 C
2j=2 kQj CO.1/ kL2 L1 kQ
j aC10 k1
. 2 k kS Œk1 k kS Œk2 X
C
3
2j=2 2. 2
"/k1
2
j.1 "/
k kS Œk1 kQ
j aC10
. 2k1 k kS Œk1 k kS Œk2 as desired. The sum over j a C 10 was estimated via Lemma 4.11. If k1 < a k2 , one proceeds similarly.
93
4.3 Bilinear estimates involving both S Œk1 and N Œk2 waves
Bilinear estimates involving both SŒk1 and N Œk2 waves
4.3
The following lemma is a crucial tool. In essence, it expresses the property N S ,! N . Lemma 4.13. 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
k kS Œk1 kF kN Œk2
(4.40)
provided Pk2 Qj F D F and under the following condition Pk Qj
C .Q<j C
F / D Pk Qj
C .Q<j Ck k1 C
F/
(4.41)
in the case k1 D k2 C O.1/ k C O.1/ j . If (4.41) fails, then one loses a factor of 1 C .k1 k/C on the right-hand side of (4.40); alternatively, one has the following weaker version of (4.40) kPk . F /kN Œk . 2k1 ^k2 2
j
k^k1 ^k2 ^0 4
k k
0; 1 2 ;1
XP k
kF kN Œk2 :
(4.42)
1 1
0; ;1 Proof. We remark beforehand that this proof will only use the XP k1 2 -norm for the elliptic regime D Pk Qk of the S Œk norm. In particular, the embedding k kS Œk1 . k k 0; 1 ;1 holds without any restrictions on the modulation, XP k
2
1
cf. (2.20). We start with the high-high case k1 D k2 C O.1/ D 0. Throughout this proof, we shall freely use Lemma 2.15 in order to remove Q<j C from various estimates. First, we consider the case where D Qj C . If j k, then by Bernstein’s and Hölder’s inequalities
Pk . F / . 2 k kPk . F /kL1 L2 . k F kL1 L1 N Œk . k kL2 L2x kF kL2 L2x . k kS Œk1 2 t
t
j 2
kF kL2 L2x t
. k kS Œk1 kF kN Œk2 which is admissible. If on the other hand j k, then we again have to consider several subcases. If D Qk , then 2
k
kQk F kL1 L2x . kQk F kL1 L1x . kQk kL2 L2x kF kL2 L2x t
t
.2
j
k 2
k kS Œk1 kF kN Œk2
t
t
94
4 Bilinear estimates involving S and N spaces
which is admissible. Hence it suffice to assume that D Qj C k . Furthermore, we can assume that the output is at modulation j . In fact, by the improved Bernstein’s inequality, k
Pk Q>j . F / .2 N Œk
X
` 2
2
kPk Q` . F /kL2 L2x t
`>j
.
X
` 2
2
2
` k 4 ^0
k F kL2 L1
`>j
.
X
2
j
` 2
2
` k 4 ^0
k kL1 L2x 2
j 2
t
kF kL2 L2x t
`>j
.2
j
k 4
k kS Œk1 kF kN Œk2
as desired. Now consider the output of modulation at most j . We also first restrict ourselves to the contributions by Qj C <<j CC . Thus, by Lemma 4.1 and Lemma 2.15,
Pk Qj .Qj C <<j CC F / N Œk X k .2 kPk Qj .Q` F /kL1 L2x t
`Dj CO.1/
.2
k
X
X
`Dj CO.1/ D2Dk
.2
k
X
X
X 0 2C
kPk Qj .PD P Q` P 0 P
D F /kL1t L2x
.`Ck/=2
X
kPD P Q` P 0 P
D F kL1t L2x
`Dj CO.1/ D2Dk 0 2C.`Ck/=2
where 0 means dist.; 0 / . diam./. Moreover, Dk is a cover of fjj 1g by disks of diameter 2k and with overlap uniformly bounded in k; the associated projections are PD . Hence, one can further estimate (recall j k)
Pk Qj . F / N Œk X X k .2 `Dj CO.1/ D2Dk
.2
k
X
X
`Dj CO.1/ D2Dk
. 2.j
k/=4
X
kPD P Q` kL2 L1 kP 0 P x t
0 2C
.`Ck/=2
X 0 2C
D F kL2t L2x
2
`C3k 4
.`Ck/=2
k kS Œk1 kF kN Œk2 :
kPD P Q` kL2 L2x kP 0 P t
D F kL2t L2x
95
4.3 Bilinear estimates involving both S Œk1 and N Œk2 waves
Next, we consider the output of modulation at most j and D Qj CC k . Then we are in the “imbalanced case” of Lemma 4.1, whence X X Pk Qj . F / D Pk; Qj .Pk1 ; 0 Ql Pk2 ; 00 F / k`j CC ; 0 ; 00 ` k
where 2 C ` k , 0 ; 00 2 C.`Ck/=2 and dist.; 0 / 2 2 , dist. 0 ; 00 / 2 2 Using (2.29) one obtains
Pk Qj . F / N Œk X X
k
Pk; Qj .Pk ; 0 Q` Pk ; 00 F / 2 1 2 NFŒ
`Ck 2
.
k`j CC ; 0 ; 00
.
X
2
k`j CC
.
X
2
k
2 2
j
k 4
`Ck 4
X
` k 2 ^0
2
j
` 4
kPk1 ; 0 Q` kS Œk1 ; kPk2 ; 00 F kL2 L2x t
0 ; 00 j 2
kQ` kS Œk1 2
kF kL2 L2x . 2
j
t
k 4
k kS Œk1 kF kN Œk2
k`j CC
as desired. To pass to the final inequality one uses (2.20) as well as kQ` kS Œk1 kQ` k 0; 1 ;1 . XP k
2
1
Now assume Q<j C D . We first dispose of outputs of modulation exceeding j C . If j k, then kPk Q>j .2 .2
C . F /kN Œk j 2 j 2
kPk Q>j
C . F /kL2t L1x
.2
j 2
k kL1 L2 kF kL2 L2
k kS Œk1 kF kL2 L2x . k kS Œk1 kF kN Œk2 t
which is admissible. On the other hand, if j k, then X ` kPk Q>j C . F /kN Œk . 2 k 2 2 kPk Q` . F /kL2 L2x t
`>j 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
k kS 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 <
C
F C /;
whence
Pk Qj
C .Q<j C
C
F C / N Œk
. k C F C kL2 L1 . k C kL1 L2 kF C kL2 L2x t
. k kS Œk1 kF kN Œk2 which is admissible. Therefore, we can now ignore the contribution of (4.3). Let us now also assume without loss of generality that D C , see Lemma 2.12. Using duality and Lemma 4.1, one obtains in view of (4.41) Pk Qj
F/ X X ˙ D Pk;˙ Qj
C .Q<j C
C .Pk1 ; 0 Q<j Ck C
Pk2 ; 00 F / (4.43)
˙ ; 0 ; 00
with caps 2 C` , 0 ; 00 2 C` satisfying dist. 0 ; 00 / 2m , dist.; 0 / 2` where ` D .j k/=2, m D .j C k/=2. Note that Lemma 4.1 also implies that j k C O.1/. Since ˙ Pk;˙ Qj
C
˙ D Pk;˙ QkC2`
(4.44)
C
the right-hand side of (4.43) represents a wave-packet decomposition in the sense of Definition 2.9. Moreover, the operators in (4.44) are disposable in the sense of Lemma 2.14. Therefore, kPk Qj
C .Q<j C
F /kN Œk
.2
k
max ˙
X
Pk ; 0 Q<j Ck 1
C
Pk2 ; 00 F kNFΠ:
0 ; 00
We could discard here since the choice of 0 leaves only a finite number of choice of . Invoking (2.29), this can be further estimated by .2
k
. 2.j
max ˙
k/=4
X 2 j Ck 4 0 ; 00
2
j
k
kPk1 ; 0 Q<j Ck
2
k kS Œk1 kF kN Œk2 :
C kS Œk1 ; 0 kPk2 ; 00 F kL2t L2x
97
4.3 Bilinear estimates involving both S Œk1 and N Œk2 waves
To pass to the final inequality here it was essential that (4.41) reduced the modulation of from < j C to < j C k C . Indeed, if (4.41) fails, then we need to write Q<j C D Q<j Ck C C Qj Ck C <j C . For the first summand here one applies the argument we just gave, whereas for the second summand the best one can do is to invoke (2.20) which results in the loss of of factor of k a claimed. This concludes the high-high case. Let us now consider the low-high case k1 < C , k2 D k D O.1/. Since (2.24) implies that kQj
C
F kL1 L2 . kQj j 2
.2 j
.2
C kL2 L1 kF kL2t L2x j
2
k1 4 ^0
2k1 k kS Œk1 kF kL2 L2x t
k1 4 ^0
2k1 k kS Œk1 kF kN Œk2
it will suffice to bound kQj C F kN Œk . Moreover, if j k1 C C , then the modulation of Qj C F is on the order of j , whence kQj
C
F kN Œk . 2 .2 .2
j 2 j 2 j 2
kQj
C
F kL2 L2x
kQj
C kL1 L1 kF kL2t L2x
t
2k1 k kS Œk1 kF kL2 L2x . 2k1 k kS Œk1 kF kN Œk2 t
as desired. We may therefore assume that j k1 C C . We first consider the case where the output has modulation j C . More precisely, let j C m k1 C C , ` D .m k1 /=2, as well as without loss of generality F D F C . Then by the balanced modulation case of Lemma 4.1, X C Qm .Qj C F / D Pk; Qm .Pk1 ; 0 Qj C F / ; 0
where ; 0 are caps of size C kQm .Qj
1 2`
and with dist.; 0 / . 2` . Therefore,
F /kN Œk X C . 2 m=2 k Pk; Qm .Pk1 ; 0 Qj C
C
F /kL2 L2x t
; 0
.2
m=2
X
2 2 kPk1 ; 0 Qj C kL 1 L1 kF k 2 2 L L
21
; 0
.2
m=2
X ; 0
22k1 2` kPk1 ; Qj
2 C kL1 L2
2 kF kL 2 L2
21
98
4 Bilinear estimates involving S and N spaces
.2
m=2
2
.k1 m/=4 k1
.2
m=2
2
.k1 m/=4 k1
2 kQj
C kL1 L2 kF kL2t L2x
2 k kS Œk1 kF kL2 L2x t
where we used Lemma 2.15 in the final step. Summing over m j that kQ>j
C .Qj C
F /kN Œk . 2k1 2
j=2
k1
2
.k1 j /=4
.k1 j /=4
.2 2
C implies
k kS Œk1 kF kL2 L2x t
k kS Œk1 kF kN Œk2
which is admissible. It therefore remains to bound kQ<j C .Qj C F /kN Œk for which we shall again apply a wave-packet decomposition as in Lemma 4.1. Since j k1 C C and k1 1, we can assume that j C in applying Lemma 4.1 (which allows us to ignore the opposing .CC/ or . / contributions in (4.3)). Without loss of generality, we assume further that D C (see Lemma 2.12). Then with caps ; 0 of size C 1 2m and separation 2m where m WD .j k1 /=2, kQ<j
C .Qj C F /kN Œk
X
. Pk; Q<j 0
X
.
Pk; Q<j
C .Pk1 ; 0 Qj C F / N Œk
k1 .Pk1 ; 0 Qj C
0
where we used Corollary 2.23 to dispose of Qj further bounded by .2
k1 2
j
2
k1 4
X 0
. 2.j
C.
21
2
F /
(4.45)
N Œk
In view of (2.29) this is
kPk1 ; 0 Qj C k2S Œk1 ; 0 kF k2L2 L2 t x
21
k1 /=4 k1
2 k kS Œk1 kF kN Œk2
as desired. It remains to consider the high-low case k D k1 CO.1/ D 0, k2 <
C . First,
kQ>j Ck2 F kN Œk . kQ>j Ck2 F kL1 L2 . kQ>j Ck2 kL2 L2 kF kL2 L1 .2
.j Ck2 /=2 k2
.2 2
j
k kS Œk1 2k2 2
k2 4 ^0
k kS 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
k kS 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 k kS Œk1 kF kN Œk2
. 2k2 2.j
k2 /=4
k kS 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
k kS 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
k kS 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 k kS 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
k 1 kS Œk1 k 2 kS Œk2 :
(4.50)
In particular, k1 2
kPk QkCC Q˛ˇ . 1 ; 2 /kL2 L2x . 2k t
Finally, for any m0
k 1 kS Œk1 k 2 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
k 1 kS Œk1 k 2 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 k 1 kS Œk1 k 2 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
k 1 kS Œk1 k 2 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
k 1 kS Œk1 k 2 kS Œk2
`mC ` k
. 2 4C 2k k 1 kS Œk1 k 2 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 "/
k 1 kS Œk1 k 2 kS Œk2
mC
.2 as desired.
` k 4
2k k 1 kS Œk1 k 2 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
k 1 kS Œk1 k 2 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
k 1 kS Œk1 k 2 kS Œk2 ı k 1 kS Œk1 k 2 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 "/
k 1 kS Œk1 k 2 kS Œk2 ık 1 kS Œk1 k 2 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 k 2 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 k 1 kS Œk1 k 2 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<`
C 1 ; Q` C 2 /kL2t L2x
(4.58)
C kPk Q` Q˛ˇ .Q<`
C 1 ; Q<` C 2 /kL2t L2x :
(4.59)
105
4.4 Null-form bounds in the high-high case
We begin with the estimate X 2 "` kPk Q` Q˛ˇ . 1 ; 2 /kL2 L2x . 2k k 1 kS Œk1 k 2 kS Œk2 t
`kCC `¤k1 CO.1/
which is stronger than what we claim – this is due to the fact that the the case of opposing .CC/ and . / waves is excluded in this sum. We first consider the case ` k1 C 0 where C 0 is large but still smaller than the constant C in (4.56)–(4.59). Then the term in (4.59) vanishes. On the one hand,
(4.56) . 2k Pk Q` Œ@ˇ .Q` C jrj 1 1 Qk2 CC @˛ jrj 1 2 /
@˛ .Q` C jrj 1 1 Qk CC @ˇ jrj 1 2 / 2 1 L t Lx
2
.2
kC`
kQ`
C 1 kL2t L2x k 2 kS Œk2 :
Here we used that kQk2 CC @ˇ jrj
1
2 kL1 L2x . k 2 kS Œk2 : t
Furthermore, X
X
2.1
"/` k k1
2
kQm 1 kL2 L2x k 2 kS Œk2 t
kCC `k1 C m` C
. 2k k 1 kXP 0;1
";2
k1
k 2 kS Œk2
. 2k k 1 kS Œk1 k 2 kS Œk2 as desired. The term (4.58) satisfies the same bound, more precisely, it can be reduced to (4.56), (4.57). Next, note that due to ` k1 C 0 it suffices to consider 1 D Qk1 CC 1 in (4.57). Consequently, X 2 "` (4.57) kCC `k1 C
.
X
2
"`
X X kCC `k1 C
x
mk1 CC
2
"`
X
` 2k 2 2 Q˛ˇ .Qm 1 ; QQ m 2 / L1 L1 t
kCC `k1 C
.
Pk Q` Q˛ˇ .Qm 1 ; QQ m 2 / 2 2 L L t
kCC `k1 C
.
X
x
mk1 CC
2
"`
X mk1 CC
. 2k k 1 kS Œk1 k 2 kS Œk2
`
2k 2 2 2m 2
2m.1 "/
k 1 kS Œk1 k 2 kS Œk2
106
4 Bilinear estimates involving S and N spaces
where we used that " < Then X
1 4
in the final step. Second, suppose that ` k1 C C 0 .
"`
Pk Q` Q˛ˇ . 1 ; 2 / L2 L2
2
t
`k1 CC 0
X
.
`k1
2
x
"`
Pk Q` Q˛ˇ .QQ ` 1 ; Q`
CC 0
X
C
`k1
`k1
"`
2
"`
Pk Q` Q˛ˇ .Q`
CC 0
Q
(4.60)
5 1 ; Q` 2 / L2 L2x t
(4.61)
X
Pk Q` Q˛ˇ .Qm 1 ; QQ m 2 /
(4.62)
CC 0
X
C
2
5 2 / L2 L2x t
L2t L2x
m` 5
which are in turn estimated as follows: X
(4.60) .
2
"` k
2
"` k `
2
Pk Q` Q˛ˇ .QQ ` 1 ; Q`
`k1 CC 0
X
.
`k1
5 2 / L2 L1x t
2 2 kQQ ` 1 kL2 L2x k 2 kL1 L2x t
t
CC 0
. 2k k 1 kS Œk1 k 2 kS Œk2 and similarly for (4.61), whereas (4.62) is bounded by X
.
2
"`
X `k1
CC 0
` 2 2 2k Q˛ˇ .Qm 1 ; QQ m 2 / L1 L1 t
`k1 CC 0
.
X
x
m` 5
2
"`
X
`
2 2 2k 2m 2
2m.1 "/
k 1 kS Œk1 k 2 kS Œk2
m` 5
k
. 2 k 1 kS Œk1 k 2 kS Œk2 as desired. It remains to consider the case j` k1 j D j`j C 0 , which gives us the weaker bound stated in the lemma. We use the decomposition (4.56)–(4.59). The terms (4.56)–(4.58) give a bound of 2k as before. The main difference lies with (4.59) which is nonzero only due to the contribution to opposing .CC/ or . / waves,
107
4.4 Null-form bounds in the high-high case
see Lemma 4.1. In fact, one has with ` D k1 C O.1/ D O.1/,
Pk QO.1/ Q˛ˇ .Q< C 1 ; Q< C 2 / 2 2 L t Lx X X
Pk QO.1/ Q˛ˇ .Q< C P 1˙ ; Q< C P 2˙ / L2 L2 . t
x
˙ 2Ck
.
k
X X
2 2 kQ<
˙ ˙ C P 1 kS Œk1 ; kQ< C P 2 kS Œk2 ;
(4.63)
˙ 2Ck
.2
k 2
X X ˙
kQ< C P 1˙ k2S Œk1 ;
12 X
2Ck
kQ<
CP
˙ 2 2 kS Œk2 ;
21
2Ck
k
. 2 2 k 2 k 1 kS Œk1 k 2 kS Œk2 : To pass to the last line, we wrote X kQ< C P 1˙ k2S Œk1 ; 2Ck
.
X
kQ<2k P 1˙ k2S Œk1 ; C
2Ck
2Ck
.
k 1˙ k2S Œk1
X
C jkj
X
X
X
. jkj
2
2kj C
kQj P 1˙ k2 0; 1 ;1 XP k
2kj C 2Ck 2
kQj P 1˙ kS Œk1 ;
(4.64)
2
1
k 1˙ k2S Œk1
and the result follows. The second statement (4.55) follows by essentially the same proof.
Remark 4.20. It is important to note that the logarithmic loss of hk1 ki2 in (4.55) only results from the case of opposing waves in the high-high case. Later we will use (4.55) without this loss in those cases where these interactions are excluded. Later, we shall also require the following technical refinement of Lemma 4.19 dealing with a further angular restriction of the first input. Corollary 4.21. Under the assumptions of Lemma 4.19 and for any m0 X X 2 21 2 "` kPk Q` Q˛ˇ .Pk1 ; 1 ; 2 /kL2 L2x
10,
t
2Cm0
`kCC k
. jm0 j 2 2 2
"k1
hk1
ki2 k 1 kS Œk1 k 2 kS Œk2
108
4 Bilinear estimates involving S and N spaces
with an absolute implicit constant. Proof. This can be seen by reviewing the proof3 of Lemma 4.19. Specifically, up 2 until (4.63), one places Pk1 ; 1 either in the XP s;b or L1 t Lx norms. The norms are amenable to square summation, in the latter case at the expense of a factor jm0 j, see Lemma 2.18. However, as far as (4.63) is concerned, we distinguish two cases: k m0 and k > 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<
C 1
D Q<2m0 1 C Q2m0 <<
C 1 :
The former is subsumed in a square function bound as in (4.63), whereas the latter leads to a loss of jm0 j as in (4.64) and the corollary is proved. Next, we obtain an improvement in case of angular alignment of the inputs. This is analogous the case of low modulations, see Lemma 4.18. Lemma 4.22. 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
X
2
1 ;2 2Cm0 `kCC dist.1 ;2 /2m0
"`
kPk Q` Q˛ˇ .Pk1 ;1 1 ; Pk2 ;2 2 /kL2 L2x t
1
ı 2. 2
"/k1
k 1 kS Œk1 k 2 kS Œk2 ; (4.65)
provided maxj D1;2 jk kj j L. The constant C in (4.65) is an absolute constant which does not depend on L or ı. Proof. The proof consists of checking that one can glean a gain from angular alignment by following the proof of Lemma 4.19. In effect, this will always be done by means of Bernstein’s inequality. The only case where this is not possible is (4.63), but that case is excluded by the angular alignment assumption. We set k1 D 0 whence jkj L and jk2 j 2L. Implicit constants here will be allowed to depend on L, but not the constants C appearing in modulation cutoffs.
3
It is important P to observe that one cannot square sum the bound of Lemma 4.19 directly due to the fact that 2Cm kPk1 ; 1 k2S Œk cannot be controlled. 0
109
4.4 Null-form bounds in the high-high case
As before, one has the decomposition kPk Q` Q˛ˇ .Pk1 ;1 1 ; Pk2 ;2 2 /kL2 L2x
(4.66)
. kPk Q` Q˛ˇ .Q`
(4.67)
t
C Pk1 ;1 1 ; Qk1 CC Pk2 ;2 2 /kL2t L2x
C kPk Q` Q˛ˇ .Q`
C Pk1 ;1 1 ; Q>k1 CC Pk2 ;2 2 /kL2t L2x
(4.68)
C kPk Q` Q˛ˇ .Q<`
C Pk1 ;1 1 ; Q` C Pk2 ;2 2 /kL2t L2x
(4.69)
C kPk Q` Q˛ˇ .Q<`
C Pk1 ;1 1 ; Q<` C Pk2 ;2 2 /kL2t L2x :
(4.70)
We first consider the case ` k1 C 0 where C 0 is large but still smaller than the constant C in (4.67)–(4.70). Then the term in (4.70) vanishes by Lemma 4.1. By Bernstein’s inequality, X 2 "` (4.67) kCC `k1 C
X
.
Pk Q` Œ@ˇ
kCC `k1 C
.Q`
C jrj
1
Pk1 ;1 1 Qk2 CC @˛ jrj
1
Pk2 ;2 2 /
1
@˛ .Q` C jrj Pk1 ;1 1 Qk2 CC @ˇ jrj 1 Pk2 ;2 2 / L2 L2 t x X 1 kQ` C Pk1 ;1 1 kL2 L2x kPk2 ;2 2 kL1 t Lx
.
t
kCC `k1 C
.2
m0 2
X
kQ`
2 C Pk1 ;1 1 kL2t L2x kPk2 ;2 2 kL1 t Lx
kCC `k1 C
ıkPk1 ;1 1 k
1 ;1 0; 2
XP 0
kPk2 ;2 2 kL1 L2x : t
Summing over 1 ; 2 now yields the desired bound by the Cauchy–Schwarz inequality (see also Lemma 2.18). The term (4.69) satisfies the same bound. Next, note that due to ` k1 C 0 it suffices to consider 1 D Qk1 CC 1 in (4.68). Consequently, X 2 "` (4.68) kCC `k1 C
.
X
X
Pk Q` Q˛ˇ .Qm Pk
1 ;1
1 ; QQ m Pk2 ;2 2 / L2 L2 t
kCC `k1 C mk1 CC
.
X
X
Q˛ˇ .Qm Pk
kCC `k1 C mk1 CC
1 ;1
1 ; QQ m Pk2 ;2 2 / L1 L2 t
x
x
110
4 Bilinear estimates involving S and N spaces
X
.
X
2m 2
2m.1 "/
kPk1 ;1 Qm 1 kL2 L2x kPk2 ;2 QQ m 2 kL2 L1 x t
t
kCC `k1 C mk1 CC
. ıkPk1 ;1 1 kXP 0;1
";2
0
kPk2 ;2 2 kXP 0;1
";2
:
0
Summing over 1 ; 2 again leads to the desired bound. Second, suppose that ` k1 C C 0 . Then X
2 "` Pk Q` Q˛ˇ .Pk1 ;1 1 ; Pk2 ;2 2 / L2 L2 X
.
2
`k1
CC 0
C
X `k1
"`
Pk Q` Q˛ˇ .QQ ` Pk1 ;1 1 ; Q`
2
"`
2
"`
Pk Q` Q˛ˇ .Q`
CC 0
X
C
x
t
`k1 CC 0
`k1 CC 0
5 Pk2 ;2 2 / L2 L2x t
Q
(4.71)
5 Pk1 ;1 1 ; Q` Pk2 ;2 2 / L2 L2x t
X
Pk Q` Q˛ˇ .Qm Pk
1 ;1
(4.72)
1 ; QQ m Pk2 ;2 2 / L2 L2 t
x
m` 5
(4.73) which are in turn estimated as follows: X (4.71) 1 ;2 2Cm0 dist.1 ;2 /2m0
.
X
X
2
"`
Pk Q` Q˛ˇ .QQ ` Pk1 ;1 1 ; Q`
1 ;2 2Cm0 `k1 CC 0 dist.1 ;2 /2m0
.
X
X
2.1
"/`
5 Pk2 ;2 2 / L2 L2x t
kQQ ` Pk1 ;1 1 kL2 L2x kPk2 ;2 2 kL1 L2x t
t
1 ;2 2Cm0 `k1 CC 0 dist.1 ;2 /2m0
ık 1 kS Œk1 k 2 kS Œk2 and similarly for (4.72), whereas X
(4.73)
1 ;2 2Cm0 dist.1 ;2 /2m0
.
X
X
1 ;2 2Cm0 `k1 CC 0 dist.1 ;2 /2m0
2
"`
X
` 2 2 Q˛ˇ .Qm Pk1 ;1 1 ; QQ m Pk2 ;2 2 / L1 L2 t
m` 5
x
111
4.5 Null-form bounds in the low-high and high-low cases
.2
m0 2
X
1
X
2. 2
"/` m
2 2
1 ;2 2Cm0 mC5`k1 CC 0 dist.1 ;2 /2m0
2m.1 "/
kPk1 ;1 1 kS Œk1 kPk2 ;2 2 kS Œk2
ık 1 kS Œk1 k 2 kS Œk2 as desired. It the remaining case j` k1 j D j`j C 0 we use the decomposition (4.67)– (4.70). The terms (4.67)–(4.69) give a bound of ı as before. The main difference lies with (4.70) which is nonzero only due to the contribution to opposing .CC/ or . / waves, see Lemma 4.1. However, this case is excluded due to the angular alignment assumption.
4.5
Null-form bounds in the low-high and high-low cases
We now derive analogues of the previous two lemmas in the high-low case, with the low-high case being completely analogous. Lemma 4.23. For any j adapted to kj with k2 k1 C O.1/ D k one has
Pk QkCC Q˛ˇ . 1 ; 2 / 2 2 . 2. 12 L L t
"/k2 "k1
x
2
k 1 kS Œk1 k 2 kS Œk2 :
Proof. We may take k D k1 C O.1/ D 0 and k2 C . Assume first that Q
P0 Qj .R˛ Q<j
j k2 CO.1/
C
X
C 1 Rˇ Q<j C 2
Rˇ Q<j
C 1 R˛ Q<j C 2 / L2 L2x t
P0 Q<j CC .R˛ Qj 1 Rˇ Qj 2
(4.74)
Rˇ Qj 1 R˛ Qj 2 / L2 L2
P0 Q<j CC .R˛ Q<j 1 Rˇ Qj 2
(4.75)
Rˇ Q<j 1 R˛ Qj 2 / L2 L2
t
x
j k2 CO.1/
C
X
t
x
j k2 CO.1/
(4.76) Each of the summands here is bounded by 2.j Ck2 /=4 . For the first, one decom-
112
4 Bilinear estimates involving S and N spaces j
poses into caps of size 2 X (4.74) .
k2 2
: X
P0 Qj .R˛ Q<j
j k2 CO.1/ 0 2C j
Rˇ Q<j X j 2
.
C P 1 Rˇ Q<j C P 0 2
k2 2
C P 1 R˛ Q<j C P 0 2 / L2 L2x t k2 4
2
k2 2
kQ<j
C P 1 kS Œk1 ; kQ<j C P 0 2 kS Œk2 ; 0
j k2 CO.1/
.2
k2 2
k 1 kS Œk1 k 2 kS Œk2
where we applied (2.29) in the last step. Note that the null-form gains a factor of the angle in this bound. As for (4.75), one performs a similar cap decomposition but without the separation between the caps: X X
P0 Q<j CC .R˛ Qj P 1 Rˇ Qj P 0 2 (4.75) . j k2 CO.1/ ; 0 2C j
.
k2 2
Rˇ Qj P 1 R˛ Qj P 0 2 / L2 L2 t x X j k2 1 2 2 kQj P 1 kL2 L2x kQj P 0 2 kL1 t Lx t
j k2 CO.1/
.
X
j
2
k2 2
j 2
2
j
2
k2 4
2k2 kQj P 1 kS Œk1 ; kP 0 2 kL1 L2x t
j k2 CO.1/
.
X
2
j Ck2 4
k 1 kS Œk1 k 2 kS Œk2 . 2
k2 2
k 1 kS Œk1 k 2 kS Œk2
j k2 CO.1/
Finally, (4.76) .
X
X
j k2 CO.1/ ; 0 2C j
.
kP0 Q<j CC .R˛ Q<j P 1 Rˇ Qj P 0 2 k2 2
Rˇ Q<j P 1 R˛ Qj P 0 2 /kL2 L2x t X j k2 2 2 kQ<j P 1 kL1 L2x kQj P 0 2 kL2 L1 x t
t
j k2 CO.1/
.
X
j
2
k2 2
j
2
k2 4
2k2 kQ<j P 1 kS Œk1 ; kQj P 0 2 kL2 L2x t
j k2 CO.1/
.
X j k2 CO.1/
2
j Ck2 4
k 1 kS Œk1 k 2 kS Œk2 . 2
k2 2
k 1 kS Œk1 k 2 kS Œk2
113
4.5 Null-form bounds in the low-high and high-low cases
If Qk2 C 2 D 2 , then we may take 1 D QC 1 , whence
P0 QO.1/ Q˛ˇ . 1 ; 2 /
. k 1 kL1 L2x kR0 2 kL2 L1 t t x X . k 1 kL1 L2x 2j kQj 2 kL2 L2x
L2t L2x
t
t
k2 j C 1
. 2. 2
"/k2
k 1 kS Œk1 k 2 kS Œk2 :
On the other hand, if 2 D QC 2 , then necessarily also 1 D QC 1 so that
P0 QO.1/ Q˛ˇ . 1 ; 2 / 2 2 L t Lx
. P0 QO.1/ Q˛ˇ . 1 ; 2 / L1 L1 t x X m Q kQm 1 kL2 L2x 2 kQm 2 kL2 L2x . t
t
mC
. k 1 kS Œk1
X
2m.1 "/ k2 . 12 "/
2m 2
2
k 2 kS Œk2
mC 1
"/k2
. 2. 2
k 1 kS Œk1 k 2 kS Œk2
and the lemma is proved. Next, we deal with the case of outputs with large modulation. Lemma 4.24. For any j adapted to kj with k2 k1 C O.1/ D k one has X
2
1 Pk Q` Q˛ˇ . 1 ; 2 / L2 L2 . 2. 2
"`
"/k2
x
t
k 1 kS Œk1 k 2 kS Œk2
`kCC
Proof. Set k D k1 C O.1/ D 0 and k2 X
2
"`
C . Then
P0 Q` Q˛ˇ . 1 ; 2 / L2 L2 t
x
`C
.
X
2
"`
P0 Q` Q˛ˇ .QQ ` 1 ; Q<`
C 2 / L2 L2x t
(4.77)
`C
C
X `C
2
"`
P0 Q` Q˛ˇ .Q`
C 1 ; Q` C 2 / L2 L2x t
(4.78)
114
4 Bilinear estimates involving S and N spaces
First, taking ˛ D 0 and ˇ D 1, X
(4.77) . 2 "` P0 Q` .R0 QQ ` 1 R1 Q<`
C 2
`C
R1 QQ ` 1 R0 Q<` X
.
"`
2
C 2 / L2 L2x t
1 .2` kQ` 1 kL2 L2x k 2 kL1 t Lx t
`C 1/ C kQQ ` 1 kL2 L2x kR0 Q<` C 2 kL1 t Lx t X 1 1 . k 1 kS Œk1 k 2 kS Œk2 2k2 C kQQ ` 1 kL2 L2x 2. 2 C"/` 2. 2 "/k2 k 2 kS Œk2 t
`C 1
"/k2
. 2. 2
k 1 kS Œk1 k 2 kS Œk2 :
To pass to the second to last line we used the estimate kR0 Q<`
1 C 2 kL1 t Lx 3j
X
. 2k2 kQk2 2 kL1 L2x C
2 2 kQj 2 kL2 L2x
t
t
k2 <j <` C 3j
X
. 2k2 kQk2 2 kL1 L2x C
222
t
.1 "/j . 12 "/k2
2
k 2 kS Œk2
k2 <j <` C 1
1
X
. 2k2 kQk2 2 kL1 L2x C
2. 2 C"/j 2. 2
t
"/k2
k 2 kS Œk2
k2 <j <` C 1
1
. 2k2 kQk2 2 kL1 L2x C 2. 2 C"/` 2. 2 t
"/k2
k 2 kS Œk2 :
On (4.78) one has the bound (again for ˛ D 0 and ˇ D 1) X
(4.78) . 2 "` P0 Q` .R0 Qm 1 R1 QQ m 2 R1 Qm 1 R0 QQ m 2 / L2 L2 t
x
m`C
X
.
1
2. 2
"/` m
2 kQm 1 kL2 L2x kQQ m 2 kL2 L2x t
t
m`C
.
X
2
. 21 "/m
1
k 1 kS Œk1 2. 2
"/k2
k 2 kS Œk2
mC 1
. 2. 2 as claimed.
"/k2
k 1 kS Œk1 k 2 kS Œk2
5
Trilinear estimates
The purpose of this section is to derive the estimates on the trilinear nonlinearities which govern the wave map system. To clarify the role that these estimates play, we include here a table which explains how these estimates relate to the trilinear terms arising in Chapter 3. Recall that the trilinear null-forms are summarized in the expression (3.14). In the table, we number the terms in this expression 1–11. The gist of the estimates to follow is that whenever one is given a trilinear null-form of the schematic form rx;t P0
1r
1
Œ
2
3
with each i localized to spatial frequency 2ki , i D 1; 2; 3, then one gains exponentially in the difference of the largest to the smallest logarithmic frequency present, unless one is in the situation where k1 D O.1/ and k2;3 1, i.e., the high-low-low case. This latter feature forces us to modify the procedure of Bahouri–Gérard in Chapter 9, and it also informs our choice of the weight w.k1 ; k2 ; k3 / below. In addition to the bilinear estimates of the previous sections, we will also heavily use the Strichartz component of the S Œk-norm, see (2.14). As already
Table 5.1. Relation of trilinear estimates to null-forms Estimate
Trilinear expression
Lemma 5.1 Corollary 5.2 Lemma 5.3 Corollary 5.4 Lemma 5.5 Remark 5.6 Lemma 5.7 Lemma 5.9 Corollary 5.10 Lemma 5.11
1, 2 3, 7 4–6, 8–11 1, 3, 7 2, 5, 6, 8–11 2 4, 9 1, 3, 7 2, 5, 6, 8–11 2, 5, 6, 8–11
116
5 Trilinear estimates
in [23], we will partially rely on Tao’s trilinear estimate from [59] which states that (relative to our norms in the S Œk-spaces)
P0 Œ
ˇ 1R
2 Rˇ
1 .k2 ^k3
3 N Œ0 . 2
k1 /^0 k1
2
3 Y
k
i kS Œki
(5.1)
i D1
for some 1 > 0. To obtain (5.1) from [59], one observes that kr kS Œk (strictly) dominates the S Œk-norms of [59], whereas kP0 kN Œ0 is dominated by the respective N Œ0-norm used in [59]. Because of this property, the trilinear bound from [59] can be adapted to this setting provided the correct scaling is taken into account. Moreover, throughout this section we define, with kmax WD k1 _ k2 _ k3 , kmin WD k1 ^ k2 ^ k3 , and kmed the median of k1 ; k2 ; k3 , 8 < 2 0 kmax 20 kmin ^0 if kmax C w.k1 ; k2 ; k3 / WD 2 0 .kmed kmin / if k1 D kmax D O.1/ : 0 .k1 Ck2 ^k3 / 2 if k1 < kmax D O.1/ where 0 > 0 is some fixed small constant. We split our argument into two cases, depending on whether all inputs are “hyperbolic” or not. This distinction is based on modulation vs. frequency.
5.1
Reduction to the hyperbolic case
The following lemma deals with the case where at least one ofPthe inputs or the interior null-form have “elliptic” character. Recall that I WD k2Z Pk QkCC and I c WD 1 I (here C is an absolute constant, C D 10 will P suffice). ThroughQ out this section, we will write Pk to denote a projection k 0 DkCO.1/ Pk 0 , and similarly with QQ k . Lemma 5.1. Let i be Schwarz functions adapted to ki for i D 0; 1; 2. Then for any ˛ D 0; 1; 2, and j D 1; 2,
P0 @ˇ A0 ŒA1 R˛
1
1
@j AQ1 Qˇj .A2
2 ; A3
3 / N Œ0
. w.k1 ; k2 ; k3 /
3 Y
k
i kS Œki
iD1
where Ai and AQ1 are either I or I c , with at least one being I c . Moreover, we impose the condition that A1 D AQ1 D I c implies ˛ ¤ 0.
117
5.1 Reduction to the hyperbolic case
Proof. Case 1: 0 k1 k2 C O.1/ D k3 C O.1/. We begin with A0 D I c and A1 D I . Then we can drop IR˛ from 1 and estimate1
P0 Q0 @ˇ Œ 1 1 @j Qˇj . 2 ; 3 / N Œ0
ˇ 1 . P0 Q0 @ Œ 1 @j Qk1 CC Qˇj .
C P0 Q0 @ˇ Œ 1 1 @j Qk1 CC Qˇj . By Lemma 4.17, placing (5.2) into XP 00;
1
1
1 ";2
@j Qk1 CC Qˇj .
.k .2
k2 2
1 kL1 L2x 2 t
k2 2
k
3 / N Œ0
(5.2) (5.3)
implies
2;
k
2 ; 3 / N Œ0 : 2;
3 / L2 L1 t x
2 kS Œk2 k 3 kS Œk3
1 kS Œk1 k 2 kS Œk2 k 3 kS Œk3
whereas X
(5.3) .
P0 Qm @ˇ ŒQm
C
1
1
@j QQ m Qˇj .
2;
3 / N Œ0
(5.4)
mk1 CC
X
C
P0 Q0 @ˇ ŒQm
1
C
1
@j Qm Qˇj .
2;
3 / N Œ0 :
mk1 CC
(5.5) Lemma 4.19 yields the following bound on (5.4): X
P0 Qm @ˇ ŒQm
1
C
1
@j QQ m Qˇj .
2;
3 / N Œ0
mk1 CC
X
.
P0 Qm @ˇ ŒQm
1
C
1
@j QQ m Qˇj .
2;
3 / XP 0; 0
mk1 CC
X
.
2
m"
k
1 kL1 L2x 2 t
k1
kPQk1 QQ m Qˇj .
2;
3 /kL2 L2x t
mk1 CC
.2
"k2
2
k1 2
hk2
k1 i
2
3 Y
k
i kS Œki :
i D1
1
It is convenient to prove the somewhat stronger bound with A0 D Q0 here.
1 ";2
118
5 Trilinear estimates
The bound on (5.5) proceeds similarly: X
P0 Q0 @ˇ ŒQ>m C 1 (5.5) .
1
@j QQ m Qˇj .
2;
3 / N Œ0
mk1 CC
.
X
X
P0 Q` @ˇ ŒQ>m
1
C
1
@j QQ m Qˇj .
2;
3 / N Œ0
mk1 CC 0`mCC
X
C
X
P0 Q` @ˇ ŒQQ `
1
1
@j QQ m Qˇj .
2;
3 / N Œ0
mk1 CC `mCC
.
X
1
X
"/`
2. 2
Q>m
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 Q0k
1 C10C
ŒQ0k1 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 Q0k C10C @ˇ ŒQ0k C10C R˛ 1 1
1
1
@j PQk1
Qk1 CC Qˇj .A2
. Q0k
1 C10C
R˛
. kQ0k1 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 Q0k C10C @ˇ ŒQ0k C10C R˛ 1 1
1
1
@j PQk1 Qk1 CC Qˇj .A2
. kQ0k1 C10C R˛ . 12
.2 .2
k1 2
"/k1
1
2. 2
1
1
@j PQk1 Qk1 CC Qˇj .A2
kQ0k1 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 Q
ˇ 1 e
C P0 Q0 @ Œ 1 @j P 0 QC Qˇj . 2 ; 3 / N Œ0 e 0 D PŒ where we write P into XP 00; 1 ";2 implies
1
1
C;C
1 1 kL1 t Lx
. 2k1 2
(5.13)
for simplicity. By Lemma 4.17, placing (5.12)
e 0 Q
.k
(5.12)
k2 2
2
3 Y
k
2;
k2 2
k
3 / L2 L2 t x 2 kS Œk2 k 3 kS Œk3
i kS Œki
i D1
whereas (5.13) .
X
P0 Qm @ˇ ŒQm
1
1
C
e 0 QQ m Qˇj . @j P
2;
3 / N Œ0
(5.14)
mC
C
X
P0 Q0 @ˇ ŒQm
C
1
1
e 0 Qm Qˇj . @j P
2;
3 / N Œ0 :
mC
(5.15) Lemma 4.19 yields the following bound on (5.14): X
P0 Qm @ˇ ŒQm
1
C
1
e 0 QQ m Qˇj . @j P
2;
3 / N Œ0
mC
.
X
P0 Qm @ˇ ŒQm
C
1
1
@j QQ m Qˇj .
2;
mC
.
X
2
m"
k
e Q
1 kP 0 Qm Q 1 kL1 ˇj . 2 ; t Lx
mC k1
.2 2
"k2
hk2
2
k1 i
3 Y i D1
k
i kS Œki
3 /kL2 L2x t
3 / XP 0; 0
1 ";2
127
5.1 Reduction to the hyperbolic case
which is admissible. The bound on (5.15) proceeds similarly: X
P0 Q0 @ˇ ŒQ>m
(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 Q010C @ˇ ŒQ010C 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 Q010C @ˇ ŒQ010C R˛
1
1
@j
e 0 QC Qˇj .A2 P
. Q010C 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
. kQ0k1 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 Œ 1 @j QkC 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
kQkC Pk Qˇj .
k
1 kL1 L2x
2
k
1 kL1 L2x
kQkC 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/
k
D
X
X
P0 Q0 @ˇ ŒQ
1
1
C
@j Ql Pk Qˇj .
2;
3/
kl>kCC
C
X
X
P0 Q0 @ˇ ŒQl
1
C
1
@j Ql Pk Qˇj .
2;
3/
kl>kCC
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 ŒQ
1
C
1
@j Ql Pk Qˇj .
2;
3 / N Œ0
kl>kCC
X
X
P0 QlCO.1/0 @ˇ ŒQ
1
1
C
@j
kl>kCC
Ql Pk Qˇj . .
X
X
2
. 21 "/l
kQ
C
1 kL1 L2x 2 t
"l
2; 1
3 /
1;
1 2 ;1
@j
kl>kCC
Ql Pk Qˇj .
XP 0
2;
3 / L2 L1 : t x
(5.23) Recalling the assumptions on the frequencies in Case 5, and using Bernstein’s inequality as well as Lemma 4.19, we can further bound the preceding by
.
X
X
2
. 21 "/l
k
22 2
"k2
k
1 kS Œk1
Y
kPkj
j kS Œkj
j D2;3
k<minfO.1/;k2 g O.1/>l>kCC
.2
"k2 _0
3 Y i D1
k
i kS Œki :
(5.24)
134
5 Trilinear estimates
The second term of (5.22) is handled similarly, using
X
X
P0 Q0 @ˇ ŒQl
C
1
1
X
P0 Q0 @ˇ ŒQl
C
1
1
@j Ql Pk Qˇj .
2;
3 / N Œ0
2;
3 / L1 HP t
kl>kCC
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 Œ 1 @j QkC 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 Q<` C 1 1 @j Pk2 CO.1/ Q` Qˇj .
2;
3/
`>k2 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 ŒQ<`
C
1
1
@j Pk2 CO.1/ Q` Qˇj .
2;
3 / N Œ0
`>k2 CC
X
P0 Q`
C 0 @
ˇ
ŒQ<`
X
.
2
. 12 "/`
kQ<`
C
O.1/>`>k2 CC
1
1
@j Pk2 CO.1/ Q`
Qˇj . 2 ; 3 / N Œ0
1 "` @j Pk2 CO.1/ Q` 1 kL1 L2x 2 t
Qˇj . 2 ; 3 / L2 L1
O.1/>`>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 . k kS 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
1
1
@j I Qˇj .I
2; I
3/
D P0 QC @ˇ Q<0
1
1
@j QC Qˇj .I
: (5.51)
2; I
3/
One can also limit the modulations of 2 ; 3 further. Indeed, by (4.42) of Lemma 4.13 and Corollary 4.14,
ˇ 1 Q P Q @ Q @ I P Q .Q I ; I /
0 C <0 1 j 2 3 k1 ˇj "k2 N Œ0
. 2 k1 k 1 rx;t jrj 1 I 3 k 0; 1 ;1 Q"k2 rx;t jrj XP k
.2
"k2
hk2
k1 i
3 Y
k
2
1
I
2
3
0;
XP k
2
1 2 ;1
(5.52)
i kS Œki
i D1
which is admissible. Note that we replaced 1 @j PQk1 by 2 k1 as explained in the paragraph preceding this lemma. Thus, assume that 1 D QC 1 , j D Q"kj j for j D 2; 3, apply the identity (5.45), and estimate the six terms on the right-hand side of (5.45) in the order in which they appear. First, by the Strichartz component (2.14),
P0 I 1 jrj 1 2 jrj 1 3
N Œ0
. P0 I 1 jrj 1 2 jrj 1 3 0; 1 ;1 XP 0 2
. 1 jrj 1 2 jrj 1 3 2 2 L t Lx
. k kL1 L2x 2 t
k2
k
2 2 kL4 L1 x t
k3
k
3 kL4 L1 x t
151
5.2 Trilinear estimates for hyperbolic S -waves 3 Y
k2 2
.2
k
i kS Œki :
i D1
Second, by (4.40) of Lemma 4.13 and Lemma 4.11,
P0 I .
1 1 jrj /jrj 1 3 2 N Œ0
Q
. hk3 i Pk3 Q"k3 1 jrj 1 3
0;
XP k
1 2 ;1
kjrj
1
2 kS Œk2
3
. hk3 i PQk3 Q"k3
3
1
1 jrj
0; 1 2 ;1
XP k
k
2 kS Œk2
3
. 2k1
k3
1
2 4 ."k3
k1 /
3 Y
hk3 i
k
i kS Œki :
i D1
Here we use that the restriction on the modulation of the output and the modulation of 2 allow us to restrict the modulation of . 1 jrj 1 3 / to size < 2"k3 . Third, by (4.42) and Lemma 4.11,
P0 I
1 .jrj
1
1
2 /jrj
. PQk3 Q"k3
1 jrj
3 1
N Œ0
3 0; 1 ;1 2 XP k
X
2
1 4 j ^0
3
Qj jrj
1
2
j "k2
. 2k1
k3
1
2 4 .3"k3
k1 /
hk3 i2
3 Y
k
0;
XP k
1 ;1 2
2
i kS Œki :
i D1
Fourth, again by (4.42) and Lemma 4.11,
P0 I .
1 1 / @ .R jrj / 1 j j 2 3 N Œ0
X `
. 2 k1 2 4 kQ` 1 k 0; 1 ;1 PQk1 QC Rj XP k
`C
.2
k1
k2 4
2
k2 4
3 Y i D1
k
2
1
i kS Œki :
2 jrj
1
3
1 ;1 0; 2
XP k
3
152
5 Trilinear estimates
Fifth, with ` D k1 k2 ,
P0 I 1 1 @j .Rj 2 jrj 1 3 / N Œ0
1 1 Q . 1 @j Pk1 Rj 2 jrj 3 2 2 L t Lx X k1
Pc Rj 2 jrj . k 1 kL1 L2x 2 t
1
P
c
3 L2 L1 t x
c2Dk2 ;`
.k
1 kL1 L2x 2
X
k1 k3
2 2 kL4 L1
kPc Rj
t
x
t
c2Dk2 ;`
X
21
kP
c
2 3 kL4 L1
.k
1 kL1 L2x 2
k1 k2 .1 2"/`
2
t
2
3k2 2
k
x
t
c2Dk2 ;`
12
2 kS Œk2 k 3 kS Œk3
which is admissible for small " > 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
Q
1 1 . hk1 ik 1 kS Œk1 Pk1 QC @j Rj 2 jrj 3 0; 1 ;1 2 XP k
1
. hk1 ik 1 kS Œk1 PQk1 QC Rj . hk1 ik
1 kS Œk1 2
5.k1 k2 / 4
2
k2 4
2 jrj
k
1
3
1 ;1 0; 2
XP k
1
2 kS Œk2 k 3 kS Œk3
which concludes Case 1. Case 2: 0 k1 D k3 C O.1/; k2 k3 C. By Lemma 4.23,
ˇ 1 P I @ Q @ I Q .I ; I /
0 0 1 j 2 3 ˇj 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
L t Lx
1
. 2. 2
"/k2
2
.1 "/k1
3 Y i D1
k
i kS Œki :
153
5.2 Trilinear estimates for hyperbolic S -waves
So it suffices to consider P0 I @ˇ Q<0
1
1
@j I Qˇj .I
2; I
D P0 QC @ˇ Q<0
3/
1
1
@j QC Qˇj .I
2; I
: (5.53)
3/
One can also limit the modulation of 3 further. Indeed, by (4.42) of Lemma 4.13 and Corollary 4.14,
ˇ 1 Q
P0 QC @ Q<0 1 @j I Pk1 Qˇj .I 2 ; Q"k3 I 3 / N Œ0
k1 Q 1 1 .2 Pk3 1 rx;t jrj I 2 0; 1 ;1 Q"k3 rx;t jrj I 3 0; 1 ;1 XP k 2 XP k 2 (5.54) 3 3 k2 k1
.2
hk1
k2 i2
3 Y
"k1
k
i kS Œki
i D1
which is admissible. As explained in Case 1, we replaced 1 @j PQk1 by 2 k1 . If 0 k2 , then we can similarly reduce the modulation of the small frequency term, cf. (5.52):
P0 QC @ˇ Q<0 1 1 @j I PQk1 Qˇj .Q"k2 I 2 ; I 3 / N Œ0
. 2 k1 PQk2 .2
k2
k1 4
2
1 1 rx;t jrj I 3
"k2
3 Y
k
0; 1 ;1 Q"k2 rx;t jrj
XP k
2
1
I
2
2
0;
XP k
1 2 ;1
2
i kS Œki
i D1
As a final preparation, we limit the modulation of the output in case k2 0. In fact, by Lemma 4.23,
ˇ 1
P0 Q.1 3"/k2 C @ 1 @j I Qˇj .I 2 ; I 3 / N Œ0
. P0 Q.1 3"/k2 C @ˇ 1 1 @j I Qˇj .I 2 ; I 3 / 0; 1 ;1 2 XP k
1 2 .1
.2 1
. 2 2 "k2 2
3"/k2
k
1 kL1 L2x
.1 "/k1
t
3 Y i D1
k
2
k1
kPQk1 I Qˇj .I
i kS Œki :
2; I
2
3 /kL2 L2x t
154
5 Trilinear estimates
Thus, for the remainder of this case we assume that 1 D QC 1 , 2 D Q"k2 ^k2 2 , and 3 D Q"k3 3 . Moreover, the output is restricted by Q.1 3"/k2 ^C . We now estimate the six terms on the right-hand side of (5.45). First, by the Strichartz component (2.14),
P0 Q.1 3"/k2 ^C 1 jrj 1 2 jrj
. P0 Q.1 3"/k2 ^C 1 jrj 1
. 2 2 .1 1
. 2 2 .1 1
. 2 2 .1
3"/k2 ^0
1 jrj
3"/k2 ^0
k
1 kL1 L2x 2
3"/k2 ^0
1
k1 4
2
2
1
k2
3 Y
k2 4
k
k
N Œ0 1
2 jrj 1
2 jrj
t
3
1
3
0;
XP 0
1 2 ;1
3 L2 L2 t x k3
2 2 kL4 L1 x t
k
3 kL4 L1 x t
i kS Œki
i D1
which is admissible. Second, by (4.40) of Lemma 4.13 and Lemma 4.11,
P0 Q.1 .2
k2 ^0
3"/k2 ^C
.
hk1 i PQk2 _0 Q.1
.2
k2 _0
.2
k2 _0
1
1 jrj
k2 _0 k1 4
1
3"/k2 ^"k2
hk1 i PQk2 _0 Q.1 hk1 i2
3 /jrj
3"/k2 ^"k2
1
2 4 Œ.1
2
N Œ0 1
1 jrj 1 jrj
3"/k2 ^"k2 k1
3
1
3 Y
3
k
1 2 ;1 2 _0
0;
XP k
0; 1 2 ;1 2 _0
XP k
kjrj
k
1
2 kS Œk2
2 kS Œk2
i kS Œki
i D1
which is again admissible. Third, by (4.42) and Lemma 4.11,
P0 Q.1 3"/k2 ^C 1 .jrj 1
. PQk2 _0 Q.1 3"/k2 ^"k2
2 /jrj
1
1 jrj
1
3 N Œ0
3 0; 1 ;1 hk2 _ 0i 2 XP k
jrj . hk2 _ 0i2
k2 _0 k1 4
2
1 4
.1 3"/k2 ^"k2 k1
2
2 _0
1
Q"k2 ^k2
"k2 ^k2 2
3 Y i D1
k
2
0;
XP k
2
i kS Œki :
1 ;1 2
155
5.2 Trilinear estimates for hyperbolic S -waves
Fourth, again by (4.42) and Lemma 4.11,
P0 I . .2
1 / k1
1
X
2 jrj
@j .Rj ` 4
2 kQ`
1k
`C
. 2k2
k1
2
3 Y
k1 4
k
3/
1
N Œ0
PQk QC ŒRj 1
0; 1 ;1 XP k 2 1
1
2 jrj
3
1 ;1 0; 2
XP k
3
i kS Œki :
i D1
Fifth,
P0 I 1 1 @j .Rj 2 jrj 1 3 / N Œ0
1 1
Q . 1 @j Pk1 Rj 2 jrj 3 L2 L2 x
t k1 1
. k 1 kL1 L2x 2 Rj 2 jrj 3 L2 L1 t
2k1
.2 .2
3k2 4
t
k
2
1 kL1 L2x kRj t
5k1 4
3 Y
k
x
k 3 kL4 L1 2 kL4 L1 x x t
t
i kS Œki :
i D1
The sixth and final term is estimated by means of (4.40) and Lemma 4.11:
P0 I
1
1
2 jrj
@j .Rj
/ 3
1
N Œ0 1
. hk1 ik 1 kS Œk1 PQk1 Q"k3
@j Rj
1
2 jrj
3
0;
XP k
1 2 ;1
1
. hk1 ik
1 kS Œk1
. hk1 i2k2
k1
2
PQk Q"k Rj 1 3
1 4 .1
"/k1
3 Y
k
2 jrj
1
3
1 ;1 0; 2
XP k
1
i kS Œki
i D1
which concludes Case 2. Case 3: 0 k1 D k2 C O.1/; k3 k2
C. This is symmetric to the preceding.
Case 4: O.1/ k2 D k3 C O.1/; k1 C. This case proceeds similarly to Case 1. Following (5.52), we begin by limiting the modulations of 2 ; 3 to
156
5 Trilinear estimates
2"k2 . Indeed, by (4.42) of Lemma 4.13 and Corollary 4.14,
P0 QC @ˇ I 1 1 @j I PQ0 Qˇj .Q"k2 I 2 ; I 3 /
N Œ0
. 2 k1 . 22k1
k2
k
k2
1
1 rx;t jrj
1 2 "k2
2
3k
I
hk2
k1 i
1 ;1 0; 2
XP k
3 Y
kQ"k2 rx;t jrj
1
2 k 0; XP k
I
3
1 2 ;1
2
k
i kS Œki
i D1
which is admissible. Next, we limit the modulation of the output: by Lemma 4.17,
ˇ 1
P0 Qk1 C @ 1 @j I Qˇj .I 2 ; I 3 / N Œ0
. P0 Qk1 C @ˇ 1 1 @j I Qˇj .I 2 ; I 3 / 0; 1 ;1 2 XP k
.2 .2
k1 2
k1
Q 1 P k 1 kL1 0 I Qˇj .I L x t
k2 2
3 Y
k
2
2 ; I 3 /
L2t L2x
i kS Œki :
i D1
We now again estimate the six terms on the right-hand side of (5.45). First, by the Strichartz component (2.14),
P0 Qk1 1 jrj 1 2 jrj 1 3 N Œ0
1 1 . P0 Qk1 1 jrj 2 jrj 3 0; 1 ;1 2 XP 0
.2 .2 .2
k1 2 k1 2
1
1 jrj
2 jrj
k kL1 L2x 2 t
k1
k2 2
3 Y
k
k2
1
k
3 L2 L2 t x
2 2 kL4 L1 x
k3
t
k
3 kL4 L1 x t
i kS Œki :
i D1
Second, by (4.40) of Lemma 4.13 and Lemma 4.11,
P0 I . 1 jrj 1 3 /jrj 1 2 N Œ0
Q
1 . hk3 i Pk3 Q"k3 1 jrj 3 / 0; XP k
3
1 ;1 2
jrj
1
2 S Œk2
157
5.2 Trilinear estimates for hyperbolic S -waves
. hk3 i PQk3 Q"k3 . 2k1
1
2 4 ."k3
k3
k1 /
hk3
3
1
1 jrj
k1 i
0; 1 2 ;1
XP k
3 Y
k
2 kS Œk2
3
k
i kS Œki :
i D1
Third, by (4.42) and Lemma 4.11,
P0 I 1 .jrj 1 2 /jrj 1 3 N Œ0
Q
1 . hk2 i Pk3 Q"k3 1 jrj 3
0; 1 ;1 XP k 2 3
. 2k1
k3
1
2 4 .3"k3
k1 /
hk2 i2 hk1 i
3 Y
k
jrj
1
2
0;
XP k
1 2 ;1
2
i kS Œki :
i D1
Fourth, again by (4.42) and Lemma 4.11,
P0 I . 1 / 1 @j .Rj 2 jrj 1 3 / N Œ0
X ` k1
4 . 2 kQ` 1 k 0; 1 ;1 PQ0 QC Rj `k1 CC k2 4
. 2 k1
XP k
2
2 jrj
1
2 jrj
3
1
3 Y
k
1 ;1 0; 2
XP k
1
i kS Œki :
i D1
Fifth, with ` D k2 ,
P0 Qk1 1
. 2k1 1
1
@j .Rj
3/
1
@j PQk1 Rj 2 jrj X
Pc Rj . 2k1 k 1 kL1 L2x t
1
N Œ0
3
L2t L2x
2 jrj
1
P
c
3 L2 L1 t x
c2Dk2 ;`
. 2k1 k
1 kL1 L2x 2
k3
X
kPc Rj
t
2 2 kL4 L1
c2Dk2 ;`
X c2Dk2 ;` k1 k2 .1 2"/`
.2
2
2
3k2 2
3 Y i D1
k
i kS Œki
t
12
x
kP
c
2 3 kL4 L1 t
x
12
3
158
5 Trilinear estimates
which is admissible for small " > 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
Q
3"/k2 @
D P0 Qk1 C.1
3"/k2
ˇ
2; I
3/
:
We now stimate the six terms on the right-hand side of (5.45). First, by the Strichartz component (2.14),
P0 Qk C.1 3"/k . 1 jrj 1 2 jrj 1 3 / 1 2 N Œ0
. P0 Qk1 C.1 3"/k2 1 jrj 1 2 jrj 1 3 0; 1 ;1 XP 0
1
. 2 2 Œ.1 1
. 2 2 Œ.1 1
. 2 2 Œ.1
3"/k2 Ck1 3"/k2 C3k1
3"/k2 C3k1
1 jrj
k 2
1
1
2 jrj
3
L2t L2x
2 1 kL1 L2x k 2 kL4 L1 t t x 3 Y
k2 4
k
2
k2 4
k
3 kL4 L1 x t
i kS Œki
i D1
which is sufficient. Second, by (4.40) of Lemma 4.13 and Lemma 4.11,
1 1 P Q . jrj /jrj
0 k1 C.1 3"/k2 1 3 2 N Œ0
. 2k2 PQ0 Q.1 3"/k2 1 jrj 1 3 0; 1 ;1 jrj 1 2 S Œk2 2 XP 0
Q . P0 Q.1 3"/k2 1 jrj 1 3 0; 1 ;1 k 2 kS Œk2 XP 0
k1
.2 2
.1 3"/k2 4
k1
^0
hk1 i
3 Y
k
2
i kS Œki
i D1
which is admissible. Third, by (4.42) and Lemma 4.11,
P0 Qk1 C.1 3"/k2 1 .jrj 1 I 2 /jrj 1 3 N Œ0
Q
1 . P0 Q.1 3"/k2 1 jrj 3 0; 1 ;1 jrj 2 XP 0
k1
.2 2
.1 3"/k2 4
k1
^0
hk1 i
3 Y i D1
k
i kS Œki :
1
I
2
0;
XP k
2
1 2 ;1
164
5 Trilinear estimates
Fourth, again by (4.42) and Lemma 4.11,
P0 Qk1 C.1 3"/k2 .Qk1 C.1 3"/k2 X ` k1 . 2 k1 2 4 kQ`
1
1 /
1 k 0; XP k
`k1 C.1 3"/k2
@j .Rj
2 jrj
1
3/
N Œ0
1 ;1 2
1
Q
P0 Qk1 CC Rj . 22k1 2 4 .1
1
3"/k2 k2
2 hk2 i
3 Y
k
2 jrj
1
3
0; 1 2 ;1
XP k
3
i kS Œki :
i D1
Fifth,
P0 Qk1 C.1 1
. 2 2 Œ.1 1
. 2 2 Œ.1 .2
1 2 Œ.1
1
. 2 2 Œ.1
1 1 @ .R jrj / 1 j j 2 3 N Œ0
3"/k2 Ck1
1 1 @j PQk1 Rj 2 jrj 1 3 2 2 L t Lx
3"/k2 Ck1 1
k 1 kL1 L2x Rj 2 jrj 3 L2 L1 3"/k2
t
3"/k2 Ck1
3"/k2 Ck1
k 2
t
1 kL1 L2x kRj
k 3 kL4 L1 2 kL4 L1 t x t x
t
3 Y
3k2 4
k
x
i kS Œki :
i D1
The sixth and final term is estimated by means of (4.40) and Lemma 4.11:
1 1 P Q Q @ .R jrj /
0 k1 C.1 3"/k2 k1 C.1 3"/k2 1 j j 2 3 N Œ0
. 2k1 k 1 kS Œk1 PQ0 Qk1 CC 1 @j Rj 2 jrj 1 3 0; 1 ;1 2 XP k
. 2k1 k 1 kS Œk1 PQk3 Qk1 Rj . 2k1 Ck2 hk2 i
3 Y
k
2 jrj
1
3
1
0; 1 2 ;1
XP k
3
i kS Œki
i D1
which concludes Case 8. Case 9: k2 D O.1/; max.k1 ; k3 /
C. Symmetric to Case 8.
Hence we are done with (5.47). Next, we turn to (5.48) which is similar; basically, one uses (5.44) instead of (5.45). First, one observes that any reductions
165
5.2 Trilinear estimates for hyperbolic S -waves
in modulation which preceded application of (5.45) to (5.47) can equally well be carried out for (5.48) since these bounds only use Lemmas 4.17 and 4.23. Second, observe that the last four terms of (5.44) reappear as the last four terms of (5.45) up to the order and the choice of signs, both of which are irrelevant. Consequently, one only needs to verify that the first two terms of (5.44) satisfy the desired bounds. Case 1: 0 k1 k2 C O.1/ D k3 C O.1/. In this case the second terms in (5.44) and (5.45) satisfy the same bounds, whence it will suffice to bound the first term in (5.44). However, by (4.42) and Lemma 4.11,
P0 I . 1 /jrj 1 2 jrj X ` 2 4 kQ` 1 k .
.2
N Œ0
0;
XP k
`C k1
k2 4
2
k2 4
3 Y
k
3
1
1 2 ;1
Q
Pk1 QC jrj
1
2 jrj
1
1
3
0; 1 2 ;1
XP k
3
i kS Œki
i D1
which is admissible. Case 2: 0 k1 D k3 C O.1/; k2 k3 C. Using the arguments from Case 2 above, we may assume that 1 D Q.1 3"/k2 ^0 k1 1 . In addition, it was shown there that it suffices to assume that 2 D Q"k2 ^k2 2 , 3 D Q"k2 3 . First,
P0 I Q.1 .2
3"/k2 ^0
.1 3"/k2 ^0 k1 4
k1
1
QC Œjrj
Q.1
1
3"/k2 ^0 k1
2 jrj
1
1
N Œ0
0;
XP k
.2
.1 3"/k2 ^0 k1 4
hk1
k2 i
k
1 ;1 2
1
QC jrj 3 Y
3
1
2 jrj
1
3
1 ;1 0; 2
XP k
1
i kS Œki
i D1
which is admissible. One may also restrict the modulation of the entire output by means of the projection Q.1 3"/k2 ^0 k1 . Applying Lemma 4.13 and
166
5 Trilinear estimates
Lemma 4.11 to the second expression in (5.44) yields
P0 Q.1 3"/k2 ^0 k1 Q.1 3"/k2 ^"k2 . 1 jrj 1 2 /jrj 1 3 N Œ0
.1 3"/k2 ^ .1 "/k2
4 .2
PQk1 Q.1 3"/k2 ^"k2 1 jrj 1 2 0; 1 ;1 XP k
.2
.2
.1 3"/k2 ^ .1 "/k2 4
.1 3"/k2 ^ .1 "/k2 4
Q.1 hk1
1 jrj
3"/k2 ^"k2
k2 i
3 Y
k
1
2
1
jrj 1 3 S Œk3
2 0; 1 ;1 k 3 kS Œk3 2 XP k
1
i kS Œki
i D1
which is admissible provided jk2 j > k1 for some > 0. When this condition is violated, we have to work a little harder. First, since we may choose > 0 arbitrarily small and the ensuing estimates will not be affected by our choice of , we may from now as well assume k2 D O.1/. With the modulation and frequency restrictions from above in place, and going back to the original (un-expanded) version of the term under consideration, write schematically rx;t Œ
1
1
@j I Qˇj .
2;
3 /
D
1 .r
1
3/ 2
where we suppress the action of convolution operators of bounded L1 -mass, as they do not affect our estimates. Then we have (for some ı1 > 0 small)
P0 I 1 Q .1 ı1 /k1 .r 1 3 / 2 N Œ0
.k .2
1
2 k 0; 1 ;1 r
ı 1 k1
XP k
2
1
Q
.1 ı1 /k1
1
3 Y
kPkj
3
0;
XP k
1 2 ;1
3
j kS Œkj :
j D1
In light of the modulation restrictions from earlier, we now reduce to estimating P0 I Q< k1 1 .r 1 Q< .1 ı1 /k1 3 / 2 : Decompose this expression into P0 I Q< k1 1 .r 1 Q< .1 ı1 /k1 3 / 2 D P0 I P< ı2 k1 ŒQ< k1 1 .r 1 Q< .1 ı1 /k1 3 / 2 C P0 I P ı2 k1 Q< ı3 k1 ŒQ< k1 1 .r 1 Q< .1 ı1 /k1 C P0 I P ı2 k1 Q ı3 k1 ŒQ< k1 1 .r 1 Q< .1 ı1 /k1
(5.61) 3 / 2
(5.62)
3 / 2
(5.63)
167
5.2 Trilinear estimates for hyperbolic S -waves
where we pick 0 < ı2 < ı3 1. For the first term on the right, suppressing the action of the convolution operator P< ı2 k1 and reverting to the original form, we have reduced to estimating the term P0 rx;t
1
1
@j I Qˇj . 2 ; X D ; 0 2C
3/
P0; 0 rx;t
ı2 k1 ;
1
1
@j I Qˇj .P0;
;
2;
3/
0
where the point is of course that we can localize the output as well as the smallfrequency input 2 to approximately the same small angular sector. If we then make the null-form expansion as at the beginning of Case 2 above, we reduce to estimating
X ; 0 2C
P0; 0 Q.1
3"/k2 ^0 k1
0 ı2 k1 ;
.Q.1
P0 Q.1
X
.
2C
3"/k2 ^"k2 .
1
1 jrj
1
P0;
2 /jrj
1
P0;
2 /jrj
1
ı4 k1
3 Y
kPkj
/ 3
N Œ0
3"/k2 ^0 k1
ı2 k1
Q.1 .2
1 jrj
3"/k2 ^"k2 .
2 3
12
N Œ0
j kS Œkj
j D1
where we have used (the proof of) Corollary 4.10, which concludes estimating the contribution of (5.61). As for that of (5.62), here we can write P0 I P
ı2 k1 Q< ı3 k1 ŒQ< k1
X
D
; 0 2C .1Cı2
D
ı C 3
1 .r
P0 I P 2
ı2 /k
X O.1/>`> ı2 k1 ; 0 2C
1
Q<
.1 ı1 /k1
1
Pk3 ; 0 Q<
P0 I P` Q< ` k1 2
3 / 2
ı2 k1 Q< ı3 k1 ŒPk1 ; Q< k1
.r
X k1 C`C
1
.r
1
1
.1 ı1 /k1
ı3 k1 ŒPk1 ; Q< k1
Pk3 ; 0 Q<
.1 ı1 /k1
3 / 2
1 3 / 2
168
5 Trilinear estimates
X
D
X
O.1/>`> ı2 k1 ; 0 2C
P0 I P`; 00 Q<
` ı3 k1 2
k1 C`C
ı3 k1 ŒPk1 ; Q< k1
; 0 1 .r
1
Pk3 ; 0 Q<
3 / 2
.1 ı1 /k1
where in the last sum 00 ranges over the O.1/ many caps in C
` ı3 k1 2
such that ei-
` ı3 k1 2
ther one of ˙ 00 is at distance . 2 from . As the operator P`; 00 Q< ı3 k1 is given by convolution with a kernel of bounded L1 -mass, we can then again suppress it and revert to estimating the expression X ; 0 2C
P0 rx;t Pk1 ;
` ı3 k1 2
k1 C`C
1
1
@j I Qˇj .
2 ; Pk3 ; 0
3/
; 0
where we have suppressed the implicit dependence on ` (coming from the suppressed action of P`; 00 Q< ı3 k1 ). Due to the preceding identity, it is easy to see that we may write X ; 0 2C
k1 C`C
` ı3 k1 2
P0 rx;t Pk1 ;
1
1
@j I Qˇj .
3/
; 0
X
D ; 0 2C
2 ; Pk3 ; 0
k1 C`C
X
` ı3 k1 2
; 0 Q 1;2 2C
P0;Q2 rx;t Pk1 ;
1
1
` ı3 k1 2
;Q 1 ˙CQ 2
@j I Qˇj .P0;Q1
2 ; Pk3 ; 0
:
3/
For fixed ; 0 , one can now again expand the null-form as at the beginning of Case 2, and as for (5.61), one then concludes that for fixed ; 0
X Q 1;2 2C
` ı3 k1 2
P0;Q2 rx;t
;Q 1 ˙CQ 2
Pk1 ; .2
ı5 k1
1
kPk1 ;
1
@j I Qˇj .P0;Q1
1 kS Œk1 kP0
2 ; Pk3 ; 0
2 kS Œk2 kPk3 ; 0
/ 3
N Œ0
3 kS Œk3
for some ı5 > 0 depending on ı3 ı2 . One can then perform the summation over ; 0 with the aid of Cauchy–Schwarz, and the summation over ` only costs
169
5.2 Trilinear estimates for hyperbolic S -waves
logarithmically. In conclusion, we obtain
X X
O.1/>`> ı2 k1 ; 0 2C
k1 C`C
P0 rx;t Pk1 ;
` ı3 k1 2
1
1
; 0
@j I Qˇj .
2 ; Pk3 ; 0
.2
3/
ı6 k1
N Œ0
3 Y
kPkj
j kS Œkj
j D1
for some small ı6 > 0, which concludes the contribution of (5.62). Finally, we consider the contribution of (5.63). Here we take advantage of Lemma 4.11, which gives
1 P Q ŒQ .r Q /
` ı3 k1 < k1 1 2 0; 1 ;1 < .1 ı1 /k1 3 2 XP l
.2
` k1 4
kQ<
k1
1 kS Œk1 kQ< .1 ı1 /k1
Then we use Lemma 4.13, which gives
P0 I PO.1/ ı2 k1 Q ı3 k1 ŒQ< k1 1 .r 1 Q< .1 ı1 /k1 3 /
X
.
P` Q ı3 k1 Q< k1 1 .r 1 Q< .1 ı1 /k1
3 kS Œk3 :
N Œ0
3 / 0;
2
k . 2.ı3
1 4 /k1
3 Y
kPkj
1 ;1 2
XP l
O.1/>`> ı2 k1
2 kS Œk2
j kS Œkj :
j D1
Case 3: 0 k1 D k2 C O.1/; k3 k2
C. This is symmetric to Case 2.
Case 4: O.1/ k2 D k3 C O.1/; k1 C. This is similar to Case 1. Indeed, the second terms in (5.44) and (5.45) satisfy the same bounds, whence it will suffice to bound the first term in (5.44). However, by (4.42) and Lemma 4.11,
P0 I .I 1 /jrj 1 2 jrj 1 3 N Œ0
X ` k1
. 2 4 kQ` 1 k 0; 1 ;1 PQ0 QC jrj 1 2 jrj 1 3 0; 1 ;1 2 2 `k1 CC
XP k
1
XP k
2
170
5 Trilinear estimates
. 2 k1
k2
3 Y
k
i kS Œki
i D1
which is admissible. Case 5: O.1/ D k1 ; k2 D k3 C O.1/. Here again it suffices to only consider the first term in (5.44). Moreover, (5.55) and (5.56) apply whence that first term is bounded by the Strichartz component (2.14):
X
1 1
P0 I Qk 1 Pk I Œjrj 2 jrj 3 N Œ0
kk2 ^0CC
X
.
P0 QkCC Qk
1 Pk I Œjrj
1
2 jrj
1
1
3
3
kk2 ^0CC
X
.
kQk
k 2 2 1 L Lx Pk I jrj
1
t
2 jrj
kk2 ^0CC
X
.
kk2 ^0CC
2k
k2 2
3 Y
k
i kS Œki . 2
i D1
k2 _0 2
3 Y
k
L1t L2x
L2t L1 x
i kS Œki :
i D1
Case 6: O.1/ D k1 k2 C O.1/ k3 C C. Here one basically starts from (5.57), which can be handled via (5.1). 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. As in Case 8 above, one first shows that one can place Qk1 C.1 3"/k2 in front of the entire output, as well as in front of 1 . So it suffices to consider P0 Qk1 C.1 D P0 Qk1 C.1
Q
3"/k2
2; I
3/
:
We now stimate the first two terms on the right-hand side of (5.44). First, by the Strichartz component (2.14),
P0 I Qk1 C.1 3"/k2 1 jrj 1 2 jrj 1 3 N Œ0
1 1 . P0 I Qk1 C.1 3"/k2 1 jrj jrj 2 3 1 2 L t Lx
.1 3"/k2 Ck1 1 1
2 1 .2 k 1 k 1 2 jrj 2 jrj 3 L t Lx
L t Lx
171
5.2 Trilinear estimates for hyperbolic S -waves
. 2.1 . 2.1
3"/k2 Ck1
3"/k2 Ck1
k 2
2 1 kL1 L2x k 2 kL4 L1 x t
k2 4
3 Y
t
k
k2 4
k
3 kL4 L1 x t
i kS Œki
i D1
which is sufficient. Second, by (4.40) of Lemma 4.13 and Lemma 4.11, and assuming first that k1 D k2 C O.1/,
P0 Qk1 C.1 3"/k2 .Qk1 C.1 3"/k2 1 jrj 1 2 /jrj 1 3 N Œ0 X
1 1 . /jrj
P0 Qk1 CC Pk .Qk1 C.1 3"/k2 1 jrj 2 3
N Œ0
kk1 CC
.
X
Pk Qk1 CC
1 jrj
1
2k Pk Qk1 CC
1 jrj
1
2
kk1 CC
.
X
2
0; XP k
kk1 CC
.
X
k
2 2
k k1 4
3 Y
k
1 2 ;1
0; 1 2 ;1
XP k
jrj
k
1
3 S Œk3
3 kS Œk3
i kS Œki
i D1
kk1 CC
which is admissible. If k2 < k1 C , then by the same lemmas,
P0 Qk1 C.1 3"/k2 .Qk1 C.1 3"/k2 1 jrj 1 2 /jrj 1 3 N Œ0
1 1 . P0 Q.1 3"/k2 CC PQk1 .Qk1 C.1 3"/k2 1 jrj 2 /jrj
. PQk1 Q.1 3"/k2 CC 1 jrj 1 2 0; 1 ;1 jrj 1 3 S Œk3 2 XP k
. 2k1 PQk1 Q.1 k1
.2 2
.1 3"/k2 4
k1
3"/k2 CC 3 Y
k
1 jrj
1
2
3
N Œ0
1 0; 1 2 ;1
XP k
k
3 kS Œk3
1
i kS Œki
i D1
which is again admissible. Finally, if k1 < k2 C , then arguing analogously yields
P0 Qk1 C.1 3"/k2 .Qk1 C.1 3"/k2 1 jrj 1 2 /jrj 1 3 N Œ0
. P0 Qk2 PQk2 .Qk1 C.1 3"/k2 1 jrj 1 2 /jrj 1 3 N Œ0
172
5 Trilinear estimates
. PQk2 Qk2
. 2k2 PQk2 Qk2 . 2k1
3 Y
k
2
1
1 jrj 1 jrj
1
2
0; 1 ;1 XP k 2 2
1 ;1 0; 2
XP k
1
jrj k
3 S Œk3
3 kS Œk3
2
i kS Œki
i D1
which concludes this case. Case 9: k2 D O.1/; max.k1 ; k3 / analysis of (5.48).
C. Symmetric to Case 8. This concludes the
Neither of the identities (5.44) or (5.45) applies to (5.49). Hence, (5.49) requires somewhat different arguments. Case 1: 0 k1 k2 C O.1/ D k3 C O.1/. As in (5.50) one sees that it suffices to consider 1 D Q0 1 . Then Q˛j D QC Q˛j and we split P0 I @ˇ IRˇ 1 1 @j I Q˛j .I 2 ; I X D P0 Q` C @ˇ Rˇ Q` C
3/
1
1
@j Q` Q˛j .I
2; I
3/
(5.64)
`C
X
C
P0 Q`1 @ˇ Rˇ Q`1
1
1
@j Q` Q˛j .I
2; I
3/
P0 Q<`2 @ˇ Rˇ Q`2
1
1
@j Q` Q˛j .I
2; I
3/
(5.65)
` C `1 C
X
C
: (5.66)
` C `2 C `
Decomposing (5.64) via Lemma 4.1 into caps of size 2 2 yields P0 Q` C @ˇ Rˇ Q` X D P0; Q`
C C@
1 ˇ
1
@j Q` Q˛j .I
Rˇ Q`
C Pk1 ; 0
2; I 1
3/ 1
@j Q` Q˛j .I
2; I
3/
0 2C `
2 `
where 0 denotes that these caps have distance about 2 2 . Hence we gain a factor of 2` from the null-form involving @ˇ and Rˇ . From (2.29) one now obtains X X
k(5.64)kN Œ0 .
P0; Q` C @ˇ `C
0 2C `
2
173
5.2 Trilinear estimates for hyperbolic S -waves
Rˇ Q` X
.
` 4
2` 2
2
C Pk1 ; 0
k1 X 2
1
1
@j Q` Q˛j .I
kPk1 ; Q`
C
2
Q` Q˛j . X
.
3`
242
k1 2
k
1 kS Œk1 2
k
i kS Œki :
k1 4C
2
k2 2
k
NFŒ
2 2k1 1 kS Π2
2C `
`C
21
2 / 3
2; I
21
2
2 ; 3 / L2 L2 t
x
2 kS Œk2 k 3 kS Œk3
`C k1
. 24
k2 2
3 Y i D1
Here we also used Lemma 2.7 as well as Lemma 4.17. The expressions in (5.65) `1
are decomposed into caps of size 2 2 but without separation. Therefore, with a gain of 2`1 from the outer null-form, X X
k(5.65)kN Œ0 .
P0; Q`1 @ˇ ` C `1 C
; 0 2C `1 2
Rˇ Q`1 Pk1 ; 0 1 1 @j Q` Q˛j .I X X `1
2 2
P0; Q`1 @ˇ
.
` C `1 C
2
P 0;
3/
2; I
3/
2
2
21
2; I
2 / 3 2
21
X0
1 2 ;1
; 0 2C `1 2
Rˇ Q`1 Pk1 ; 0 1 1 @j Q` Q˛j .I X X
P0; Q`1 @ˇ
.
` C `1 C
12
2; I
L t L2x
; 0 2C `1 2
Rˇ Q`1 Pk1 ; 0
1
1
@j Q` Q˛j .I
L t L1x
:
(5.67) To pass to (5.67) one invokes the improved Bernstein estimate of Lemma 2.1. Hence, this can be further bounded by .
X ` C `1 C
2`1
X 0 2C `1 2
kQ`1 Pk1 ; 0
2 1 kL1 L2 t
x
174
5 Trilinear estimates
21
2 Q` Q˛j .I 2 ; I 3 / L2 L2 t x 21 ` k1 2 2 k1 2 4C 1 kS Œk1 ; 0 2k1
2 X
.
2`1
X 0 2C
` C `1 C
kQ`1 Pk1 ; 0
`1 2
k2 2
2 k1 .2
k1 4
k2 2
3 Y
k
kQ` Q˛j .I
2; I
3 /kL2 L2x t
i kS Œki :
i D1
For (5.66) one proceeds similarly, performing a cap decomposition and placing the entire expression in L1t L2x . We skip the details. Case 2: 0 k1 D k3 C O.1/; k2 k3 C. This is essentially the same as the preceding with Lemma 4.23 replacing Lemma 4.17. Case 3: 0 k1 D k2 C O.1/; k3 k2
C. This is symmetric to the preceding.
Case 4: O.1/ k2 D k3 C O.1/; k1 C. This is very similar to Case 1. First, one checks that the entire output can be restricted by Qk1 . This implies that due to the I -operator in front of 1 , the decomposition (5.64)–(5.66) continues to hold but with ` k1 C C : P0 I @ˇ IRˇ 1 1 @j I Q˛j .I 2 ; I X D P0 Q` C @ˇ Rˇ Q`
3/ C
1
1
@j Q` Q˛j .I
2; I
3/
(5.68)
`k1 CC
C
X
P0 Q`1 @ˇ Rˇ Q`1
1
1
1
1
@j Q` Q˛j .I
2; I
3/
` C `1 k1 CC
C
X
ˇ
P0 Q<`2 @ Rˇ Q`2
(5.69) @j Q` Q˛j .I 2 ; I 3 / :
` C `2 k1 CC
(5.70) One can again decompose (5.68) into caps, but of size 2
X X
k(5.68)kN Œ0 .
P0; Q` C @ˇ `k1 CC
0 2C `
Rˇ Q`
` k1 2
. Therefore,
k1 2
C Pk1 ; 0
1
1
@j Q` PQ0 Q˛j .I
2; I
2
3/
NFŒ
21
175
5.2 Trilinear estimates for hyperbolic S -waves
X
.
2
3.` k1 / 4
k1 2
2
`k1 CC
X
kPk1 ; Q`
C
2
3 / L2 L2 t x
2
Q 1 kSŒ Q` P0 Q˛j . 2 ;
12
2C `
2
X
.
2
3.` k1 / 4
k1 2
2
k
1 kS Œk1 2
k2 2
k
2 kS Œk2 k 3 kS Œk3
`k1 CC
.2
k1 2
k2 2
3 Y
k
i kS Œki
i D1
which is admissible. Furthermore, k(5.69)kN Œ0 is bounded by
X
.
` C `1 k1 CC
; 0 2C `1
Rˇ Q`1 Pk1 ; 0 X `1 2 2
.
P0; Q`1 @ˇ
X 2
k1
1
1
@j Q` Q˛j .I 2 ; I
X
P0; Q`1 @ˇ
; 0 2C `1
` C `1 k1 CC
2
0 2C `1
` C `1 k1 CC
X0
1 2 ;1
12
k1
Rˇ Q`1 Pk1 ; 0 1 1 @j Q` Q˛j .I 2 ; I X X `1 2 2 k1 kQ`1 Pk1 ; 0
.
2 / 3 0; P
2
2 / 3 2
21
L t L2x
2 1 kL1 L1 x t
k1
12
2
2 ; I 3 / L2 L2 t x X `1 `1 k 1 222 4 kQ`1 Pk1 ; 0
Q` PQ0 Q˛j .I X
.
0 2C `1
` C `1 k1 CC
Q` PQ0 Q˛j .I .2
3k1 4C
k2 2
3 Y
k
2 1 kS Œk1 ; 0
21
`
2 4C
k2 2
k1
2
2 ; I 3 / L2 L2 t
x
i kS Œki :
i D1
Finally, (5.70) is similar to the previous estimate and we skip it. Case 5: O.1/ D k1 ; k2 D k3 C O.1/. We apply (5.55) and reduce the modulation
176 of
5 Trilinear estimates 1
via (5.56) to
1
D Qk
1.
Furthermore,
P0 @ˇ I Rˇ Qk 1 1 @j Pk I Q˛j . 2 ; 3 / D P0 I jrj 1 Qk 1 1 @j Pk I Q˛j . 2 ; C P0 I Rˇ Qk 1 1 @j @ˇ Pk I Q˛j . 2 ;
3/
(5.71)
3/ :
(5.72)
Lemmas 4.13 and 4.17 imply the following bound on (5.71):
X
P0 I jrj 1 Qk 1 1 @j Pk I Q˛j . 2 ;
3/
kk2 ^0CC
X
.
X
2
m k 4
jrj
1
Qm
1
kk2 ^0CC mk 1
X
.
2
.2
1 2 k2 _0
k
2;
1 2 ;1
0;
XP 0
3/
0; 1 2 ;1
XP k
i kS Œki
i D1
kk2 ^0CC 3 Y
3 Y
k k2 2
@j Pk I Q˛j .
N Œ0
k
i kS Œki
i D1
which is admissible. The second term (5.72) needs to be expanded as follows: 2Rˇ Qk 1 1 @j @ˇ Pk I Q˛j . 2 ; 3 / D Qk jrj 1 1 1 @j Pk I Q˛j . Qk jrj Qk jrj
1
1
2;
(5.73)
3/
1
1
@j Pk I Q˛j .
2;
3/
(5.74)
1
1
@j Pk I Q˛j .
2;
3 /:
(5.75)
We just dealt with the term (5.74). Since the modulation of the entire output is . 2k , one concludes that
X
(5.73) .
Qk jrj 1 1 1 @j Pk I Q˛j . 2 ; 3 / 0; 21 ;1 XP 0
kk2 ^0CC
.
X
k
22 k
1 kL1 L2x t
1
@j Pk I Q˛j .
2;
3 / L2 L1 t x
kk2 ^0CC
.
X kk2 ^0CC
2
k k2 2
3 Y i D1
k
i kS Œki . 2
1 2 k2 _0
3 Y i D1
k
i kS Œki
177
5.2 Trilinear estimates for hyperbolic S -waves
as well as, from Lemma 4.13, X
(5.75) .
Qk jrj
1
k
1
1
@j Pk I Q˛j .
2;
3 / N Œ0
kk2 ^0CC
X
.
1 kL1 L2x t
1
@j Pk I Q˛j .
2;
3/
X
.
2
3k k2 2
3 Y
k
i kS Œki
1 2 k2 _0
.2
i D1
kk2 ^0CC
3 Y
0;
XP 0
kk2 ^0CC
k
1 ;1 2
i kS Œki
i D1
which is sufficient. Case 6: O.1/ D k1 k2 C O.1/ k3 C C. As before, one reduces the modulation of 1 to 1 D Qk2 1 . Furthermore, P0 @ˇ I ŒRˇ Qk2 1 1 @j PQk2 I Q˛j . 2 ; 3 / D P0 I jrj 1 Qk2 1 1 @j PQk2 I Q˛j . C P0 I Rˇ Qk2 1 1 @j @ˇ PQk2 I Q˛j .
2;
3/
(5.76)
2; 3/ :
(5.77)
Lemmas 4.13 and 4.23 imply the following bound on (5.76):
P0 I jrj 1 Qk2 1 1 @j PQk2 I Q˛j . X m k2
2 4 jrj 1 Qm 1 0; . XP 0
mk2 1
. 2. 2
"/.k3 k2 /
3 Y
k
2; 1 2 ;1
3/
N Œ0
1
@j PQk2 I Q˛j .
2;
3/
1 ;1 0; 2
XP k
2
i kS Œki
i D1
which is admissible. The second term (5.77) needs to be expanded as follows: 1 @j @ˇ PQk2 I Q˛j . 2 ; 3 / D Qk2 jrj 1 1 1 @j PQk2 I Q˛j . 2 ; 3 / Qk2 jrj 1 1 1 @j PQk2 I Q˛j . 2 ; 3 / Qk2 jrj 1 1 1 @j PQk2 I Q˛j . 2 ; 3 /:
2Rˇ Qk2
1
(5.78) (5.79) (5.80)
We just dealt with the term (5.79). Since the modulation of the entire output
178
5 Trilinear estimates
is . 2k2 , one concludes that
(5.78) . Qk2 jrj .2
k2 2
k
1
1 kL1 L2x t
"/.k3 k2 /
. 2. 2
1
1
1 1
3 Y
k
@j PQk2 I Q˛j .
@j PQk2 I Q˛j .
i kS Œki .
i D1
3 Y
0;
XP 0
1 2 ;1
2;
k
/ 3
2;
3 / L2 L1 t x i kS Œki
i D1
as well as, from Lemma 4.13, 1
(5.80) . Qk jrj .k
1 kS Œk1 1
. 2. 2
1 1
"/.k3 k2 / k2
2
1
@j PQk2 I Q˛j .
@j PQk2 I Q˛j .
3 Y
k
2;
2;
3 / N Œ0
3/
1 2 ;1
0;
XP 0
i kS Œki
i D1
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 / duced to Qk1 :
P0 @ˇ IQk1 Rˇ Qk1 .2 .2
k1 2
k1 2
C. The modulation of the output can be re-
1
1
1 k 1 kL1 t Lx
1
1
2. 2
"/k3
3 Y
k
@j PQ0 I Q˛j .
2;
3/
N Œ0
@j PQ0 I Q˛j . 2 ; 3 / L2 L2 t
x
i kS Œki :
i D1
Similarly, the input
1
can be reduced to Qk1
1.
As in Case 6,
P0 @ˇ Qk1 ŒRˇ Qk1 1 1 @j PQ0 Qk1 Q˛j . 2 ; 3 / D P0 Qk1 jrj 1 Qk1 1 1 @j PQ0 Qk1 Q˛j . C P0 Qk1 Rˇ Qk1 1 1 @j @ˇ PQ0 Qk1 Q˛j .
2;
3/
2; 3/ :
(5.81) (5.82)
179
5.2 Trilinear estimates for hyperbolic S -waves
Lemmas 4.13 and 4.23 imply the following bound on (5.81):
P0 Qk1 jrj 1 Qk1 1 1 @j PQ0 I Q˛j . 2 ; 3 /
N Œ0
.
X
2
m k1 4
1
jrj
Qm
1
mk1
.2
k1 2
1
2. 2
"/k2
3 Y
k
0; 1 ;1 XP k 2 1
PQ0 Qk Q˛j . 1
2;
3/
1 ;1 0; 2
XP 0
i kS Œki
i D1
which is admissible. The second term (5.82) needs to be expanded as follows: 1 @j @ˇ PQ0 I Q˛j . 2 ; 3 / D Qk1 jrj 1 1 1 @j PQ0 I Q˛j . 2 ; 3 / Qk1 jrj 1 1 1 @j PQ0 I Q˛j . 2 ; 3 / Qk1 jrj 1 1 1 @j PQ0 I Q˛j . 2 ; 3 /:
2Rˇ Qk1
1
We just dealt with the term (5.84). Next,
(5.83) . Qk1 Qk1 jrj 1 1 .2 .2
k1 2
k1 2
jrj
1
1
"/k3
2. 2
1 L1 L1 x t 3 Y
k
1
1
@j PQ0 I Q˛j .
@j PQ0 I Q˛j .
2;
2;
3/
(5.83) (5.84) (5.85)
0;
XP 0
1 2 ;1
3 / L2 L2 t x
i kS Œki
i D1
as well as, from Lemma 4.13,
(5.85) . Qk1 jrj 1 1 1 @j PQ0 Qk1 Q˛j . 2 ;
. k 1 kS Œk1 1 @j PQ0 Qk1 Q˛j . 2 ; 3 / .2
. 21 "/k3
2
k1 2
3 Y
k
3 / N Œ0 0;
XP 0
1 2 ;1
i kS Œki
i D1
which concludes Case 8. Case 9: k2 D O.1/; max.k1 ; k3 /
C. Symmetric to Case 8.
180
5 Trilinear estimates
Remark 5.6. It follows from the high-low-low interaction case of the proof of Lemma 5.5 that for some > 0,
ˇ 1 P I P I @ @ P I Q .P ; P /
0 1 j k1 k ˇj k2 2 k3 3 N Œ0
k
.2
w.k1 ; k2 ; k3 /
3 Y
k
i kS Œki
(5.86)
i D1
provided k1 D O.1/, k k2 D k3 C O.1/ O.1/. In effect, for later use, we also mention the following lemma, which is proved using identical reasoning: Lemma 5.7. Assume k1 D O.1/. Then we have the bounds
P0 IRˇ
1 @ˇ P k
1
@j Q˛j . 2 ; 3 /
N Œ0 k
.2
w.k1 ; k2 ; k3 /
3 Y
kPki
i kS Œki ;
i D1
P0 IRˇ
1 @˛ P k
1
@j Qˇj .
2;
3/
N Œ0
. 2 k w.k1 ; k2 ; k3 /
3 Y
kPki
i kS Œki
i D1
for suitable > 0.
5.3
Improved trilinear estimates with angular alignment
We conclude this section on trilinear bounds with a technical result which we shall require in several instances, such as the blow-up criterion of the following section. By Corollary 5.4, one gains extra smallness outside of the parameter range (5.43); note that the latter describes precisely Case 5 in the proof of Lemmas 5.1 and 5.5 which is the high-low-low case of interactions. In fact, the exact same gain as in that corollary can also be obtained for the trilinear expressions of Lemma 5.5. Corollary 5.8. The nonlinearities of Lemma 5.5 satisfy the estimates of Corollary 5.4. I.e., given ı > 0 there exist L; L0 large so that the ı-gains in the sum
181
5.3 Improved trilinear estimates with angular alignment
P P over 0k1 ;k2 ;k3 as well as 00k1 ;k2 ;k3 with k k2 types of trilinear null-forms in Lemma 5.5.
L0 , are obtained for the three
Proof. As in the case of Corollary 5.4, this follows from the form of the weights w.k1 ; k2 ; k3 / as well as from the fact that an extra gain in Case 5 of Lemma 5.5 was obtained when k < k2 L0 . However, one cannot gain smallness in the high-low-low case without further assumptions. In this section we shall prove that angular alignment between the Fourier support of at least two of the inputs implies smallness in this case. We start with the contributions by I c Qˇj . In Corollary 5.4 we isolated one case where smallness P00 cannot be obtained without any further assumptions.0 It was given by the sum k1 ;k2 ;k3 2Z over the range (5.43) together with k2 L k k2 C O.1/. Recall that L and L0 are very large depending on ı. Throughout this section, i will be Schwartz functions satisfying X max kPk i k2S Œk K 2 i D1;2;3
k2Z
for some constant K. We shall use defined earlier.
P00
k1 ;k2 ;k3 2Z
repeatedly in the sense that it was
Lemma 5.9. Given any ı > 0 there exists m0 .ı/ large and negative such that X 2 ;3 2Cm0 dist.2 ;3 /2m0
k2 CO.1/ X
X00 k1 ;k2 ;k3 2Z
2 X
P0 r t;x Pk1
1
kDk2 L0
1
@j I c Pk Qˇj .Pk2 ;2
2 ; Pk3 ;3
j D1
ı K 2 sup max 2 k2Z i D1;2;3
as well as X 1 ;2 2Cm0 dist.1 ;2 /2m0
X00 k1 ;k2 ;k3 2Z
k2 CO.1/ X kDk2 L0
3/
0 jkj
N Œ0
kPk
i kS Œk
182
5 Trilinear estimates 2 X
P0 r t;x Pk1 ;1
1
1
2 ; Pk3
/ 3
2 ; Pk3 ;3
/ 3
@j I c Pk Qˇj .Pk2 ;2
N Œ0
j D1
C
k2 CO.1/ X
X00
X
k1 ;k2 ;k3 2Z
1 ;3 2Cm0 dist.1 ;3 /2m0
2 X
P0 r t;x Pk1 ;1
1
kDk2 L0
1
@j I c Pk Qˇj .Pk2
N Œ0
j D1
ı K 2 sup max 2
0 jkj
k2Z i D1;2;3
kPk
i kS Œk :
Proof. The proof simply consists in verifying that the argument in Case 5 of Lemma 5.1 allows for this extra gain. We first consider angular alignment between 1 and 2 . In this case, we will need to repeat the argument of Case 5, obtaining the gain from Bernstein’s inequality. First, restrict the output by Q0 and assume that 1 D Pk1 ;1 1 and 2 D Pk2 ;2 2 with fixed caps 1 ; 2 . In the end, one verifies that it is possible to sum over these caps. Then
P0 Q0 @ˇ .
k2 CO.1/ X kDk2
C
1
1
@j I c Qˇj .
P0 Q0
1
1
2;
3/
N Œ0
@j QkC Pk Qˇj .
2;
/ 3
L0
k2 CO.1/ X
X
2
P0 Qm Qm
"m
C
1
1
(5.87)
L2t L2x
@j QQ m Pk Qˇj .
2;
kDk2 L0 mC
3/
L2t L2x
(5.88) C
k2 CO.1/ X kDk2
L0
X
P0 Q0 Q>m
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
QkC 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<`
10
1 kL1 Lx
kQ`
10
1 kL1 L2x
t
/ 3
XP 0
1;
1 2 ;1
(5.90) D I 1 . Then,
CL1t HP
1
1 @j Q` Qˇj . 2 ; 3 / L2 L1 t
x
O.1/>`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 <`
P0 Q0 @ˇ I
1
1
@j Q` Qˇj .
2;
3/
N Œ0
: (5.91)
Here we have chosen C1 large enough depending on ı, while C is as in the definition of I c . Then using Lemma 4.19, we infer that the first two amongst the last
185
5.3 Improved trilinear estimates with angular alignment
three preceding terms are bounded by .ı
3 Y
kPkj
j kS Œkj
j D1
and summation over the angular sectors/frequencies is straightforward to give the bound of the lemma. On the other hand, for the last expression X kP0 Q0 @ˇ ŒI 1 1 @j Q` Qˇj . 2 ; 3 /kN Œ0 ; k2 CC <`
we use Lemma 4.17 to give the same bound. To conclude the case of angular alignment between 1 ; 2 , we sum over 1 ; 2 using Cauchy–Schwarz, Lemma 2.18, and Corollary 4.21. Finally, consider the case where 2 and 3 are aligned on the Fourier side. Using Lemmas 4.18 and 4.22 instead of Lemmas 4.17 and 4.19, respectively, one immediately verifies that the desired gain can indeed be obtained. The only exception here is the estimate (5.26). But this case is excluded here as it involves I Qˇj and not I c Qˇj . Next, we need to obtain an analogous statement in the hyperbolic regime of the inner null-form. As in Corollary 5.4, Lemma 5.5 implies the following result. Corollary 5.10. Let ı > 0 be small. Then X00
X
k1 ;k2 ;k3 2Z
kk2 L0 2
X
P0 @ˇ R˛ Pk1
@j IPk Qˇj .Pk2
2 ; Pk3
3/
N Œ0
@j IPk Qˇj .Pk2
2 ; Pk3
/ 3
N Œ0
2 ; Pk3
3/
N Œ0
1
1
1
1
1
1
j D1
C
X00
X
k1 ;k2 ;k3 2Z
kk2 L0
2
X
P0 @˛ Rˇ Pk1
j D1
C
X00
X
k1 ;k2 ;k3 2Z
kk2 L0
2
X
P0 @ˇ Rˇ Pk1
j D1
@j IPk Q˛j .Pk2
186
5 Trilinear estimates
ı K 2 sup max 2
0 jkj
k2Z i D1;2;3
kPk
i kS Œk
where L0 D L0 .L; ı/ is a large constant. Next, we need to obtain an improvement in the range (5.43) under the additional assumption of angular alignment. Lemma 5.11. For any ı > 0 there exists m0 .ı/, a large negative constant, such that k2 CO.1/ X
X00
X
k1 ;k2 ;k3 2Z
2 ;3 2Cm0 dist.2 ;3 /2m0
2
X
P0 @ˇ R˛ Pk1
kDk2 L0
1
1
@j IPk Qˇj .Pk2 ;2
2 ; Pk3 ;3
3/
j D1
ı K 2 sup max 2
0 jkj
k2Z i D1;2;3
N Œ0
kPk
i kS Œk
as well as k2 CO.1/ X
X00
X 2 ;3 2Cm0 dist.2 ;3 /2m0
k1 ;k2 ;k3 2Z
kDk2 L0
2
X
P0 @˛ Rˇ Pk1
1
1
@j IPk Qˇj .Pk2 ;2
N Œ0
N Œ0
2 ; Pk3 ;3
3/
2 ; Pk3 ;3
3/
j D1
C
X 2 ;3 2Cm0 dist.2 ;3 /2m0
X00 k1 ;k2 ;k3 2Z
2
X
P0 @ˇ Rˇ Pk1
k2 CO.1/ X kDk2 L0
1
1
@j IPk Qˇj .Pk2 ;2
j D1
ı K 2 sup max 2
0 jkj
k2Z i D1;2;3
for any ˛ D 0; 1; 2. An analogous statement holds in case similarly aligned.
1;
2
kPk or
i kS Œk 1;
3
are
187
5.3 Improved trilinear estimates with angular alignment
Proof. We begin with the first trilinear form, and also assume alignment between 2 and 3 . We first reduce ourselves to the purely hyperbolic case, i.e., when all inputs are restricted by the operator I , as well as the entire output. Without further mention, implicit constants are allowed to depend on L; L0 . In particular, we assume that k; k1 ; k2 ; k3 are fixed in the range we are summing over. In the notation of Lemma 5.1, if A0 D I c , then A1 D I c and by Lemma 4.18,
P0 I c @ˇ I c r t;x
X
1
1
@j I Qˇj .Pk2 ;2
2 ; Pk3 ;3
3/
N Œ0
2 ;3 2Cm0 dist.2 ;3 /2m0
X
.
X
2
P0 Qm QQ m r t;x
"m
1
1
2 ;3 2Cm0 m0 dist.2 ;3 /2m0
Qˇj .Pk2 ;2 X
.
X
2.1
"/m
kQQ m
/ 3
2 ; Pk3 ;3
1 kL2 L2x 2
@j Pk I
L2t L2x
k
t
2 ;3 2Cm0 m0 dist.2 ;3 /2m0
Pk I Qˇj .Pk ; 2 ; Pk ; 3 / 1 1 2 2 3 3 L t Lx
k k 1 kS Œk1 2 2 IPk Qˇj .Pk2 ;2 2 ; Pk3 ;3 3 / L2 L2
X
.
x
t
2 ;3 2Cm0 dist.2 ;3 /2m0 k
. ı2 2 k ık
1 kS Œk1 2
k2 2
1 kS Œk1 kPk2
kPk2
2 kS Œk2 kPk3
2 kS Œk2 kPk3
3 kS Œk3
3 kS Œk3 :
Summing over k1 D O.1/, k2 D k3 C O.1/ yields the desired gain. 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 I
. I
1
1
@j I Qˇj .I c Pk2 ;2
2; I
c
Pk3 ;3
3/
N Œ0
1
@j IPk Qˇj .I c Pk2 ;2 2 ; I c Pk3 ;3 3 / L1 L2 t x
k2 c c . k 1 kL1 L2x 2 IPk Qˇj .I Pk2 ;2 2 ; I Pk3 ;3 3 / L1 L1 t t x X
k2 . k 1 kL1 L2x 2 Qˇj .Qm Pk2 ;2 2 ; QQ m Pk3 ;3 3 / L1 L1 : 1
t
t
x
mk2 CC
In the last inequality we use that we may assume k D k2 C O.1/. Splitting the
188
5 Trilinear estimates
modulations of the last two inputs dyadically yields X . k 1 kL1 L2x 2m 2k2 kPk2 ;2 Qm 2 kL2 L1 kPk3 ;3 QQ m x t
t
3 kL2 L1 x t
mk2 CC
. 2m0 Ck2 k
X
1 kL1 L2x t
2m
k2
2
2.1 "/m .1 2"/k2
2
mk2 CC
kPk2 ;2 Qm . 2m0 k
2k
1 kS Œk1 kPk2 ;2
XP k
1 C";1 ";1 2 2
2k
XP k
kPk3 ;3 QQ m
1 C";1 ";2 2 2
kPk3 ;3
3k XP 3k XP k
1 2 C";1 ";1 k3
1 2 C";1 ";2 3
:
Summing over the caps 2 ; 3 and k1 D O.1/, k2 D k3 C O.1/ yields the desired gain. We may therefore assume that A0 D A1 D A2 D A3 D I , which reduces us to the trilinear null-form expansion (5.45) restricted to Case 5 of Lemma 5.5. Beginning with the first of the trilinear nonlinearities and for the case of aligned 2 ; 3 , we now modify the analysis of Case 5 from that lemma. For ease of notation we will fix caps 2 ; 3 and drop the projections Pki ;i . In the end, an application of the Cauchy–Schwarz inequality will allow for summation over the caps. We first limit the modulation of 1 :
P0 @ˇ Q>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 <
i
/
4
N Œ0
10
i D0
h
rx;t Pk0
0r
1
Pr1 R0 Pk1 1 r
. 2ık0 2ıŒminfrj ;kj g
1
2r
Pr2 Pk2
maxfrj ;kj g
4 Y
1
Pr3 Qj k .Pk3
kPki
i kS Œki ;
3 ; Pk4
i
4 //
N Œ0
r1 2 Π10; 10
i D0
h
rx;t P0 Pk0 0 r 1 Pr1 R0 Pk1 1 r .2
1
Pr2 Pk2
ık0 ıŒminfrj ;kj g maxfrj ;kj g
2
2r 4 Y i D0
1
Pr3 Qj k .Pk3
kPki
i kS Œki ;
3 ; Pk4
i
4/
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 PΠ5;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 PΠ5;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 Q
100
1r
1
Pr2
l>10
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
100
1 kL2
t;x
. 2.l
k1 /
2
k1 2
kPk1 kS Œk1
199
6 Quintilinear and higher nonlinearities
where > 0 is as in the definition of S Œk, which implies that kR0 Pk1 Q
100
From this we infer that
h
rx;t PΠ5;5 Ql P0 0 r r .2
l
1
Pr2 Pk2 Q>l
1
1 kL1 L2x t
Pr1 R0 Pk1 Q
10
k 2 1
r 0 L Lx
kP0
1
t
Pk2 Q>l
10
. 2.l
2r
1
1
l
2
100
Pr1 R0 Pk1 Q
Pr3 Qj k .Pk3
. 2 2 2minfk1 minfr1;2 g;0g 2
r 1 Pr2 Pk2 Q>l
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
1
i
3 ; Pk4
Pr3 Qj k .Pk3
L2t L1 x 100
1 kL1 L2x t
3 ; Pk4
4/
1
L2t HP x2
:
Substituting the bounds from before, this is bounded by .2
"l
2.l
r1
2 2 2minfk1 k1 /
l
2
k1 2
minfr1;2 g;0g
2
minfminfr1;2 g k1 ;0g 2
kPk1 kS Œk1 2.1
/.k2 l/ Œminfr2;3 ;k2;3;4 g maxfr2;3 ;k2;3;4 g
2
4 Y
kPkj
j kS Œkj :
j D2
This is equivalent to an estimate of the form claimed in the lemma, with an extra gain 2 l which allows us to sum over l > 10. (i2): maxfk1 ; : : : ; k4 g l, R0 Pk1 1 D R0 Pk1 QŒl 100:lC100 1 . The estimate here is similar except that square summation over l is made possible since we have 1 C;1 ;2 Pk1 Q>k1 1 2 XP k12 : (i3): maxfk1 ; : : : ; k4 g l, R0 Pk1 1 D R0 Pk1 Q lC100 1 . This is again similar. Fixing the modulation of R0 Pk1 Q lC100 1 to size 2l1 , l1 > l C 100, there is at least one other input which has modulation at least comparable to 2l1 . Then one proceeds as in case (i1). (i4): maxfk1 ; : : : ; k4 g > l C O.1/, R0 Pk1
1
D R0 Pk1 Q<maxfk1;2;3;4 g
1.
200
6 Quintilinear and higher nonlinearities
Here we obtain a gain in minfr1;2;3 ; k1;2;3;4 g maxfr1;2;3 ; k1;2;3;4 g, which suffices to offset the loss due to the possibly large modulation of R0 Pk1 Q<maxfk1;2;3;4 g 1 . Specifically, write h rx;t Ql P0
1
0r
1
1
Pr2 Pk2 2 r 1 Pr3 Qj k .Pk3 1 Pr1 R0 Q
r h D rx;t Ql P0
Pr1 R0 Q<maxfk1;2;3;4 g Pk1
3 ; Pk4
i
4/
i Pr2 Pk2 2 r 1 Pr3 Qj k .Pk3 3 ; Pk4 4 / h C rx;t Ql P0 0 r 1 Pr1 R0 QŒk1 ;maxfk1;2;3;4 g Pk1 1 1
r
r
1
2r
Pr2 Pk2
1
Pr3 Qj k .Pk3
i
3 ; Pk4
4/
:
Here we use the inequalities
r
1
Pr2 Pk2
1
2r
Pr3 Qj k .Pk3
.2
3 ; Pk4
1 2 Œminfr2;3 ;k1;2;3;4 g
4/
1
L2t HP x2
maxfr2;3 ;k1;2;3;4 g
4 Y
kPki
i kS Œki ;
kPki
i kS Œki :
i D1
r
1
Pr2 .Pk2
1
2r
Pr3 Qj k Pk3
3 ; Pk4
. 2minfr2;3 ;k1;2;3;4 g
4/
2 L1 t Lx
maxfr2;3 ;k1;2;3;4 g
4 Y i D1
Then we can estimate
h
rx;t Ql P0 r .2
l
1
kP0
0r
1
Pr1 R0 QŒk1 ;maxfk1;2;3;4 g Pk1
Pr2 Pk2 0 kL1 L2x t
r
1
i Pr3 Qj k .Pk3 3 ; Pk4 4 / k XP 0
1 r Pr1 R0 QŒk1 ;maxfk1;2;3;4 g Pk1 2r
1
1
Pr2 Pk2
2r
1
Pr3 Qj k .Pk3
1 2 C; 1 ;2
1
3 ; Pk4
4/
L2t L1 x
:
To conclude the contribution of this term, one then checks, using standard
201
6 Quintilinear and higher nonlinearities
Littlewood–Paley trichotomy, that
r
1
Pr1 R0 QŒk1 ;maxfk1;2;3;4 g Pk1 Pk2
.2
minfr1;2 ;k1 g 2
2
1
1
2r
minfr1;2 ;k1 g maxfr1;2 ;k1 g 2
r
1
1r
Pr2 Pk2
Pr2
Pr3 Qj k .Pk3
3 ; Pk4
4/
kR0 QŒk1 ;maxfk1;2;3;4 g Pk1 1
2r
Pr3 Qj k .Pk3
3 ; Pk4
L2t L1 x
1 1k L2t HP x2
4/
2 L1 t Lx
:
Combining with the bound from above, and furthermore assuming the in the definition of k:kS to be small enough, we conclude that for suitable ı > 0 we have
h
rx;t Ql P0 r
1
1
0r
Pr1 R0 QŒk1 ;maxfk1;2;3;4 g Pk1 2r
Pr2 Pk2
1
Pr3 Qj k .Pk3
3 ; Pk4
1
i
/
4
XP 0
l ıŒminj ¤0 frj ;kj g maxj ¤0 frj ;kj g
.2
2
1 2 C; 1 ;2
4 Y
kPki
i kS Œki
i D0
and square summing over 10 < l < maxfk1;2;3;4 g yields the desired bound. For the term h rx;t Ql P0
1
0r
Pr1 R0 Q
1
1
Pr2 Pk2
2r
1
Pr3 Qj k .Pk3
i / 4
3 ; Pk4
from further above, estimate
h
rx;t Ql P0 r .2
l
1
kP0
0r
1
Pr1 R0 Q
Pr2 Pk2 0 kL1 L2x t
1
i
1 r P Q .P ; P /
12 C; 2 r3 j k k3 3 k4 4 XP 0
r 1 Pr1 R0 Q
1
Pr3 Qj k .Pk3
3 ; Pk4
1 ;2
2
/
4
L2t L1 x
;
202
6 Quintilinear and higher nonlinearities
and we have
1
r Pr1 R0 Q
r .2
minfr1;2 ;k1 g 2
2
1r
1
Pr2 Pk2
Pr3 Qj k .Pk3
minfr1;2 ;k1 g maxfr1;2 ;k1 g 2
r
1
Pr2 Pk2
2
/
4
3 ; Pk4
kPk1
1 kL1 L2x t
1
2r
L2t L1 x
Pr3 Qj k .Pk3
3 ; Pk4
/ 4
1
L2t HP 2
;
which in conjunction with the bound from above
r
1
Pr2 Pk2
1
2r
Pr3 Qj k .Pk3
/ 4
3 ; Pk4
1
. 2 2 Œminfr2;3 ;k1;2;3;4 g
1
L2t HP x2
maxfr2;3 ;k1;2;3;4 g
4 Y
kPki
i kS Œki ;
i D1
implies that
h
rx;t Ql P0 r .2
l
2
1
1
0r
2r
Pr2 Pk2
minfr1;2 ;k1 g 2
Pr1 R0 Q
2
1
1
Pr3 Qj k .Pk3
3 ; Pk4
i
4/
XP 0
1 C; 1 ;2 2
minfr1;2 ;k1 g maxfr1;2 ;k1 g 2
1
2 2 Œminfr2;3 ;k1;2;3;4 g
maxfr2;3 ;k1;2;3;4 g
4 Y
kPki
i kS Œki :
i D0
Square summing over l > 10 yields a bound as claimed in the lemma with ı D 21 . (i5): maxfk1 ; : : : ; k4 g > l C O.1/, R0 Pk1
1
D R0 Pk1 Q maxfk1;2;3;4 g
1.
Freeze the modulation of R0 Pk1 1 to dyadic value 2l1 2maxfk1;2;3;4 g . Here there must be at least one other input with modulation at least comparable to 2l1 . Let this input be Pk2 2 for definitiveness’ sake, the other cases being treated similarly. Thus consider the term h rx;t Ql P0
0r
1
Pr1 R0 Ql1 Pk1
1r
1
Pr2 Pk2 Ql1 CO.1/ r
1
Pr3 Qj k .Pk3
2
3 ; Pk4
i
4/
:
203
6 Quintilinear and higher nonlinearities
Assuming a high-low frequency cascade r1 k1 k2 , we can estimate this by (using Bernstein’s inequality)
h
rx;t Ql P0 0 r 1 Pr1 R0 Ql1 Pk1 1 r 1 Pr2 Pk2 Q>l1 CO.1/ 2 i
r 1 Pr3 Qj k .Pk3 3 ; Pk4 4 / 1 C; 1 ;2 2 XP 0
1
. 2l. 2
/
kP0
0 kL1 L2x t
r
Pk2 Q>l1 CO.1/ 1
. 2l. 2
2r
1 Pr1 R0 Ql1 Pk1 1
Pr3 Qj k .Pk3
1
1r 3 ; Pk4
Pr2
/
4
L1t L1 x
/Cr1 minfr3 ;k3;4 g maxfr3 ;k3;4 g k1 k2
2
2
kR0 Ql1 Pk1
1 kL2
t;x
kPk2 Q>l1 CO.1/
kP0 0 kL1 L2x Y t kPkj j kS Œkj 2 kL2 t;x
j D3;4 1
. 2l. 2
/ minfr3 ;k3;4 g maxfr3 ;k3;4 g .l1 k1 / r1
2
2
2
4 Y
k1 Ck2 2
2k1
kPkj
k2 .1 /.k2 l1 /
2
j kS Œkj :
j D0
Summing over l1 maxfk1;2;3;4 g > l C O.1/, one obtains a bound of the form claimed in the lemma with ı D 12 " in the particular case at hand. The remaining frequency interactions, while keeping our assumptions on the modulations, are treated similarly. This concludes the elliptic case (i). (ii): Output in hyperbolic regime. Now we consider the expression h rx;t PŒ 5;5 Q<10 P0 0 r 1 Pr1 R0 Pk1 1
We decompose this into h rx;t PΠ5;5 Q<10 P0
r
1
0r
1
Pr2 Pk2
2r
Pr1 R0 Pk1
1
Pr3 Qj k .Pk3
1
Pr2 Pk2
2r
i
4/
:
1
r 1 Pr2 Pk2 2 r 1 Pr3 Qj k .Pk3 h D rx;t PΠ5;5 Q<10 P0 0 r 1 Pr1 R0 Pk1 Q
3 ; Pk4
1
Pr3 Qj k .Pk3
3 ; Pk4
i / 4
3 ; Pk4
i / 4
(6.1)
204
6 Quintilinear and higher nonlinearities
C rx;t PŒ
h Q 5;5 <10 P0
0r
1
Pr1 R0 Pk1 QŒk1 ;maxfk1;2;3;4 gCO.1/
r 1 Pr2 Pk2 2 r 1 Pr3 Qj k .Pk3 3 ; Pk4 h C rx;t PΠ5;5 Q<10 P0 0 r 1 Pr1 R0 Pk1 Q maxfk1;2;3;4 g 1 1
r
2r
Pr2 Pk2
1
Pr3 Qj k .Pk3
3 ; Pk4
(6.2)
1
i / 4 (6.3) i : 4/
To estimate the first expression (6.1) on the right, we exploit the fact that we control sharp Strichartz norms, in addition to the basic null-form bilinear estimate controlling Qj k .Pk3 3 ; Pk4 4 /. The key is the fact that we have the almost sharp Klainerman–Tataru norm built into S . To see this, consider the most difficult case, a high-low frequency cascade corresponding to r1 k1 k2 . We estimate the expression by starting from the inside:
r
1
2r
Pr2 Pk2
Pr3 Qj k .Pk3 X
r2
D2
1
3 ; Pk4
4/
4
L t3 L2x
c1;2 2Dk2 ;r2 ;dist.c1 ; c2 /.r2
Pr2 Pk2 ;c1 .2
r2
.
X
2r
1
Pr3 ;c2 Qj k .Pk3
3 ; Pk4
1 2
1 Pr Pr Qj k .Pk 2 kL4 L1 / 2 r 3 3 3
kPk2 ;c1
3 ; Pk4
x
t
c1 2Dk2 ;r2
/ 4
4
L t3 L2x
4 / L2 t;x
Here we have used Cauchy–Schwarz and Plancherel’s theorem. Then using the definition of k:kS , we can bound this by
2
r2
.
X
kPk2 ;c1
1 2
1 Pr Pr Qj k .Pk 2 kL4 L1 / 2 r 3 3 3
t
c1 2Dk2 ;r2
.2
r2
2
k2 4
2
x
r 2 k2 2C
2
minfr3 ;k3;4 g maxfr3 ;k3;4 g 2
3 ; Pk4
4 Y
kPkj
4 / L2 t;x
2 kS Œkj :
j D2
Turning to the full expression further above, we then have for the contribution of
205
6 Quintilinear and higher nonlinearities
this term to the hyperbolic part of the output
h
rx;t Q<10 P0
1
0r
Pr1 R0 Q
r
. kP0
. kP0
1
1
1
Pr2 Pk2 2 r
1 Pr1 R0 Q
1r
t
0 kL1 t
Pk2
2r
1
r
Pr1
1
Pr3 Qj k .Pk3
Pr3 Qj k .Pk3 X
1
1
Pr2 ;c2 Pk2
2r
1
i
4/
L1t HP x 1
Pr2
4/
3 ; Pk4
L1t L1 x
R0 Q
c1;2 2Rk1 ;r1 ; dist.c1 ; c2
r
3 ; Pk4
1
/.2r1
Pr3 Qj k .Pk3
3 ; Pk4
We intend to substitute the intermediate bound from above for
1
1 r P P P r Q .P ; P /
r2 r3 j k k2 2 k3 3 k4 4
4/
L1t L1 x
:
4
L t3 L2x
where we can exploit that, by Minkowski’s and Plancherel’s inequality, we have X
21
kPr2 ;c2 F k2 4
c1 2Dr2 ;c2
. kPr2 F k
4
L t3 L2x
L t3 L2x
:
Thus we can estimate, using Cauchy–Schwarz, Bernstein’s inequality and the preceding observation
r
1
Pr1
X
1
r .
X
Pr2 ;c2 Pk2 t
Pk2 r 1 k1 2C
2
3 4 k1
2
r2
2
2r k2 4
2
1
x
1
2r
21
r 1
2 1 kL4 L
kPk1 ;c1
c1 2Dk1 ;r1
.2
R0 Q
1
c1;2 2Dk1 ;r1 ; dist.c1 ; c2 /.2r1
1
2
3 ; Pk4
4/
L1t L1 x
Pr2
Pr3 Qj k .Pk3
r2 k2 2C
Pr3 Qj k .Pk3
3 ; Pk4
4/
minfr3 ;k3;4 g maxfr3 ;k3;4 g 2
4
L t3 L2x
4 Y j D1
kPkj
2 kS Œkj :
206
6 Quintilinear and higher nonlinearities
But by our assumption r1 k1 k2 we have r2 D k1 C O.1/, whence we can replace the above bound by
h
rx;t Q<10 P0 0 r 1 Pr1 R0 Q
r 1 Pr2 Pk2 2 r 1 Pr3 Qj k .Pk3 3 ; Pk4 4 / 1 1 P L t Hx
.2
r 1 k1 2C
2
r 2 k2 4C
2
minfr3 ;k3;4 g maxfr3 ;k3;4 g 2
4 Y
kPkj
2 kS Œkj ;
j D0
and this is again enough to yield the statement of the lemma (here with ı D The remaining frequency interactions can be handled similarly. Next, consider the second term (6.2) above, i.e., h rx;t PŒ 5;5 Q<10 P0 0 r 1 Pr1 R0 Pk1 QŒk1 ;maxfk1;2;3;4 gCO.1/ r This is much simpler: Indeed,
h
rx;t PΠ5;5 Q<10 P0 0 r
. kP0
1
2r
Pr2 Pk2
1
Pr3 Qj k .Pk3
1
i / : 4
3 ; Pk4
1
Pr1 R0 Pk1 QŒk1 ;maxfk1;2;3;4 gCO.1/ i
r 1 Pr2 Pk2 2 r 1 Pr3 Qj k .Pk3 3 ; Pk4 4 / 1 L t HP 1 Pr1 R0 Pk1 QŒk1 ;maxfk1;2;3;4 gCO.1/ 1 0 kL1 L2x r
1 4C ).
1
1
t
r
1
Pr2 Pk2
1
2r
Pr3 Qj k .Pk3
3 ; Pk4
/
4
L1t L1 x
:
For sake of concreteness, we again assume that r1 k1 k2 , the remaining cases being similar. Therefore,
1
r Pr1 R0 Pk1 QŒk1 ;maxfk1;2;3;4 gCO.1/ 1 r 1 Pr2
Pk2 2 r 1 Pr3 Qj k .Pk3 3 ; Pk4 4 / 1 1 L t Lx
r1
. 2 kR0 Pk1 QŒk1 ;maxfk1;2;3;4 gCO.1/
1
r Pr Pk 2 2 .2
r1 k1 .maxfk1;2;3;4 g k1 /
2
2
r2
1 kL2
t;x
2r
maxfk3;4 g 2
4 Y j D0
1
Pr3 Qj k .Pk3
kPkj
j kS Œkj :
3 ; Pk4
4/
L2t L2x
207
6 Quintilinear and higher nonlinearities
This corresponds to a bound as in the lemma with ı D 21 , where we recall is as in the definition of k kS Œk . The remaining frequency interactions for this term are treated similarly. Finally, consider the last term above h rx;t PŒ 5;5 Q<10 P0 0 r 1 Pr1 R0 Pk1 Q maxfk1;2;3;4 g r
1
Pr2 .Pk2
2r
1
1
Pr3 Qj k .Pk3
3 ; Pk4
i // : 4
Here we again need to compensate for the losses coming from estimating the term R0 Pk1 Q maxfk1;2;3;4 g 1 . Freeze its modulation to dyadic size 2l . Then either at least one other input has at least comparable modulation, or else the output has modulation 2l (in which case necessarily l < O.1/. In the latter case, one then estimates (where l maxfk1;2;3;4 g and we assume all other inputs to be at much lower modulation)
h
rx;t PΠ5;5 Q<10 P0 0 r 1 Pr1 R0 Pk1 Ql 1 i
r 1 Pr2 Pk2 2 r 1 Pr3 Qj k .Pk3 3 ; Pk4 4 / N Œ0
h
1 D rx;t PΠ5;5 Ql P0 0 r Pr1 R0 Pk1 Ql 1 i
r 1 Pr2 Pk2 2 r 1 Pr3 Qj k .Pk3 3 ; Pk4 4 / N Œ0
h
rx;t PΠ5;5 Ql P0 0 r 1 Pr1 R0 Pk1 Ql 1 i
r 1 Pr2 Pk2 2 r 1 Pr3 Qj k .Pk3 3 ; Pk4 4 / 1; 1 ;1 2 XP 0
l
. 2 2 kP0 0 kL1 L2x r 1 Pr1 R0 Pk1 Ql 1 t
r 1 Pr2 Pk2 2 r 1 Pr3 Qj k .Pk3 3 ; Pk4 4 / 2 1 : L t Lx
Here the second factor above is estimated by
1
r Pr1 R0 Pk1 Ql 1 r 1 Pr2 Pk2 2 r 1 Pr3 Qj k .Pk3 .2
minfr1;2 ;k1 g 2
2
minfr1 ;r2 ;k1 g maxfr1 ;r2 ;k1 g 2
2
3 ; Pk4
/
4
minfr2;3 ;k2;3;4 g maxfr2;3 ;k2;3;4 g 2
4 Y j D1
kPkj
L2t L1 x
2.l
k1 /
j kS Œkj :
208
6 Quintilinear and higher nonlinearities
Inserting this bound into the last inequality but one and summing over l maxfk1;2;3;4 g results in a bound as in the lemma with ı D 12 . The case when at least one further input has at least modulation at least comparable to 2l is similar, one places the output into L1t HP 1 . This completes the proof of the first inequality of the lemma. The remaining ones are treated by an identical procedure. In a similar vein, one has estimates controlling the second kind of quintilinear term. We state the Lemma 6.2. For the second type of quintilinear null-form, we have the following estimates for suitable ı > 0:
rx;t P0 r 1 Pk1 1 Ps1 r 1 Qj .Pk2 2 ; Pk3 3 /
Pr1 r 1 IQj .Pk4 4 ; Pk5 5 / N Œ0
.2
ı r1 ıŒminf0;k1;2;3 ;s1 g maxf0;k1;2;3 ;s1 g
2
2ıŒminfr1 ;k4;5 g
maxfr1 ;k4;5 g
5 Y
kPkj
j kS Œkj ;
r1 <
10
j D1
rx;t P0 Ps1 Œr
1
.Pk1
Pr1 r .2
1
1 Ps2 r 1
Qj .Pk2
IQj .Pk4
2 ; Pk3
3/
4 ; Pk5
5/
N Œ0
ıs1 ıŒminfs1;2 ;k1;2;3 g maxfs1;2 ;k1;2;3 g
2
2ıŒminfr1 ;k4;5 g
maxfr1 ;k4;5 g
5 Y
kPkj
j kS Œkj ;
r1 2 Π10; 10
j D1
rx;t P0 Ps1 Œr
1
.Pk1
1 Ps2 r
Pr1 r .2
1
1
Qj .Pk2
2 ; Pk3
3/
IQj .Pk4
4 ; Pk5
/ 5
N Œ0
ıs1 ıŒminfs1;2 ;k1;2;3 g maxfs1;2 ;k1;2;3 g
2
2ıŒminfr1 ;k4;5 g
maxfr1 ;k4;5 g
5 Y
kPkj
j kS Œkj ;
r1 > 10
j D1
Proof. We verify this again for the first inequality above, the other ones following a similar pattern. As usual, we distinguish between elliptic and hyperbolic output components:
209
6 Quintilinear and higher nonlinearities
(i): Output in elliptic regime. This is the expression rx;t PΠ5;5 Q>10 P0 r 1 Pk1 1 Ps1 r 1 Qj .Pk2 Pr1 r
1
2 ; Pk3
3/
IQj .Pk4
4 ; Pk5
:
5/
As usual the only slight complication arises due to the fact that w may have D 0. Freeze the modulation of the output to dyadic size 2l , l > 10. Then one re-iterates the same steps as in the preceding proof: (i1): maxfk1;2;3 g l, time derivative falls on term with modulation < 2l 100 . In this case at least one additional input (which is not hit by a time derivative) has modulation > 2l 10 . For example, assume this is Pk1 1 D Pk1 Q>l 10 1 , the other cases being treated similarly. Then assuming a high-low scenario, say, i.e., k1 1, we have (using Bernstein’s inequality)
h i
P0 r 1 Pk1 Q>l 10 1 Ps1 r 1 Qj .Pk2 2 ; Pk3 3 / 1 2 L t Lx
. kPk1 Q>l .2
10
1 kL2
t;x
kPs1 r
k1 .1 /.k1 l/ .l k2 /
2
1
2
3 Y
Qj .Pk2
kPkj
2 ; Pk3
3 /kL2
t;x
j kS Œkj :
j D1
Substituting this into the full expression, we obtain for the output the bound
rx;t PΠ5;5 Ql P0 r 1 Pk1 Q>l 10 1 Ps1 r 1 Qj .Pk2 2 ; Pk3 3 /
Pr1 r 1 IQj .Pk4 4 ; Pk5 5 / 1 C; 1 ;2 XP 0 2
h i
. 2 l P0 r 1 Pk1 Q>l 10 1 Ps1 r 1 Qj .Pk2 2 ; Pk3 3 / 2 L t;x
kPr1 r 1
. 2. 2
h /l
P0 r
1
Pk1 Q>l
kPr1 r 1
. 2. 2
/l
2
1
IQj .Pk4
10 1
1 Ps1 r
IQj .Pk4
k1 .1 /.k1 l/ .l k2 / r1
2
2
2 2
4 ; Pk5 1
5 /kL1 t;x
Qj .Pk2
4 ; Pk5
2 ; Pk3
i
3/
L1t L2x
5 /kL1 t;x
minfr1 ;k4;5 g maxfr1 ;k4;5 g 2
5 Y
kPkj
j kS Œkj :
j D1
Summing over l maxfk1;2;3 g, the desired inequality of the first type of the lemma follows in this case. The remaining frequency interactions within h i P0 r 1 Pk1 Q>l 10 1 Ps1 r 1 Qj .Pk2 2 ; Pk3 3 /
210
6 Quintilinear and higher nonlinearities
are handled similarly. (i2): maxfk1;2;3 g l, time derivative falls on term with modulation 2l . In this case, we place the time derivative term into L2t;x , and are guaranteed gains in the maximal occurring frequency: for example, consider the term (arising upon unraveling the inner Qj null-structure with D 0) h i P0 r 1 Pk1 1 Ps1 r 1 .Pk2 QlCO.1/ R0 2 Pk3 3 / : In the high-high case k2 s1 , one can then estimate
h i
P0 r 1 Pk1 1 Ps1 r 1 .Pk2 QlCO.1/ R0 2 Pk3 3 /
L2t;x
.2
minf0;k1 ;s1 g maxf0;k1 ;s1 g 2
kPk1
2
minfs1 ;k2 ;k3 g maxfs1 ;k2 ;k3 g 2
1 kL1 L2x kPk2 QlCO.1/ R0
2 k 2 P 1 kPk3 Lt H 2
t
From here one estimates the full expression by
rx;t PΠ5;5 Ql P0 r 1 Pk 1 Ps r 1 .Pk QlCO.1/ R0 2 Pk 1 1 2 3 1
Pr r IQj .Pk 4 ; Pk 5 / 1 1
. 2 l P0 Œr
1
l .l k2 /
2
2
1
1 Ps1 r
.Pk1
kPr1 r .2
4
1
XP 0
.Pk2 QlCO.1/ R0
IQj .Pk4
minf0;k1 ;s1 g maxf0;k1 ;s1 g 2
kPk1
5
2
2r1 2minfr1 ;k4;5 g
t
3/
2 C; 1 ;2
3 // L2 t;x
5 /kL1 t;x
minfs1 ;k2 ;k3 g maxfs1 ;k2 ;k3 g 2
1 kL1 L2x kPk2 QlCO.1/ t
4 ; Pk5
2 P k3
3 kL1 L2x :
2k
XP k
1 2 C;1 ;1 2
Y
maxfr1 ;k4;5 g
kPk3
kPkj
3 kL1 L2x t
j kS Œkj :
j D4;5
One may sum here to obtain a bound of the type as in the first inequality of the lemma, with ı D 21 ". The remaining frequency interactions within P0 Œr
1
.Pk1
1 Ps1 r
1
.Pk2 QlCO.1/ R0
2 Pk3
3 //
are again handled similarly. (i3): maxfk1;2;3 g l, time derivative falls on term with modulation 2l . In this case at least one additional term has at least comparable modulation, and one argues as in case (i1).
211
6 Quintilinear and higher nonlinearities
(i4): maxfk1;2;3 g > l C O.1/, time derivative falls on term with modulation < maxfk1;2;3 g C O.1/. Here the losses coming from the time derivative are easily counteracted by the gains in the large frequencies: first, one reduces the inputs Pk2;3 2;3 to the elliptic regimes. To do so, note that we have
P0 Œr 1 .Pk 1 Ps r 1 .Pk QŒk ;maxfk
3 // 2 g R0 2 Pk 1
1
.2
2
2
1;2;3
minf0;s1 ;k1;2;3 g maxf0;s1 ;k1;2;3 g 2
L t;x
3
2.maxfk1;2;3 g
3 Y
k2 /
kPkj
j kS Œkj ;
j D1
and inserting this into the full expression is easily seen to yield the desired inequality. Hence we have reduced this case to the expression rx;t PŒ
5;5 Ql
1
P0 Œr
.Pk1
1
1 Ps1 r
Qj .Pk2 Q
Pr1 r
2 Pk3 Q
IQj .Pk4
3 //
4 ; Pk5
:
5/
Of course in the present case at least one of k2;3 > l C O.1/. Assume that we have a high-high-low type situation in 1
Ps1 r
Qj .Pk2 Q
2 Pk3 Q
3 /;
i.e., s1 k2 , this being the most delicate case. We distinguish between two cases: (a): modulation of Ps1 : : : is less than 2lCO.1/ . In this case, we may “pull out” a (time)- derivative from the Qj -null-form, using the simple identity R
1
2
Rj
1
Rj
R
2
D @ Œr
1
1
Rj
2
Hence in this case we can estimate
rx;t PŒ 5;5 Ql P0 Œr 1 .Pk 1 Ps Q
l
kP0 Œr
1
.Pk1
Qj .Pk2 Q
2 Pk3 Q
kPr1 r .2
l l k2
2
r1
222
1
1
@j Œr
1
R
2 :
1
4 ; Pk5
5/
XP 0
1 2 C; 1 ;2
1
3 //kL1 L2x t
IQj .Pk4
minfr1 ;k3;4 g maxfr1 ;k3;4 g 2
4 ; Pk5
2minfs1 ;k1 ;0g
5 /kL2 L1 x t
5 Y j D1
kPkj
j kS Œkj :
212
6 Quintilinear and higher nonlinearities
One can sum over l < maxfk1;2;3 g to arrive at the desired first inequality of the lemma in the case at hand. (b): modulation of Ps1 : : : is 2l . In this case the modulation of the first input Pk1 1 needs to be comparable to that of Ps1 r
1
Qj .Pk2 Q
2 Pk3 Q
3 /:
Hence we can write this contribution as X
rx;t PŒ
5;5 Ql
P0 Œr
1
.Pk1 Ql1 CO.1/
1 Ps1 Ql1 r
1
l1 l
Qj .Pk2 Q
2 Pk3 Q
3 // 1
Pr1 r
IQj .Pk4
4 ; Pk5
:
5/
To estimate it, we use
Ps Ql r 1 1
1
Qj .Pk2 Q
2 Pk3 Q
3 / L2 t;x
. 2l1
k2
2
s1 2
Y
kPki Q
2 kS Œki :
i D2;3
We the insert this bound into the full expression. In case that k1 > O.1/, we can estimate
rx;t PŒ
5;5 Ql
P0 Œr
1
.Pk1 Ql1 CO.1/
Qj .Pk2 Q
l
l
2 Pk3 Q
2 2 kPk1 Ql1 CO.1/ Qj .Pk2 Q
1 kL2
t;x
1
4 ; Pk5
kPs1 Ql1 r
2 Pk3 Q
kPr1 r
1
1 Ps1 Ql1 r
5/
XP 0
1 2 C; 1 ;2
1
3 /kL2
t;x
IQj .Pk4
4 ; Pk5
: 5 /kL1 t;x
213
6 Quintilinear and higher nonlinearities
In case that l1 < k1 C O.1/, we can bound this by 2
l
l
2 2 kPk1 Ql1 CO.1/
1 kL2
t;x
1
kPs1 Ql1 r
.2
l
l
22 2
kPk1
1 kS Œk1 2
l1 k2
2
.2
l
l
22 2
l1 2
2l1
k2
2
s1 2
3 /kL2
t;x
IQj .Pk4 4 ; Pk5 5 /kL1 t;x Y kPki Q
s1 2
kPr1 r
2 Pk3 Q
1
kPr1 r l1 2
Qj .Pk2 Q
i D2;3 1
2r1 2minfk3 ;k4 g
IQj .Pk4
maxfk3 ;k4 g
4 ; Pk5
5 Y
kPki
5 /kL1 t;x
i kS Œki :
i D1
Summing over k1 C O.1/ > l1 l and then over l > O.1/ results in a bound as in the first inequality of the lemma with ı D 21 C . Next, still in the case k1 > O.1/, if l1 > k1 C O.1/, one proceeds as before but uses kPk1 Ql1 CO.1/
1 kL2
t;x
.2
k1 2
2.1
One obtains a final bound with the same ı D
1 2
/.k1 l1 /
kPk1 kS Œk1 :
C as in the preceding case.
In the case k1 < O.1/, one simply places Pk1 Ql1 CO.1/ gaining an additional factor 2k1 . We omit the details.
1
into L2t L1 x , thereby
(i5): maxfk1;2;3 g > l C O.1/, time derivative falls on a term with modulation >> maxfk1;2;3 g. This is similar to the preceding an omitted. This concludes case (i), when the output is in the elliptic regime. (ii): Output in hyperbolic regime. This is the expression rx;t PŒ
5;5 Q<10
P0 Œr
1
.Pk1
1 Ps1 r
1
Qj .Pk2 r
1
2 ; Pk3
3 //Pr1
IQj .Pk4
4 ; Pk5
:
5/
To treat it, we decompose P0 r
1
P k1
1 Ps1 r
D P0 r
1
Qj .Pk2
2 ; Pk3 3 / 1 Pk1 1 Ps1 r 1 I c Qj .Pk2 2 ; Pk3 3 / C P0 r 1 Pk1 1 Ps1 r 1 IQj .Pk2 2 ; Pk3
:
3/
214
6 Quintilinear and higher nonlinearities
(iia): contribution of the elliptic type term. This is the expression rx;t PŒ
5;5 Q<10
1
P0 Œr
1 Ps1 r
.Pk1
1 c
I Qj .Pk2 1
r
2 ; Pk3
3 //Pr1
IQj .Pk4
4 ; Pk5
:
5/
We shall treat the case s1 10, i.e., the case of a high-low interaction within P0 r 1 Pk1 1 Ps1 r 1 Qj .Pk2 2 ; Pk3 3 / : The remaining cases are again more of the same. Now freeze the modulation of the expression Ps1 r 1 I c Qj Pk2 2 ; Pk3 3 / to size 2l , l s1 . Then decompose the corresponding full expression into the following: rx;t PŒ 5;5 Q<10 P0 r 1 Pk1 1 Ps1 Ql r 1 I c Qj .Pk2 2 ; Pk3 3 / Pr1 r 1 IQj .Pk4 4 ; Pk5 5 / D rx;t PŒ
5;5 Q<10 1 c
r
P0 Œr
1
.Pk1 Q>l
I Qj .Pk2 Pr1 r
2 ; Pk3 1
1 Ps1 Ql
10 3 //
IQj .Pk4
4 ; Pk5
5/
(6.4)
1
C rx;t PŒ
.Pk1 Q
(6.5)
The first term (6.4) on the right can then be estimated by krx;t PΠr
5;5 Q<10
1 c
I Qj .Pk2
. kPk1 Q>l
10
1
P0 Œr 2 ; Pk3
1 kL2
t;x
.Pk1 Q>l 3 //
Pr1 r
kPs1 Ql r
1 Ps1 Ql
10 1
IQj .Pk4
1 c
I Qj .Pk2
kPr1 r
1
4 ; Pk5
5/
kL1 HP
2 ; Pk3
3 /kL2 L1 x
IQj .Pk4
4 ; Pk5
t
1
t
: 5 /kL1 t;x
Then from Lemma 4.18 and Bernstein’s inequality we infer that provided k2 s1 , we have 2
s1 2
kPs1 Ql r
1 c
I Qj .Pk2 l
.2 2
2 ; Pk3
maxfs1 ;k2;3 g
3 /kL2 L1 x t
maxfk2
s1 ; 1g2
Y j D2;3
kPkj
j kS Œkj :
215
6 Quintilinear and higher nonlinearities
Inserting this into the preceding bound we infer that kPk1 Q>l
10
1 kL2
t;x
kPs1 Ql r 1
kPr1 r .2
s1 l 2
2l 2
maxfs1 ;k2;3 g
1 c
I Qj .Pk2
IQj .Pk4
2 ; Pk3
4 ; Pk5
3 /kL2 L1 x t
5 /k
L1 t;x
s1 ; 1g2 2r1 2minfr1 ;k4;5 g
maxfk2
5 Y
maxfr1 ;k4;5 g
j kS Œkj :
kPkj
j D1
Summing over l > s1 yields the bound of the first inequality of the lemma with ı D . On the other hand, when k2 D s1 C O.1/, say, one can use Lemma 4.23 instead, which then gives the desired inequality with ı D 21 . Next consider (6.5). Here we distinguish between the cases l < r1 C O.1/ and l r1 . In the former case, as before assuming s1 < 10, we have
rx;t PŒ
5;5 Q<10
P0 r
1
Pr1 r . kPk1 Q
10
Pk1 Q
10
1 c
1 Ps1 Ql r
I Qj .Pk2
1 P1 5/
1
2 ; Pk3
3/
IQj .Pk4 4 ; Pk5 Lt H
1 c
I Qj .Pk2 2 ; Pk3 3 / L2 L1 1 kL1 L2x Ps1 Ql r t
t x Pr1 r 1 IQj .Pk4 4 ; Pk5 5 / L2 L1 : t
x
Using Lemma 4.18-4.23 again, we obtain the bound .2
s1 Cr1 2
2.l
maxfk2 ;k3 g/
js1
r1
k2 j2 2 2 2
minfr1 ;k4;5 g maxfr1 ;k4;5 g 2
5 Y
kPkj
j kS Œkj :
j D1
One may sum here over s1 < l < r1 C O.1/ to obtain the desired first inequality of the lemma with ı D . Next, consider the case l r1 . But in this case we can write rx;t PŒ
5;5 Q<10 P0 r 1 Pk1 Q
Pr1 r D rx;t PŒ
5;5 QŒl 10;10 1 Pk1 Q
P0 r
1 Ps1 Ql r 1 IQj .Pk4
1 c
2 ; Pk3
3/
1 Ps1 Ql r
1 c
2 ; Pk3
3/
I Qj .Pk2 4 ; Pk5 5 /
I Qj .Pk2
Pr1 r
1
IQj .Pk4
4 ; Pk5
:
5/
216
6 Quintilinear and higher nonlinearities
But this we can then estimate via the k:k
XP 0
1;
1 2 ;1
-norm of the output, i.e., it suf-
fices to bound krx;t PŒ
5;5 QŒl 10;10 P0 r 1 Pk1 Q
3/
0
.2
l 2
1 c
kPs1 Ql r
I Qj .Pk2
kPk1 Q
2 ; Pk3
kPr1 r
1
IQj .Pk4
1 kL1 L2x t
4 ; Pk5
: 5 /kL1 t;x
From here the estimates are continued in a fashion identical to the ones used to control (6.4). This completes estimating the contribution of Ps1 r 1 I c Qj .Pk2 2 ; Pk3 3 //. (iib): Contribution of the hyperbolic type term. Next we consider the contribution of Ps1 r 1 IQj Pk2 2 ; Pk3 3 / which is the expression rx;t PŒ
5;5 Q<10
P0 r
1
Pk1
1 Ps1 r
1
IQj .Pk2
Pr1 r
1
2 ; Pk3
3/
IQj .Pk4
4 ; Pk5
:
5/
We shall again make the reduction s1 < 10, the remaining frequency interactions being treated analogously. This is accomplished using Lemma 4.16. We obtain
rx;t PΠ5;5 Q<10 P0 r 1 Pk 1 Ps r 1 IQj .Pk 2 ; Pk 3 / 1 1 2 3 Pr1 r 1 IQj .Pk4 4 ; Pk5 5 / L1 HP 1 t
. kPk 1 1 2 kPs r 1 IQj .Pk 2 ; Pk 3 / 2 1 L t Lx
1
1
2
kPr1 r .2
s1 Cr1 2
2
minfs1 ;k2;3 g maxfs1 ;k2;3 g 2
2
3
1
L t Lx
IQj .Pk4
minfr1 ;k4;5 g maxfr1 ;k4;5 g 2
4 ; Pk5 5 Y
5 /kL2 L1 x
kPkj
t
j kS Œkj :
j D1
This is as desired with ı D 12 .
6.1 Error terms of order higher than five
6.1
217
Error terms of order higher than five
Here we consider the errors generated by repeated application of Hodge decompositions, which are of higher than quintic degree. We recall that they arise when we apply repeated Hodge decompositions to the second and third input in rx;t Πr or else to the second and third input in h rx;t r 1 r 1 .
2
1
.
2
/
/r
1
i IQj . ; / :
To simplify the discussion, we shall call terms that arise in the first situation “of the first type”, while those in that arise in the second situation will be called of “second type”. In either case, we associate a binary graph with each such expression as in the discussion above, see Chapter 3. We call expressions whose associated graph has only directed subgraphs of length at most three “short”, and those with directed graphs of length at least four “long”. For technical reasons, it will be most convenient to organize the “short” and “long” higher order terms into suitable sums, which are easier to estimate. Specifically, note that each of these higher order terms consists of nested terms of the form : : : r 1 Ps1 Pk1 R Pr1 r 1 Pk2 Ps2 r 1 Ps2 Œ: : : ; (6.6) here the case of a node with one outgoing edge, or alternatively : : : r 1 Ps1 Ps2 r 1 Pk1 r 1 Pr1 Œ: : : r 1 Pk2 Ps3 r 1 Œ: : :
(6.7)
in case of two outgoing edges. It is the first type of expression which may cause some mild difficulties due to the presence of the R -operator, which for D 0 may be formally unbounded. However, re-combining a term of type (6.6) with a suitable term of the form (6.7) and using the relation X R C D ; D Rk k ; kD1;2
we replace each such “intermediate” gradient term (i.e., not contributing to one of the innermost Qj null-forms in case of “short” expressions) R by its nongradient counterpart . We shall call the resulting expressions “reduced”. Thus, for example, the (short) quintilinear expression h i rx;t Pk1 1 r 1 Pr1 Pk2 R 2 Pr2 ŒPk3 r 1 Pr3 Qj .Pk4 4 ; Pk5 5 /
218
6 Quintilinear and higher nonlinearities
has reduced version h rx;t Pk1 1 r 1 Pr1 Pk2
r
2 Pr2 ŒPk3
1
Pr3 Qj .Pk4
4 ; Pk5
i
5 /
:
Now we can formulate Proposition 6.3. Let l D 2; 3; 4
P0 F2lC1 . /;
be short reduced higher order expression of first type at frequency 1. We can write it in nested form P0 F2lC1 . / h D rx;t P0 Pk1
1r
1
h Pr1 : : : r
r
1
Œ: : : r
1
1
Prj Pkj C1
Pr2l
1
j C1 r
Qj .Pk2l
1
Prj C1 Pkj C2
2l ; Pk2lC1
j C2
ii / : 2lC1
Then we have the following bounds: (1) If r1
10, we have
kP0 F2lC1 . /kN Œ0 . 2ıŒminfk2 ;:::;k2lC1 ;r1 ;:::;r2l
1g
maxfk2 ;:::;k2lC1 ;r1 ;:::;r2l
1 g
2lC1 Y
kPkj
j kS Œkj
j D1
for a suitable constant ı > 0. (2) If r1 2 Œ 10; 10, we obtain the bound kP0 F2lC1 . /kN Œ0 . 2ık1 2ıŒminfk2 ;:::;k2lC1 ;r1 ;:::;r2l
1g
maxfk2 ;:::;k2lC1 ;r1 ;:::;r2l 2lC1 Y
kPkj
1 g
j kS Œkj :
j D1
(3) If r1 > 10, we have kP0 F2lC1 . /kN Œ0 .2
ık1 ıŒminfk2 ;:::;k2lC1 ;r1 ;:::;r2l
2
1g
maxfk2 ;:::;k2lC1 ;r1 ;:::;r2l 2lC1 Y j D1
kPkj
1 g
j kS Œkj :
219
6.1 Error terms of order higher than five
The proof of this follows the exact same pattern as the one for Proposition 6.1, and is omitted. In fact, for l > 2, one no longer needs to use the sharp improved Strichartz endpoint as in the case l D 2. In a similar vein, we have the analogue of Proposition 6.2. A short reduced expression of the second type can be written as P0 F2lC3 . / D rx;t P0 r r
1
:::r
1
1
Pr1 : : : r 1 Prj Pkj C1 j C1 r
1
Prj C1 Pkj C2 j C2 Qj .Pk2l 2l ; Pk2lC1 2lC1 / Ps1 1 1 Qj r .Pk2lC2 2lC2 ; Pk2lC3 2lC3 / (6.8)
Pr2l
where l D 1; 2; 3. Then we have Proposition 6.4. Using the representation (6.8), let P0 F2lC3 . / be a short reduced term at frequency 1 of the the second type. Then the following hold: (1) If s1 <
10, we have
kP0 F2lC3 . /kN Œ0 . 2ıs1 2ıŒminfs1 ;k2lC2 ;k2lC3 g ıŒminfk2 ;:::;k2l ;r1 ;:::;r2l
1g
2
maxfs1 ;k2lC2 ;k2lC3 g
maxfk2 ;:::;k2l ;r1 ;:::;r2l
1 g
2lC3 Y
kPkj
j kS Œkj :
j D1
(2) If s1 2 Œ 10; 10, we have kP0 F2lC3 . /kN Œ0 . 2ı r1 2ıŒminfs1 ;k2lC2 ;k2lC3 g ıŒminfk2 ;:::;k2l ;r1 ;:::;r2l
2
1g
maxfs1 ;k2lC2 ;k2lC3 g
maxfk2 ;:::;k2l ;r1 ;:::;r2l
1 g
2lC3 Y
kPkj
j kS Œkj :
j D1
(3) If s1 > 10, we have kP0 F2lC3 . /kN Œ0 . 2
ı r1 ıŒminfs1 ;k2lC2 ;k2lC3 g maxfs1 ;k2lC2 ;k2lC3 g
ıŒminfk2 ;:::;k2l ;r1 ;:::;r2l
2
2
1g
maxfk2 ;:::;k2l ;r1 ;:::;r2l
1 g
2lC3 Y j D1
kPkj
j kS Œkj :
220
6 Quintilinear and higher nonlinearities
Again the proof is similar to the one of Proposition 6.2. Note that in order to estimate the expressions of short type, we still need to to use a little bit of null-structure to make them amenable to estimation by the S -spaces. This is no longer the case for ‘long’ expressions P0 F11 . / of reduced type: Write such an expression as P0 F11 . / D rx;t P0 Pk1 Pkj C2
Pr1 : : : r 1 Prj Pkj C1 1 : : : r 1 Pr9 .Pk10 j C2 r 1r
1
j C1 r
1
10 Pk11
Prj C1 11 /
if it is of first type or P0 F11 . / D rx;t P0 r 1 Pr1 : : : r 1 Prj Pkj C1 j C1 r Pkj C2 j C2 r 1 : : : r 1 Pr9 .Pk10 10 Pk11 Ps1 r
1
1
Prj C1 11 /
.Pk12
12 Pk13
13 /
if it is of the second type. Note that the innermost bilinear expressions r
1
Pr9 .Pk10
10 Pk11
11 /
are no longer null-forms. Proposition 6.5. Let P0 F11 . / be a long expression of either first or second type, written as in immediately preceding. The if P0 F11 . / is of the first type, we have if r1 < 10 kP0 F11 . /kN Œ0 . 2ıŒminfk2 ;:::;k9 ;r1 ;:::;r9 g 2
maxfk2 ;:::;k9 ;r1 ;:::;r9 g
ıŒminfk10 ;k11 g maxfk10 ;k11 g
11 Y
kPkj
j kS Œkj :
j D1
Thus by contrast to Proposition 6.3 case (1), we have an extra factor 2ıŒminfk10 ;k11 g
maxfk10 ;k11 g
;
whence we cannot gain in case of high-high interactions in the innermost expression r
1
Pr9 .Pk10
10 Pk11
11 /:
221
6.1 Error terms of order higher than five
Similarly, if P0 F11 is of the second type, we obtain kP0 F11 . /kN Œ0 . 2ıŒminfk2 ;:::;k9 ;r1 ;:::;r9 g
maxfk2 ;:::;k9 ;r1 ;:::;r9 g
2ıŒminfk10 ;k11 g 2
ıjs1 j ıŒminfs1 ;k12 ;k13 g maxfs1 ;k12 ;k13 g
2
13 Y
maxfk10 ;k11 g
kPkj
j kS Œkj :
j D1
Proof. This is purely an application of our available Strichartz norms: Indeed, for expressions of the first type, we have for suitable ı > 0 kPr8 ŒPk9
9r
1
Pr9 .Pk10
10 Pk11
3
. 2. 8 C/r8 2ıŒminfr8;9 ;k9 g
4C 11 /k L8t Lx3
maxfr8;9 ;k9 g ıŒminfk10 ;k11 g maxfk10 ;k11 g
2
11 Y
kPkj
j kS Œkj
j D9
where we define
4 . C/ 3
3 4
D
1
:
Further, we have for p D 1; 2; : : : ; 7 and suitable ıp > 0 2
pC1 r1
kPr1 ŒPk r
1
Pr2 F k
8 pC1
Lt
4C
Lx3
. 2ıp Œminfr1;2 ;kg
maxfr1;2 ;kg
Œ2
p r2
kPr2 F k
8 pC1
Lt
4C
Lx3
where scaling dictates p 4 2. C/ 1 : 8 3 The proposition follows by applying these two inequalities sufficiently often. The argument for expressions of the second type is similar. p D 2
Remark 6.6. We note that in the estimates above, we have not used wave-packet atoms. Remark 6.7. Using the arguments of the present subsection, it is straightforward to obtain the following refinement: given ı > 0, there exists C > 1 large enough,
222
6 Quintilinear and higher nonlinearities
such that if P0 F2iC1 D P0 ŒPk1 1 Pr1 r specialize to k1 D O.1/ and X P0 FQ2i C1 WD P0 Pk1
1 Œ: : :
is of the first type, i 2, and we
1 Pr1 Q>r1 C2C r
1
Œ: : : ;
r< C
then we can improve the bounds of the preceding propositions by a factor ı.
7
Some basic perturbative results
This section develops some of the basic perturbative theory required for our work. More precisely, we introduce a norm locally on some time interval . T0 ; T1 / which we denoted by k kS. T0 ;T1 / with the property that its finiteness insures that the gauged wave map can be continued outside of that time interval. The second topic we discuss is the issue of defining wave maps with data which are merely of energy class. This is accomplished by means of passing to the limit in energy of smooth wave maps.
7.1
A blow-up criterion
Assume we are given a wave map u W . T0 ; T1 / R2 ! H2 with Schwartz data at time t D 0, by which we mean that the derivative components ˛i , i D 1; 2, ˛ D 0; 1; 2, and thus also the Coulomb components ˛i , are Schwartz functions at time t D 0. These functions will then also be Schwartz on fixed time slices on the maximal interval of existence . T0 ; T1 / R2 . The following norm will provide us with sufficient control for long time existence and scattering. Definition 7.1. For any Schwartz function on . T0 ; T1 / R2 set k kS WD
X
kPk k2S Œk..
21 T0 ;T1 /R2 /
:
k2Z
Here kPk kS Œk..
T0 ;T1 /R2 /
WD
sup T
kPk
i kS Œk.Œ T 0 ;T R2 /
where the local norms are those from (2.68) using the jjj jjj-norm. The goal of this section is to prove the following result. Proposition 7.2. Let . T0 ; T1 / be the maximal interval of existence for the wave map u in the smooth sense. If k kS < 1 then necessarily T0 D T1 D 1. Moreover, the wave map scatters at infinity, i.e., the components ; approach free waves in the energy topology as t ! ˙1.
224
7 Some basic perturbative results
The strategy for proving the theorem will be to demonstrate an a priori bound k .t; /kH s < 1
sup t 2. T0 ;T1 /
for some s > 0, using the assumption k kS < 1. By the Klainerman–Machedon local well-posedness theory, this implies that u may be extended smoothly to some interval . T0 ; T1 C "/ for " > 0 provided T1 < 1, which contradicts minimality, and similarly for T0 . Once we know that u exists for t 2 . 1; 1/, scattering will follow by using a similar argument. To obtain a priori control over sub-critical norms, we use Tao’s device of frequency envelope: for some ı1 > 0 depending only on certain a priori parameters specified later, define ck WD
X
2
ı1 jk `j
2 kP` .0; /kL 2
12
x
:
(7.1)
`2Z
Here as always we let D f ˛i g the vector of derivative components. Proposition 7.2 now follows from the following result. Proposition 7.3. Let . T0 ; T1 / be the maximal interval of existence for the wave map u in the smooth sense. If k kS < 1, then there exists a number C1 D C1 .u/ < 1 (which may depend in a complicated fashion on the wave map u, and not just its energy)1 such that kPk kL1 ..
T0 ;T1 /IL2x /
kPk kS Œk..
T0 ;T1 /R2 /
t
C1 ck :
In fact, C1 ck :
To establish the existence of C1 , we shall cover the time interval . T0 ; T1 / by a finite number of shorter open intervals Ij (which can still be very large): Let k kS < C0 . Then 1 . T0 ; T1 / D [jMD1 Ij ;
M1 D M1 .C0 /
(7.2)
where jIj will satisfy a suitable smallness property. The idea then is to bootstrap certain bounds on each Ij , beginning with the interval containing the time slice t D 0. More precisely, the intervals Ij will be chosen so that the wave map 1
This is an artifact of the proof to follow and will be improved in the following sections.
225
7.1 A blow-up criterion
restricted to each Ij is well approximated by a free wave. While the error can be treated perturbatively, the free wave has better dispersive properties which we can exploit. All functions will be smooth in space and time and Schwartz functions on fixed time slices.
7.1.1 Splitting the wave map on shorter time intervals We first derive a simple estimate on the nonlinearities appearing in (1.12) and (1.13). It will be based entirely on the Strichartz estimates, see Lemma 2.17. We will keep the time interval . T0 ; T1 / from above fixed throughout. Lemma 7.4. Let maxi D1;2;3 k
P0
1 jrj
1
.
2
i kS
< C0 . Then
3/
2 LM t .. T0 ;T1 /ILx /
. C02 sup
jkj M
sup 2
i D1;2;3 k2Z
kPk
i kS Œk.. T0 ;T1 /R2 / ;
provided M is large and with an absolute implicit constant. Alternatively, one has the bound
P0
1 jrj
1
.
2
.k
3/
2 LM t .. T0 ;T1 /ILx /
2 kL1 L2x k 3 kL1 L2x t
t
sup 2
jkj M
kPk
k2Z
1 kS Œk.. T0 ;T1 /R2 /
with an absolute implicit constant. Proof. Assume to begin with that i is adapted to ki 2 Z. As in Chapter 5, we now consider all possible cases of interactions. Also, we shall drop the time interval . T0 ; T1 / from our notation with the understanding that integration in time is to be restricted to this interval. Moreover, replacing each i by a globally defined Schwartz function Q i with the property that k Q i kS Œki 2k
i kS Œki .Œ T 0 ;T R2 / ;
Q i jŒ
D
T 0 ;T
for some T 0 ; T as above, allows us to assume that the initially. Finally, fix any M 100.
i
i jΠT 0 ;T
are globally defined
226
7 Some basic perturbative results
Case 1: 0 k1 k2 C O.1/ D k3 C O.1/. Then
P0 1 jrj 1 . 2 3 / M 2 . kP0 . 1 jrj L L .k
k1
2 1 kLM L1 x t
1
x
t
k
2 kL1 L2x k t
.
3 //kLM L1x
2
t
3 Y
k1 M
3 kL1 L2x . 2
k
t
i kS Œki
i D1
where the final estimate is from (2.37). Case 2: 0 k1 D k2 C O.1/; k3 k2 C. If k3 0, one proceeds as in Case 1. Otherwise,
P0 1 jrj 1 . 2 3 / M 2 . kP0 . 1 jrj 1 . 2 3 //k M 1 L Lx L L x
t
.k
1 kL1 L2x
2
.2
k1 k3 .1
1 M
t
2
k1
/
k
t
2 kL1 L2x k 3 kLM L1 x t t
3 Y
k
i kS Œki
i D1
by (2.37) and Bernstein’s inequality. Case 3: 0 k1 D k3 C O.1/; k2 k3 one.
C: This case is symmetric to the previous
Case 4: O.1/ k2 D k3 C O.1/; k1 C. Here
P0 1 jrj 1 . 2 3 / M 2 . P0 1 jrj L L x
t
.k
1 kLM L1 x t
k
1
2 kL1 L2x k 3 kL1 L2x t t
PQ0 .
.2
.1
2 1 M
3/
/k1
1 LM t Lx
3 Y
k
i kS Œki :
2
3/
i D1
Case 5: k1 D O.1/; k2 D k3 C O.1/. In this case we estimate X
P0 1 jrj 1 Pk .
P0 1 jrj 1 . 2 3 / M 2 . L L t
x
kk2 ^0CC
X
.
k
1 kLM L2C kjrj
k
1 kS Œk1 2
t
1
x
Pk .
2
3 /kL1 LN x t
kk2 ^0CC
X
.
.1
2 N
kk2 ^0CC .1
.2
2 N
/k2 ^0
3 Y i D1
k
i kS Œki :
/k
k
2
3 kL1 L1x t
2 LM t Lx
227
7.1 A blow-up criterion
Case 6: O.1/ D k1 k2 C O.1/ k3 C C. Here one has
P0 1 jrj 1 . 2 3 / M 2 . P0 1 jrj 1 PQk . 1 L t Lx
1 Q
. k 1 kLM L2C jrj Pk1 . 2 3 / L1 LN x
t
.k
.1
.2
2 N
/k3
3 Y
k
3 kL1 L2x t
.k
3/
2 LM t Lx
x
t
1 kS Œk1 k 2
2
1 kS Œk1 k 2 kL1 L2x k 3 kL1 LN x t t
i kS Œki :
i D1
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. Finally, in this case the estimate reads
P0 1 jrj 1 . 2 3 / M 2 . P0 1 jrj 1 PQ0 . 2 3 / M 2 L L L L x
t
.k
k 2 1 kLM L1 x t
t
3 kL1 L2x t
. 2k1 .1
1 M
/ k3
2
3 Y
k
x
i kS Œki :
i D1
Case 9: k3 D O.1/; max.k1 ; k2 /
C. This is symmetric to the preceding case.
We now drop the assumption on the frequency support of the inputs. Summing over all these cases yields the bound
P0 1 jrj 1 . 2 3 / M 2 L Lx i h jkjt X max kPk j k2S Œk ; . sup sup 2 M kPk i kS Œk j D1;2;3
i D1;2;3 k2Z
k2Z
which proves the first bound. The proof of the second estimate is implicit in the preceding and the lemma is proved. Remark 7.5. If 2 , 3 are gauged wave maps with energy bounded by E, then the second bound of the preceding lemma becomes
P0
1 jrj
1
.
2
3/
2 LM t .. T0 ;T1 /ILx /
. E 2 sup 2 k2Z
with an absolute implicit constant.
jkj M
kPk
1 kS Œk.. T0 ;T1 /R2 /
(7.3)
228
7 Some basic perturbative results
Our main goal here is to prove the following decomposition of the gauged wave map. Lemma 7.6. Let k kS < C0 . Given "0 > 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 Πr . / "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 ΠT 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 Q<2m0 C` in modulation, and the inputs are in the hyperbolic regime, i.e., Pki i D Qki CC Pki i where C is large depending on L. Then we
232
7 Some basic perturbative results
bound (7.14) (and (7.15)) as follows, first on the whole time axis R (assuming as we may that the inputs have been suitably extended): X
X r t;x Q<2m0 C` C P` Pk1 ;1 1 jrj 1 Pk I c
`
k;k1 ;k2 ;k3 1 ;2 ;3
Q.Pk2 ;2 X X
.
` 2Cm0
X
2 ; Pk3 ;3
P`; Q<2m0 C`
C
2
3/
N Œ`
Pk1 ;1
1 jrj
1
Pk I c
k;k1 ;k2 ;k3 1 ;2 ;3
Q.Pk2 ;2
2 ; Pk3 ;3
2
3/
(7.16)
NFŒ
Note that the 22` -factor produced by the output is canceled against the scaling factor which is part of the N Œ`-norm, see Definition 2.9. By the usual arguments involving disposable multipliers, we may replace jrj 1 Pk I c by 2 k2 (implicit constants are allowed to depend on L). Since the inputs are hyperbolic, we may also ignore the null-form Q. For any , max dist.; i / & 2m0 :
i D2;3
Let us assume that this happens for i D 3. Then by (2.29), followed by (2.30) and Cauchy–Schwarz (recall that k2 D k3 C O.1/), (7.16) C.L; m0 /
X
2
k2
kPk1 ;1
1 Pk2 ;2
2 2 kL2 L2 kPk3 ;3 t
k;`;k1 ;k2 ;k3 1 ;2 ;3
C.L; m0 /
X
kPk1 ;1
2 1 kS Œk1 ;1 kPk2 ;2
x
2 3 kS Œk3
2 2 kS Œk2 ;2
k;`;k1 ;k2 ;k3 1 ;2 ;3
kPk3 ;3
2 3 kS Œk3 ;3
X 3 C.L; m0 / kPk k2S Œk C.L; m0 /C06 : k2Z
Note that we are not assuming that Pki ;i i are wave-packets, i.e., localized in modulation to < 22m0 Cki but only to modulations < 2ki CC . Therefore, to pass to the last line one needs to use Lemma 2.7 for the modulations between these two cut-offs. However, this only costs a factor of . jm0 j which is admissible. We
233
7.1 A blow-up criterion
now rewrite the first line in this estimate in the form Z (7.16) C.L; m0 /
X
R3
k2 ˇ
ˇ
2
Pk1 ;1
1 .t; x/Pk2 ;2
ˇ2 ˇ
2 .t; x/
k;`;k1 ;k2 ;k3 1 ;2 ;3 2 3 kS Œk3 dt dx
kPk3 ;3
C.L; m0 /C06 :
By the dominated convergence theorem, we can cover the line (and especially . T0 ; T1 /) into finitely many intervals Ij such that Z C.L; m0 /
X
Ij R2
2
k2 ˇ
ˇ Pk1 ;1
1 .t; x/Pk2 ;2
ˇ2 ˇ
2 .t; x/
k;`;k1 ;k2 ;k3 1 ;2 ;3 2 3 kS Œk3 dt dx
kPk3 ;3
< "0 C06 (7.17)
for each Ij . Moreover, the number of these intervals is M1 ."0 /. Unfortunately, in the preceding calculations we suppressed the action of the nonlocal operator Pk I c , which is given by convolution with a kernel (in both t; x) of bounded L1 mass. Although this is only a technical nuisance, we quickly explain here how to deal with it for completeness’ sake: Consider the schematically written expression X
`
X
X
k1 D`CO.1/ k2 Dk3 CO.1/ k1 CO.1/
Z R2C1
m.a/ Pk1 ;1
1 Pk2 ;2
2 .
a/Pk3 ;3
3 .
2
a/
N Œ`
where we have km./kL1 .R2C1 / D O.1/. Under the same assumptions on the frequency localizations as above, we estimate this by .
X
X
` k1 D`CO.1/
Z R2C1
X k2 Dk3 CO.1/ k1 CO.1/
m.a/ Pk1 ;1
1 Pk2 ;2
2 .
a/ L2 da kPk3 ;3 t;x
3 kS Œk3
2
(7.18)
where we have used Minkowski’s inequality as well as the translation invariance of the norms used. We proceed by using Cauchy–Schwarz twice in a row, to
234
7 Some basic perturbative results
bound the preceding by
X
.
k1 D`CO.1/
Pk .
R2C1
k2 Dk3 CO.1/ k1 CO.1/ 1 Pk2 ;2
1 ;1
Z
X
hZ
X
R2C1
k1 D`CO.1/
X
jm.a/j
i 12
2 a/ L2 da kPk3 ;3
2 .
t;x
jm.a/j Pk1 ;1
kPk3 ;3
2 3 kS Œk3
1 Pk2 ;2
2 .
3 kS Œk3
2
2 a/ L2 da t;x
k3
.
k1 k2 CO.1/
X
Z
Z
X
R2C1
kPk3 ;3
jm.a/j
R2C1
2 3 kS Œk3
ˇ ˇPk
1 ;1
1 Pk2 ;2
2 .
ˇ2 a/ˇ dt dx da
:
k3
Thus, properly speaking, instead of smallness of Z X ˇ 2 k2 ˇPk1 ;1 1 .t; x/Pk2 ;2 Ij R2
k1 k2 CO.1/
we need to achieve smallness of Z Z X jm.a/j k1 k2 CO.1/
ˇ2 ˇ ;
2 .t; x/
Ij R2
R2C1
ˇ ˇPk ; 1 1
1 Pk2 ;2
2 .
ˇ2 a/ˇ dt dx da;
which of course can be achieved identically, via divisibility of the inner integral and Fubini’s theorem. We shall henceforth suppress this technicality and stick with the notationally simpler condition (7.17) as well as similar ones in the sequel, it being understood that we sometimes suppress harmless convolution operators. Retracing our steps shows that the intervals Ij have the desired properties (7.14) and (7.15) under the modulation assumptions Pki i D Qki CC Pki i , and the additional assumption that the output is limited to size . 22m0 C` (the Schwartz extensions implicit in (7.14) and (7.15) are simply obtained by multiplying the L2tx functions by smooth bump functions). The remaining cases where these modulation assumptions are violated are handled similarly. For example, consider (7.15) for outputs of modulations & 2`CC but again on the whole time axis;
235
7.1 A blow-up criterion
here, it is easy to see that we may assume k D kj C O.1/, j D 1; 2; 3, whence we reduce to estimating (with the preceding frequency restraint implicit in the summation)
X
r t;x P` Q`CC Pk1 ;1 R 1 jrj 1 Pk I k;`;k1 ;k2 ;k3 1 ;2 ;3
.
X
Q.Pk2 ;2
3/
1
Pk I
2 ; Pk3 ;3
3/
r t;x P` Q`CC Pk1 ;1 R
1 jrj
k;`;k1 ;k2 ;k3 1 ;2 ;3
Q.Pk2 ;2 .
X
X
k;`;k1 ;k2 ;k3 1 ;2 ;3
.
X
2
. 21 "/` .1 "/m
X
2
`
N Œ`
2
P
1 2 C"; 1 ";2
X`
kPk1 ;1 Qm
1 kL2 L2x
2
t
m`CC
2 2k Pk I Q.Pk2 ;2 2
2`
kPk1 ;1
k;`;k1 ;k2 ;k3 1 ;2 ;3
.
2
2
2 ; Pk3 ;3
2 1k XP
1 2 C";1 ";2
k;`;k1 ;k2 ;k3 1 ;2 ;3
2`
kPk1 ;1
2
3 / L1 L1 x t
2k
`
Pk I Q.Pk ; 2 2 2
2 ; Pk3 ;3
2 1k XP `
1 2 C";1 ";2
2
3 / L2 L2 t x
2 ; Pk3 ;3
2k kL.Pk2 ;2
2 ; Pk3 ;3
2 3 /kL2 L2 t
x
C.L; m0 /C06 where the final bound again follows from (2.29) (L stands for the usual averaged space-time translation operator which arises via removal of disposable multipliers). Writing out the L2t L2x -norm explicitly in the previous estimate allows us again to choose intervals Ij with the desired properties. The remaining case of output modulations Qm with 2m0 C ` m ` C C is similar:
X
2
r t;x P` Qm Pk1 ;1 R 1 jrj 1 Pk I Q.Pk2 ;2 2 ; Pk3 ;3 3 / N Œ`
k;`;k1 ;k2 ;k3 1 ;2 ;3
.
X
r t;x P` Qm Pk1 ;1 R
1 jrj
1
Pk I
k;`;k1 ;k2 ;k3 1 ;2 ;3
Q.Pk2 ;2
2 ; Pk3 ;3
2 / 3 P
X`
1 2 C"; 1 ";2
236
7 Some basic perturbative results
X
.
`
2
2 1 kL1 L2
kPk1 ;1 Qm
x
t
k;`;k1 ;k2 ;k3 1 ;2 ;3
Pk I Q.Pk ; 2 2
2
3 / L2 L2 t x
2 ; Pk3 ;3
C.L; m0 /C06 Due to the L2t L2x -norm one can now proceed as before. Finally, suppose that the output as well as 1 are hyperbolic, but that 2 and 3 are elliptic. Then, restricting the inner sum to k D ` C O.1/ k2 D k3 C O.1/ as we may, we have X
X r t;x P` Q`CC Pk1 ;1 Qk1 CC R 1 jrj 1 Pk I
`
k;k1 ;k2 ;k3 1 ;2 ;3
Q.Pk2 ;2 .
X
2 2`
`
X
X
r t;x P` Q`CC Pk1 ;1 Qk1 CC R
k;k1 ;k2 ;k3 mk2 CC 1 ;2 ;3
Q.Pk2 ;2 Qm
2 / 3
2 ; Pk3 ;3
N Œ` 1
1 jrj
Q
2 ; Pk3 ;3 Qm
Pk I
2
1
3/
L t L2x
which can be further estimated as (using Cauchy–Schwarz for the third inequality) X X
X . P` Q`CC Pk1 ;1 Qk1 CC R 1 jrj 1 Pk I
`
k;k1 ;k2 ;k3 mk2 CC 1 ;2 ;3
Q.Pk2 ;2 Qm .
X `
X
X
kPk1 ;1
X
X
k;`;k1 ;k2 ;k3 m1;2 k2 CC 1 ;2 ;3
1
L t L2x
t
Pk I Q.Pk2 ;2 Qm
kPk1 ;1
2 / 3 1
1 kL1 L2x
k;k1 ;k2 ;k3 mk2 CC 1 ;2 ;3
r .
Q 2 ; Pk3 ;3 Qm
2 1 kL1 L2 t
x
Q 2 ; Pk3 ;3 Qm
22.m1
k2 /
kPk3 ;3 QQ m2
2
/ 3 L1 L2
kPk2 ;2 Qm1
t
x
2 2 kL2 L2 t
x
2 3 kL2 L1 t
x
C.L; m0 /C06 by Bernstein’s inequality and the definition of S Œk; to pass to the second line use that P` Q`CC is disposable. Partitioning R into finitely many intervals on which X 2 22.1 "/m 2 .1 2"/k kPk Qm 2 kL 2 mk
t;x
237
7.1 A blow-up criterion
is small allows us to obtain the desired conclusion as before. Alternatively, one can gain smallness here by taking C in m k2 C C large; this will be important later (see Remark 7.8). We leave the analogous analysis of (7.14) to the reader. The proof of the claim (7.13) for the higher degree nonlinearities is outlined in the Appendix. A crucial feature of the construction of the intervals fIj g1j M1 above is that it is universal, i.e., it does not depend on the choice of the underlying frequency scale. We now conclude the proof of Lemma 7.6. Fix some Ij and localize .j / Pk L satisfies the to frequency 2k . If jIj j < "1 2 k , then Pk NL WD Pk bound (7.4) by the analysis in Case 1. Otherwise, one represents the solution via (2.73). The bounds in Case 1 above then imply the estimate ˇ k.Pk /ˇŒt
j
k ;t C" 1 j
"1 2
2
k
kS Œk . kPk kS Œk :
The free wave Pk L at dyadic frequency 2k is now defined as the free evolution in (2.73) with data at some t0 2 Œtj "1 2 k ; tj C "1 2 k where
.Pk
˛;
1
˛ /.t0 ; / L2 HP
@ t Pk
1
. "0 M ak
.j /
whereas Pk NL is the sum of the other two terms in that formula. That the preceding choice of t0 is possible follows from Lemma 7.4. Summing over k now yields the claimed local splitting of in Lemma 7.6. Finally, the proof of (7.6) is implicit in the preceding and we skip the details. Remark 7.7. Later we will apply (7.6) in the following context. If X
kPk k2S Œk
21
< ı3
k>k0
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
where m0 is sufficiently large depending on the energy E of a bound of the form
. Then we can infer
k kS . C2 .E; m0 ; /: This is done by a bootstrap, with the desired smallness coming either from the intervals Ij or the gains from the angular alignment. Moreover, assume that for each k1 > 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
<
2
< ık kS
t;x
k
C
X 2.1
"/l . 21 "/k
kQl ik kL2
t;x
(7.20)
k
where ı > 0 is small depending on the energy and the integer , but independent of k kS . Then one can again conclude k kS . C2 .E; m0 ; /: Note the the time intervals Ij are determined only by means of the fk1 ;k2 and not the gk1 ;k2 .
239
7.1 A blow-up criterion
7.1.2 Proof of Proposition 7.3 Recall that we are making the assumption k kS < C0 . We first show that the wave map cannot break down in finite time, i.e., T D T 0 D 1. Assume for example that T < 1. For "0 > 0 a sufficiently small but absolute constant (which will be specified later), pick the M1 .C0 ; "0 /-many intervals Ij as in Lemma 7.6. It will suffice to consider that interval Ij0 which has T as its endpoints. Alternatively, starting with that interval Ij containing the initial time slice t D 0, one can inductively obtain control over the frequency-localized constituents of , the Pk . Lemma 7.9. Let Ij D .tj ; tj C1 / be an interval as in Lemma 7.6. Introduce the frequency envelope X
2 1 ck WD 2 0 jk `j P` .tj ; / L2 2 x
`2Z
where 0 > 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 Πr 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
r
1
Pk .Pk2 ;2
L
Pk3 ;3
kDk2 L0
L / L1 L1 2 t x .Jj R /
< "0 :
It is furthermore clear that we also obtain X 1 ;2 ;3 2Cm0 maxi¤j dist.i ;j />2m0 k2 CO.1/ X
X00
k1 ;k2 ;k3 2Z k2 2A
P0 rx;t Pk ; r 1 1
1
Pk .Pk2 ;2
L
Pk3 ;3
L/
N Œ0.Jj R2 /
kDk2 L0
"0 2
0 jk1 j
kPk1 kS Œk1 A.C0 /c0 ;
which completes the bootstrap for the trilinear source terms. The contribution of the higher order terms is dealt with in the appendix. By applying the above bootstrap argument on each of the finitely many intervals Jj i comprising each Ij , the proof of Lemma 7.9 now follows. The proof of Proposition 7.3 can easily be concluded. Indeed, one infers from the Klainerman–Machedon criterion that T D T 0 D 1. Moreover, we obtain a global a priori bound kPk kS Œk C1 ck were the constant C1 depends implicitly on L . As the above argument is fairly crude, we have no a priori way here of controlling this number. In Chapter 9 we will refine this type of estimate. Finally, Lemma 7.6 implies the scattering for large times.
244
7.2
7 Some basic perturbative results
Control of wave maps via a fixed L2 -profile
A fundamental issue that we need to address is the very definition of wave maps with data that are in some sense only of energy class. To propagate such data under the wave map evolution, we shall use approximations by smooth wave maps each of which can be continued canonically. The following lemma justifies this procedure. Lemma 7.10. Let ˛n be the derivative components of a sequence of Schwartz class2 wave maps un W . T0n ; T1n / R2 ! H2 on their maximal time interval of existence and assume that the Coulomb components ˛n .0; / satisfy lim k
n!1
n ˛ .0; /
V˛ kL2x D 0
for some V˛ 2 L2 .R2 /. Denoting the collection of components V˛ by V , there is a time T0 D T0 .V / > 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 ΠT0 ; T1 R2 ! H2 be an admissible wave map with gauged derivative components denoted by . Assume that k kS.ΠT0 ;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 ΠT0 ; T1 and satisfies k Q kS.ΠT0 ;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.ΠT0 ;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;0R2 /
< 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<2r0 logŒ yy .0;/ 0 : (7.24) .x1 ; y1 / WD Œjx x0 j
Œjx x0 j<2r0
250
7 Some basic perturbative results
¬ R with B WD jBj 1 B . Note that we have chosen the cutoffs on the two components differently – the one on the second component is slightly larger than the first. This is merely a technical convenience which amounts to y1 D yy0 when r Œjx x0 j
2 .Rd / ,! BMO.Rd /. Lemma 7.19. BP 2;1 d
Proof. By duality, it suffices to prove that H1 ,! BP 2;12 .Rd / for the Hardy space H1 . Thus, we need to show that X d 2 j 2 kPj k2 C.d / j 2Z
for any which is an H1 .Rd / atom. Here Pj are the usual Littlewood–Paley projections to frequencies of size 2j . By scalingRand translation invariance we may assume that supp. / B.0; 1/, j j 1 and .x/ dx D 0. If j 0, then we use that kPj k2 k k2 C.d /: If j 0, then writing Pj D 2jd .2j / we conclude that Z jPj .x/j C
2
j.d C1/
Z
B.0;1/
0
1
jr j 2j .x
ty/ dtj .y/j dy;
which implies that Z kPj k2 C
B.0;1/
2j.d C1/ kr k2 2
j
d 2
d
j .y/j dy C.d /2j. 2 C1/
and we are done.
The importance of BMO in this context lies with the fact that exponentiation maps small balls in BMO into the Ap -class. Recall that w is an Ap -weight in the sense of Muckenhaupt (see Chapter 7 in [8] or Stein [46] for all this standard material) provided Z Z p 1 0 1 1 jQj w.x/ dx jQj w 1 p .x/ dx Ap .w/ (7.25) Q
Q
7.2 Control of wave maps via a fixed L2 -profile
251
uniformly for all cubes Q Rd for some constant Ap .w/. Here 1 < p < 1 and p 0 D pp 1 as usual. Note that Ap Aq if p q. The A1 class is defined as all w with M w C w a.e., where MSis the Hardy–Littlewood maximal operator. At the other end one has A1 WD 1p<1 Ap , which is characterized by the estimate jSj ı w.S / C (7.26) w.Q/ jQj for all S Q (this is deep and requires the “reverse Hölder inequality”). Here C and ı > 0 only depend on w. From the John–Nirenberg inequality, w D e is an Ap weight for some 1 < p < 1 provided k kBMO < r0 is small enough and the Ap -constant Ap .w/ in (7.25) only depends on r0 . 1
Lemma 7.20. Let k'kL2 A and set w WD e . / 2 ' . Then for any 1 < p < 1 one has Ap .w/ C.p; A/ where the latter constant only depends on p and A. Proof. For any ı > 0, ˚ # j 2 Z j kPj 'k2 ı ı
2
A2 :
In particular, for any ı > 0 there is a decomposition ' D e ' C .' 1 2 3 2 k. / e ' k1 C ı A and, by Lemma 7.19,
. /
1 2
.'
e ' / so that
e ' / BMO C ı:
By the John–Nirenberg inequality one may choose ı small depending on p 2 1 ' / 2 Ap with some absolute Ap -constant. .1; 1/ such that exp . / 2 .' e Since
. / 21 e
2 3 '
e eC ı A 1
we are done.
The importance of Ap weights lies with the fact that the Hardy–Littlewood maximal operator M as well as Calderón–Zygmund operators T are bounded on Lp .w dx/ with constants that only depend on the dimension and the Ap constant from (7.25) (and T in case of a singular integral) provided 1 < p < 1. In the present context, we will require a version of Poincaré’s inequality with A2 weights. Now for the small energy lemma.
252
7 Some basic perturbative results
Lemma 7.21. Let .xn1 ; yn1 / be as in (7.24) applied to .xn ; yn /. Then given "0 > 0, we can pick r0 > 0 small enough such that
rxn
ryn
1
n 2 C n1 2 "0 : y 1 Lx y 1 Lx Here r is the spatial gradient and r0 does not depend on n. Since one can clearly also arrange
Œjx x j
we have now achieved smallness of the energy of these data. Moreover, r0 > 0 can be chosen uniformly in x0 2 R2 . Proof. We assume as we may (by rescaling) that k ˛1 k2 C k ˛2 k2 1 for ˛ D 0; 1; 2. We shall also suppress the time dependence and drop the superscript n. In view of (7.23) it suffices to estimate the contributions of those terms in which the derivatives falls on the cutoff in (7.24). Starting with the component y1 D y.x/ Œjx x0 j
B
˛D1;2
B
uniformly in n provided r0 is small enough. Here we used the relation (1.4) and that the gauge change is given by multiplication by a unimodular factor. For the x1 -component, we make the preliminary observation that y 2 Ap for any 1 < p < 1. Indeed, by Lemma 7.19, for any 1 M < 1 we can find C D C.M / so large that X 1 @j j2 kBMO M 1 ; kPRnΠC;C log ykBMO D kPRnΠC;C j D1;2
which implies that y2 WD exp PRnŒ C.p/;C.p/ log y 2 Ap for any 1 < p < 1 with a suitable C.p/. Since Lemma 7.19 implies that kyy2 1 k1 C , the claim follows. We now use the following weighted Poincaré inequality, see Theorem 1.5 in [9]: for any w 2 A2 , and ball B of radius r > 0, Z Z 2 2 jf .x/ .f /B j w.x/dx C.w/r jrf .x/j2 w.x/dx; B B − .f /B WD f .x/ dx: B
7.2 Control of wave maps via a fixed L2 -profile
253
Consequently, with w D y 2 2 A2 , and in view of our definition of x0 , Z ˇ Z ˇ ˇ ˇ ˇ x.x/ x0 ˇ2 ˇ rx.x/ ˇ2 2 r0 ˇ ˇ dx1 dx2 . ˇ ˇ dx1 dx2 y.x/ B B y.x/ X Z ˇ ˇ ˇ 1 .x/ˇ2 dx1 dx2 : . j j D1;2 B
By our choice of cutoffs in (7.24) we are done. To obtain the final statement of the proof, simply note that we can always find r0 > 0 such that sup
2 Z X
x0 2R2 ˛D0 Dx0 ;r0
ˇ ˇ ˇV˛ .x/ˇ2 dx1 dx2 "2 : 0
Consequently, for all sufficiently large n, sup x0
2R2
2 Z X ˛D0 Dx0 ;r0
ˇ ˇ
ˇ2 n ˇ ˛ .x/ dx1 dx2
"20 ;
and therefore also sup
2 Z X
x0 2R2 ˛D0 Dx0 ;r0
ˇ n ˇ2 ˇ .x/ˇ dx1 dx2 "2 ˛ 0
for all large n, which is all that is needed for the proof.
We will also require an analogous result on small energy outside of a big ball. Thus, let R0 1 be large and define x .x2 ; y2 / WD Œjxj>R0
x0 y0
;e
R Œjxj> 20
y 0
logΠy .0;/
:
(7.27)
Here Œjxj>R0 is a smooth cutoff to the set fjxj > R0 g which equals one on jxj > 2R0 , say, and − x0 WD x.x/ dx1 dx2 ; ŒR0 <jxj<2R0 − y0 WD exp log y.x/ dx1 dx2 : R Œ
0 2 <jxj
In analogy to (7.24) the construction here is such that y2 D fr Œjxj>R0 ¤ 0g.
y y0
on the set
254
7 Some basic perturbative results
Lemma 7.22. Let .xn2 ; yn2 / be as in (7.27) applied to .xn ; yn /. Then given "0 > 0, we can pick R0 > 0 large enough such that
rxn
ryn
2
C
n 2 n2 2 "0 : y 2 Lx y 2 Lx Here r is the spatial gradient and R0 does not depend on n. Since one can clearly also arrange k Œjxj>R0 0n kL2x "0 we have now achieved smallness of the energy of these data. Proof. The argument is completely analogous to the one for Lemma 7.21. The only difference is that one uses the following Poincaré inequalities on annuli instead of disks: For any R0 > 0, Z Z ˇ ˇ ˇ ˇ ˇf .x/ .f /R ˇ2 w.x/dx CR2 ˇrf .x/ˇ2 w.x/dx 0 0 R0 <jxj<2R0
R0 <jxj<2R0
for any A2 -weight w and a constant C which only depends on the A2 constant of w. As usual .f /R0 denotes the average of f over the annulus. For w D 1 this is of course standard, and for general w it follows from [9]. Next, we wish to establish control over the ˛n in the S -norm on a nonempty time interval . T0 ; T0 / uniformly in n. The idea is to apply Theorem 7.18 to the finitely many small energy maps given by Lemma 7.21 and then to reconstruct and also bound the original sequence ˛n in terms of these constituents. The latter of course relies on finite propagation speed and involves smooth partitions of unity. In order to handle partitions of unity, we need to derive estimates of the form k kS C. /k kS for Schwartz functions and some constant C. /. Due to issues having to do with the slow decay as well as limited regularity of the logarithmic potential 1 @ which appears in the phase of the gauge change, we will need to allow for a larger class of functions . The following lemma is tailored to such purposes. Lemma 7.23. Let 2 C 1 .R2C1 / satisfy the following properties:4 For some constant A 4
The logarithmic potential in (1.11) decays like jzj 1 (but in general no faster) which explains why we need p > 2 in the first condition. Since one in fact has asymptotic equality with jzj 1
7.2 Control of wave maps via a fixed L2 -profile
255
ı maxkD0;1 maxj˛j100 k@kt rx˛ kLqt Lpx A for all 2 < p 1, 1 q 1, ı kh i max.jj; jj100 /b . ; /kLq L1 A for all 2 q 1. Then there exists an absolute constant C0 such that k kS C0 Ak kS for any Schwartz function . The S -norm here is defined in terms of the jjj jjj-norm, and it can be either localized to some interval in time or be defined globally in time. Proof. It suffices to consider global in time estimates. We need to prove X 12 X 12 k kS WD jjjPk . /jjj2S Œk . A jjjP` jjj2S Œ` : k2Z
(7.28)
`2Z
Written out, the left-hand side here means 21 X
Pk . / 2 1 2 C Pk . / 2 N Œk L L
(7.29)
x
t
k2Z
and we shall write k k instead of jjj jjj. We begin with the energy component of the norm. If k C , then by Bernstein’s inequality
2k
Pk . / 1 2 . 2 2k 3 k k 1 3 k k 1 2 . 2 3 k k 1 3 k k S L Lx L Lx L Lx L L t
x
t
.2
2k 3
t
t
Ak kS :
(7.30)
Here we used that k kL1 L2x
X
t
k2Z
kPk k2L1 L2 t
21
x
X
kPk k2S Œk
21
k kS :
k2Z
On the other hand, if k C , then
Pk . / 1 2 L L
t x Pk .Pk 10 PQk / L1 L2 C kPk .P>k t
x
10
/kL1 L2x t
Qk k 1 2 C kP>k 10 kL1 k k 1 2 1 kP . k kL1 L t Lx L t Lx t Lx t;x Qk kS ŒkCO.1/ C 2 k k kS 1 C kr k 1 1 k P . k kL1 L L L x x t t . A kPQk kS ŒkCO.1/ C 2 k k kS up to a multiplicative constant, it follows that the Fourier transform of this potential around zero exhibits a jj 1 -singularity, which explains the second condition. Finally, we cannot control more than one time derivative of (1.11), and showing that one time derivative can be controlled in terms of the energy alone is nontrivial and requires the div-curl system for , see Corollary 7.25.
256
7 Some basic perturbative results
where we used the reverse Bernstein inequality kP>k
1 10 kL1 t Lx
k
.2
1: kr kL1 t Lx
Square-summing now implies the desired bound. It remains to bound the second term in (7.29). First, X X
Pk QkCC . / 2
Pk QkCC . / 2 1; 1 ;1 . N Œk kC
XP k
kC
.
X
2
2 2 2 2k Pk . / L2 L2 . k kL 2 1 k kS L t
x
t
kC 2
x
. A k k2S : Second, X
Pk Q>kCC . / 2 N Œk kC
.
X
Pk Q>kCC . / 2
XP k
kC
.
X
X
2
1 2 C"; 1 ";2
.1 2"/k 2j.1 "/
2
2 Pk Qj . / L2 L2 t
x
kC j >kCC
.
X
X
22"k 22j.1
"/
2 Pk Qj . /
4
L2t Lx3
kC j >kCC
:
(7.31)
P The sum kCC <j
Pk Qj . / 4 . k k 2 4 k k 1 2 . Ak kS L Lx L Lx t
L2t Lx3
t
which can be summed in this range. So we restrict our attention to j C . We can assume furthermore that D Pj C as otherwise
j
Pk Qj .P>j C / krP>j C kL2 L4x k kL1 L2x . 2 j Ak kS 4 . 2 t
L2t Lx3
t
makes a summable contribution to (7.31). We now split D Qj Q<j C . On the one hand,
Pk Qj .Qj C / 4 . kQj C k 2 4 k k 1 2 L Lx L Lx t
L2t Lx3
.2
j
.2
j
t
kr t;x kL2 L4x k kL1 L2x t
Ak kS
t
C
C
7.2 Control of wave maps via a fixed L2 -profile
257
which makes a summable contribution to (7.31), and on the other hand,
Pk Qj .Q<j
C
2 /
4 L2t Lx3
2 . Pk Qj .Q<j C Pj C QQ j / 4 L2t Lx3 X . A2 .1 C 2` / 2 kP` QQ j k2L2 L2 : t
`j C
Substituting this bound into (7.31) yields X X X 22"k 22j.1 "/ A2 .1 C 2` / kC j >kCC
kP` QQ j k2L2 L2 t
`j C
X X
. A2
2
22j.1
"/
.1 C 2` /
2
X
x
kP` QQ j k2L2 L2 t
` j `CC
. A2
x
x
kP` k2S Œ` . A2 k k2S
`
which is admissible. We may therefore assume that k C . To proceed we need to control kP` kS Œ` . If ` 0, then kP` kS Œ` . kP` Q` k
0; 1 2 ;1
XP `
C kP` Q>` k
XP `
1 2 C";1 ";2
`
. 2 2 kP` Q` kL2 L2x t
C2 Z `
1 2 C"/`
.
kP` Q`<<0 kL2 L2x C 2. t 21 2 2 . ; /j d sup jj jb
. 22
1 2 C"/`
kP` Q>0 @ t kL2 L2x t
jj2`
C 2.
1 2 C"/`
Z
12 jj2 jb . ; /j2 d
h i2 sup jj2`
. A 2.
1 2 C"/`
;
(7.32)
whereas if ` 0, then kP` kS Œ` . kP` Q` k
1 ;1 0; 2
XP `
C kP` Q>` k
`
XP `
1 2 C";1 ";2
1
. 2 2 kP` Q` kL2 L2x C 2. 2 C"/` kP` Q>` @ t kL2 L2x t t Z 12 . A2 . 2 10` h i2 sup jj22 jb . ; /j2 d jj2`
10`
: (7.33)
258
7 Some basic perturbative results
Using these bounds and applying Lemma 4.11 one obtains X kPk QC . /k2N Œk k>C
.
X
kPk QC . /k2
XP k
k>C
.
X X k>C
3
2 4 k1 ^k2 2
1 ;1 2
1;
k k1 _k2 4
.
XX k>C
kPk Qj . /k
j C
kPk1 kS Œk1 kPk2 kS Œk2
2
2 1 ;1 0; 2
XP k
. A2 k k2S :
k1 ;k2
(7.34) The sum over k; k1 ; k2 here respects the usual trichotomy. We remark that one needs to limit the output here to modulations QC as one would otherwise encounter logarithmic divergences. Next, one estimates X X X
Pk Qk . / 2
Pk Qj . / 2 1 C"; 1 ";2 . N Œk k>C
XP k
k>C j k
.
XX
2
.1 2"/k 2.1 "/j
2
2
kPk Qj . /k2L2 L2 : t
k>C j k
x
(7.35) The contribution of Pj
Pk Qj .Pj C /
C
is bounded by
L2t L2x
. kPj
k C kL2t L1 x
kL1 L2x . A 2
10j
k kS
j
k kS
t
which can be summed in (7.35). Furthermore,
Pk Qj .Pj
C Qj C
.2
j
kPj
/ L2 L2 t
x
k C Qj C @ t kL2t L1 x
kL1 L2x . A 2 t
which is again sufficient for (7.35). Finally,
Pk Qj .Pj C Qj C / 2 2 L t Lx
. Pk Qj .Pk C Qj C PQk QQ j / L2 L2 t x
C Pk Qj .Pk C <j C Qj C Pj QQ j / L2 L2 : t
x
Substituting (7.36) into (7.35) yields the estimate XX 2 Qk QQ j /k2 2 2 . A2 k k2 : 2 .1 2"/k 22.1 "/j k kL 1 kP S L L t;x
k>C j k
t
x
(7.36) (7.37)
7.2 Control of wave maps via a fixed L2 -profile
259
Similarly, after a further frequency decomposition of , substituting (7.37) into (7.35) leads to the same estimate. In summary, we are left to consider the output under the modulation constraint QC <
Pk Qk .Q 3k / > 4 N Œk
. Pk Qk .Q> 3k / 4
k . 2 2 Pk .P>k
XP k
1;
1 ;1 2
k . 2 2 Pk .Q> 3k 4
/ L2 L2 t
x
/ L2 L2
10 Q> 3k
t
4 k 2
C 2 Pk .Pk
k . ; / L1 L2 k kS . 2 2 Œjj&2k b
x
10 Q> 3k 4
PQk / L2 L2 t
x
k 2
. ; /kL1 L2 kPQk kS Œk C 2 k 3k b Œjj j jjj&2 4
k 100
.2 jj b . ; / L2 L1 k kS
C2 . A2
k 4
k 4
kh ijj _ jj100b . ; /kL2 L1 kPQk kS Œk
k kS
which is admissible. We now estimate each of the three terms on the right-hand side of
Pk Q>C .Q 3k / . Pk Q>C .Q 3k / N Œk 4 N Œk 4
C Pk Q>C .@˛ Q 3k @˛ / N Œk C Pk Q>C .Q 3k / N Œk : (7.38) 4
4
First,
Pk Q>C .Q 3k 4
.2 .2
k
/ N Œk
kQ 3k 4
k 4
max max
kD0;1 j˛j1
kL1 L2x . 2 t
k
kQ 3k kL1 L1 k kS x
k@kt r ˛ kL1 L1 k t x
4
kS . A 2
t
k 4
k kS
which is admissible. Second, by estimate (29) in [59] as well as (7.32) and (7.33),
260
7 Some basic perturbative results
and with k; k1 ; k2 respecting the usual trichotomy,
Pk Q>C .@˛ Q 3k @˛ / 4 N Œk X .2 k 2k1 Ck2 kPk1 kS Œk1 kPk2 kS Œk2 k1 ;k2
X
k
. A2
2k1 Ck2 min.2
. 21 "/k1
10k1
;2
/kPk2 kS Œk2
k1 ;k2 k
. A.2
k kS C kPk kS Œk /
which is again square-summable in k. As for the third term in (7.38) we are reduced to showing the bound X
Pk Q>C .Q
3k 4
X
2 F / N Œk . A2 kP` F k2N Œ` :
(7.39)
`2Z
kC
This bound in turn follows via Schur’s lemma from the following claim: 1 4 jk
Pk Q>C .Q 3k P` F / . A2 N Œk
`j
4
kP` F kN Œ`
(7.40)
If j ` and j C , then by (2.32) one always has the bound
Pk Q>C .Q 3k P` Qj F / 4 N Œk
k 2 Pk .Q 3k P` Qj F / L1 L2 t x 4
k
.2 Q 3k kL2 L1 P` Qj F kL2 L2x t x t 4
j . 2` k 2 2 b . ; /kL2 L1 P` Qj F kN Œ` . A 2`
k
kF kN Œ`
which agrees with (7.40) provided ` k C C . On the other hand, if ` k C C but still j C the same estimate holds with an additional high-high gain of 2 100` coming from which is of course more than sufficient for (7.40). Finally, if C j `, then an additional Bernstein gain yields
Pk Q>C .Q 3k P` Qj F / 4 N Œk
2 k Pk .Q 3k P` Qj F / L1 L2 . 2 t
4
.2
` 2
k
kb . ; /k
4
L3 L2
x
k
kQ 3k kL2 L4x kP` Qj F kL2 L4x 4
` k
kP` Qj F kL2 L2x . A 2 t
t
kF kN Œ`
t
7.2 Control of wave maps via a fixed L2 -profile
261
as desired. Therefore, the claim (7.40) holds provided F D QC F . Let us now verify (7.40) for each of the four types of N Œ`-atoms with the additional assumption that F ¤ QC F . If F is an energy atom, then
Pk Q>C .Q
3k 4
P` F / N Œk . 2
k
kQ 3k P` F kL1 L2x t
4
.2
` k
1 kP F k kQ 3k kL1 ` N Œ` t Lx 4
which is sufficient if ` k C C and if ` > k C C then
Pk Q>C .Q
P` F / N Œk 3k 4
.2
k
kQ 3k P` F kL1 L2x t
4
.2
` k
1 kP F k kQ 3k kL1 ` N Œ` t Lx 4
which is more than sufficient. Here we used the estimate
kQ 3k kL1 . b . ; / L1 L1 . h ijj _ jj100b . ; / L2 L1 . A: L1 x t 4 (7.41) For the remaining atoms we first make the simplifying assumption that b . ; / is supported on j j C jj . 1. Now suppose that P` Qj F D F with j > C . If j
kF kL2 L2x 2` 2 2 and j `, then essentially does not change the Fourier t support of F . Thus, ` D k C O.1/ and
Pk Q>C . F /
N Œk
.2
`
2
j 2
k F kL2 L2x . AkF k t
XP `
1;
1 ;1 2
(7.42)
1
as desired. On the other hand, if j > ` and kF kL2 L2x 2`. 2 "/ 2j.1C"/ , then we t need to distinguish the case ` C from ` > C . In the latter case, one argues as in (7.42). In the former case, the modulation of the output is essentially 2j and k CPwhich is excluded. It remains to consider the null-frame atoms. Thus, F D 2Cm F where F D P`; Q`C2m F and m 100. Due to F ¤ QC F , one has ` C m ` C 2m C which implies P that the Fourier support of F is essentially that of F . Therefore, F D F can be treated as a wave-packet atom satisfying the bounds X X k F k2NFŒ . A2 kF k2NFŒ :
Since k D `CO.1/ we are done. Next, suppose that b . ; / is supported on j j n n 1 1 2 with n 10 and jj . 1. Then k kL t Lx . 2 A. Start with a wave-packet
262
7 Some basic perturbative results
atom F of the type we just considered. If n k C 2m C 10, then F has essentially the same Fourier support as F , whence k F kN Œk . 2
k
X
k F k2NFŒ
12
. A2
n
2
`
X
kF k2NFŒ
12
n
.2
AkF kN Œ`
which is summable in n 10. If n > k C 2m C 10, then F has modulation of size 2n . If k n, then k F kN Œk . k F k
1;
XP k
3n 2
. A2
k
1 ;1 2
.2
n 2
k
k F kL2 L2x t
3n 2
kF kL2 L2x . A 2
k
XP `
t
n
. A2
3`
2 2 kF k
1;
1 ;1 2
kF kN Œ`
where we used (2.32) and ` D k C O.1/. If k < n, then k F kN Œk . k F k
XP k
1 2 C"; 1 ";2
. A2
n.2C"/
. A2
n
2
n.1C"/
.2
k. 21 "/
2
k. 12 "/
k F kL2 L2x t
kF kL2 L2x . A 2
n.2C"/ k.1C"/
2
t
kF k
XP `
1;
1 2 ;1
kF kN Œ` : 1
1; 2 ;1 Now suppose that F is a XP ` -atom with F D P` Qj F . If j > n C 10, then F is the same kind of atom and one argues as before gaining a factor of 2 n . If j D n C O.1/, then
k F kN Œk . 2 .2
k
k F kL1 L2x . 2
`
t
n
j 2
A2 kF k
XP `
1;
k kL2 L1 kF kL2 L2x x t
1 ;1 2
.2
t
n 2
A:
Finally, if n > j C 10, then F has modulation of size 2n . If n `, then k F kN Œk . k F k
XP k
1;
1 ;1 2
.2
`
n 2
k F kL2 L2x . 2 t
whereas in case n `, one checks similarly that k F kN Œk . k F k
XP k
1 C"; 1 ";2 2
.2
n
A
n
A
7.2 Control of wave maps via a fixed L2 -profile
263
1
C"; 1 ";2 as desired. If F is a XP ` 2 -atom with F D P` Qj F , then analogous arguments lead to a bound of k F kN Œ` . 2 "n A which is again summable in n 0. Finally, one needs to consider the case where b . ; / is supported on jj 2n with n 10, say. However, this is easier due to the rapid decay of b in . We leave those details to the reader.
Remark 7.24. Lemma 7.23 of course applies to any space-time Schwartz function . Moreover, one can check that the exact same conclusions of Lemma 7.23 hold for any Schwartz function which only depends on t and x alone; the only difference is that C0 A needs to be replaced by C. /. It is now a simple matter to prove that the ˛n have uniformly controlled S norms on some time interval . T0 ; T0 / where T0 D T0 .V /. Corollary 7.25. Under the assumptions of Lemma 7.10 there exists a time T0 D T0 .V / > 0 such that max k
˛D0;1;2
n ˛ kS.. T0 ;T0 /R2 /
C.V / < 1
(7.43)
uniformly in large n. Proof. Pick r0 > 0 small enough and R0 large enough according to Lemmas 7.21 and 7.22, respectively. In view of (7.24), Theorem 7.18, and finite propagation speed, patching up the local evolutions of .xn1 ; yn1 / shows that the evolution of .xn ; yn / exists on some time interval . T0 ; T0 / uniformly in large n; in fact, one can take T0 D r0 . Note that this part of the argument does not require .xn2 ; yn2 /. These functions are needed to obtain uniform control over k ˛n kS.. T0 ;T0 /R2 / , to which we now turn. The ˛n of the original sequence agree with the Q ˛n obtained from (7.24) on the cone Kx0 ;r0 WD f.t; x/ j jx x0 j < r0 t; 0 t < r0 g. This follows from the construction of .xn1 ; yn1 / and finite propagation speed. Note that the Q ˛n exist globally in R1C2 but agree with ˛n only on Kx0 ;r0 . A similar observation applies to .xn2 ; yn2 / on the set KR0 ;T0 WD fjxj > R0 C t; 0 t < T0 g. Cover R2 by finitely many Dj WD D.xj ; r0 / as well as the complement of D0 WD D.0; R0 /. This can be done PJ is such a fashion that2 there exists a smooth and finite partition of unity 1 D j D1 j on Œ0; T0 R such that each j is entirely supported in either a cone Kxj ;r0 or within KR0 ;T0 . Thus X X n n i 1 @Re. Q n;j n / D D j en;j : j ˛ ˛ ˛ e j
j
264
7 Some basic perturbative results
n;j Here Q ˛ are the derivative components of the small energy wave maps which n;j were constructed by means of Lemmas 7.21 and 7.22, and Q ˛ are their gauged counterparts. If j has compact support, we now claim that
e jn WD j e i
1 @Re. Q n;j
n/
satisfies the hypotheses of Lemma 7.23 with a constant A that can be chosen uniformly in n. The compact support assumption in time can of course be fulfilled. Since for each j and all n j . Q ˛n;j
˛n / D 0;
n;j it follows from the uniform L2 bound on Q ˛ and ˛n that Z 1 x y 1 n;j n ˛ /.t; x/ D @Re. Q ˛ Re. Q ˛n;j ˛n /.t; y/ dy 2 jx yj2
(7.44)
is a smooth function relative to x on the support of j with uniform L1 bounds on the derivatives (uniform here means relative to large n). Indeed,
j r ˛ 1 @Re. Q n;j n /.t; x/ 1 C˛ . Q n;j n /.t; x/ 2 C˛ E x ˇ ˇ ˇ ˇ Lx Lx (7.45) where E governs the energy uniformly in t . It turns out that we can also incorporate one time derivative into these bounds (but not necessarily any higher regularity in time). This follows from the div-curl system for ˛ , see (1.6)–(1.9). Indeed, if ˛ ¤ 0 then plugging (1.6) into Z x y 1 @0 Re. Q ˛n;j ˛n /.t; y/ dy @ t 1 @Re. Q ˛n;j ˛n /.x/ D 2 jx yj2 leads to an expression which is of the schematic form Z Z x y Q n;j 2 x y n;j n Q @ Re. /.t; y/ dy C . / ˛ 0 0 jx yj2 jx yj2
. n /2 .t; y/ dy:
Integrating by parts in the first integral moves the derivative from the ’s onto the kernel which allows for the same estimate as in (7.45). As for the second integral on the right-hand side, one has
h Z x y i
˛ Q n;j /2 . n /2 .t; y/ dy .
rx j
1 Lx jx yj2
n;j 2
C˛ . Q / . n /2 .t / L1 C˛ E x
7.2 Control of wave maps via a fixed L2 -profile
265
as desired. If ˛ D 0, then one uses (1.9) to arrive at the same conclusion. This establishes our claim concerning the hypotheses of Lemma 7.23; in fact, we obtained stronger conclusions as far as the conditions for large x or small are concerned. Now let us consider the cut-off function j with unbounded support, which we may assume is 0 . We can arrange the partition of unity so that 0 .t; x/ D 00 .x/ 01 .t / with 01 smooth and supported in . 1; 1/ and with 1 00 smooth and compactly supported in R2 . With e n0 defined as above, we now claim that 1 Q n;j n /.t;x/ 01 .t/ e n0 .t; x/ D 01 .t / 1 00 .x/ e i @Re. satisfies the requirements of Lemma 7.23 with a constant A that is controlled uniformly in n. First, 01 .t / 00 .x/ Re. Q ˛n;j
˛n /.t; x/ D 0
which shows as before that e n0 .t; x/ is smooth in x with derivatives that are uni1 formly bounded in Lx relative to n. In addition, the same arguments involving the div-curl system allow us to place one @ t on e n0 .t; x/ without destroying these conclusions. As for the asymptotic behavior in x ! 1 and ! 0, one simply expands x y x y D C O. / jx yj2 jxj2 jxj2 inside the integral in (7.44) which is sufficient due to jyj . R0 . In conclusion, by Lemma 7.23 and Remark 7.24 X X k ˛n kS. T0 ;T0 / ke jn en;j C.e jn /ken;j ˛ kS. T0 ;T0 / ˛ kS j
j
X
C.e jn /C "0
j
is finite uniformly in n.
The preceding corollary concludes the proof of Lemma 7.10 up to the assertion about the frequency envelope at the end. This will be proved in Chapter 9.5. We close this section with an important strengthening of the bound on L from Lemma 7.6. More specifically, we prove that the intervals Ij can be chosen in such a way that the estimate (7.5) only depends on the energy of . This will play an important role later on. In order to achieve this property, we require an improvement over Lemma 7.4. We begin with the following technical statements which allow us to make a better choice of the intervals Ij in the proof of Lemma 7.6.
266
7 Some basic perturbative results
Lemma 7.26. Let k kS < C0 and "0 > 0 be arbitrary, with defined on R2C1 . Then there exists a partition of R into intervals fIj gjMD1 which depend on but with M D M."0 ; C0 / and which satisfy X
Pk . jrj 1 2 / 2 max 1 "0 2 P 1j M
L t .Ij IH
k2Z
2/
where r D rx and jrj 1 2 is schematic notation which stands for any one of the nonlinearities appearing on the right-hand side of the div-curl system (1.12), (1.13). Proof. It of course suffices to show that X
Pk
1
jrj
1
.
2
3/
2
L2t .RIHP
k2Z
We begin with the case where
Pk .I c
2
2
.
is replaced by I c
Then by the usual trichotomy, X
2 k Pk . 1 jrj 1 .I c
3
1 2/
3 / L2 t;x
k
. 22 k
3 Y
k
2 i kS :
(7.46)
i D1 2
3.
It is easy to see that
2 kS k 3 kS :
2
3 // L2 t;x
2
1
P
k2Z
.
X
2
k
Pk
1
jrj
5 .I
c
2
2
3 / L2 t;x
k2Z
C
X
2
k
X
Pk PQ`
k2Z
C
X k2Z
1 jrj
1
P` .I c
2
2
/ 3 L2 t;x
`>k
2
k
P
5
1 jrj
1
Pk .I c
2
3 Y
2
/ . k 2 3 L t;x
2 i kS :
i D1
Hence, we may assume that the two inner inputs are both hyperbolic, i.e., i D Qki i for i D 2; 3. Now implement the Hodge decomposition for the inputs of jrj 1 . 2 /, i.e., write ˛ D R˛ C ˛ : We begin by considering the resulting trilinear expressions, more specifically the one where the inner null-form is hyperbolic: Suppressing the indices on for
7.2 Control of wave maps via a fixed L2 -profile
267
simplicity, X
1
kPk . jrj
I Q˛ˇ . ; //k2 2
1 2/
L t .RIHP
k2Z
C k k6S
(7.47)
where Q˛j is the null-form from Definition 4.16. As usual, this splits into the high-low, high-high, and low-high cases: X
2
k
jrj
Pk
1
2 I Q˛ˇ . ; / L2
t;x
k2Z
.
X
2
k
Pk PQk jrj
1
P
5 I Q˛ˇ .
2 ; / L2
t;x
k2Z
C
X
2
k
X
Pk P` PQ` jrj
k2Z
C
X
1
2 I Q˛ˇ . ; / L2 t;x
`>k
2
k
Pk P
5
PQk jrj
1
2 I Q˛ˇ . ; / L2
t;x
k2Z
DW A C B C C: Next, one writes A A1 C A2 C A3 reflecting the high-high, high-low, and low-high decomposition of the Q˛ˇ -null-form. Thus, by Lemma 4.17, A1
X
2
k2Z
k
X
X
k0 k0 5
2
Pk PQk jrj 1 Pk I Q˛ˇ .Pk ; Pk / 2 0 2 3 L t;x X X X . 2 k kPQk kL1 L2x t
k2Z
k0 k0 5
2
Pk I Q˛ˇ .Pk ; Pk / 2 0 2 3 L t;x X X X . 2 k kPQk k2L1 L2 k2Z
t
x
k0 k0 5
kPk2 kS Œk2 kPk3 kS Œk3 . k k6S :
2
2k0
k2 2
268
7 Some basic perturbative results
Similarly, by Lemma 4.23, X X A2 2 k k2Z
.
X
X
k2 CO.1/Dk0
2
k2Z
k
Pk PQk jrj X
1
2 Pk0 I Q˛ˇ .Pk2 ; Pk3 / L2 t;x X
k2 CO.1/Dk0
2 kPQk kL1 L2x kPk0 I Q˛ˇ .Pk2 ; Pk3 /kL2 t;x t X X X k Q 2 . 21 "/k3 "k0 2 . 2 kPk kL1 L2 2 t
k2Z
x
k2 CO.1/Dk0
kPk2 kS Œk2 kPk3 kS Œk3
2
. k k6S : This concludes the bound on A since A3 is of course symmetric to A2 . Next, with B B1 C B2 C B3 via the same trichotomy, X X X B1 . 2 k 2k kP` kL1 L2x 2 ` t
k2Z
`>k
k2 Dk3 CO.1/>` 5
2
PQ` I Q˛ˇ Pk ; Pk / 2 2 2 3 L t Lx X X X . 2 k 2k kP` kL1 L2x 2 ` t
k2Z
`>k
k2 Dk3 CO.1/>` 5
2`
k2 2
kPk2 kS Œk2 kPk3 kS Œk3
by Lemma 4.17, whereas X X B2 . 2 k 2k kP` kL1 L2x 2
`
2
. k k6S
X
t
k2Z
`>k
`Dk2 CO.1/>k3 5
2
PQ` I Q˛ˇ .Pk ; Pk / 2 2 2 3 L t Lx X X X . 2 k 2k kP` kL1 L2x 2 ` t
k2Z
`>k
`Dk2 CO.1/>k3 5 1
2. 2
"/k3 "k2
2
kPk2 kS Œk2 kPk3 kS Œk3
2
. k k6S
7.2 Control of wave maps via a fixed L2 -profile
269
by Lemma 4.23. The low-high case of (7.47) is treated in an analogous fashion and we skip it. Next, we treat the case where the inner null-form is elliptic. Then the desired bound reads X 6
Pk jrj 1 I c Q˛ˇ . ; / 2 (7.48) 1 Ck k : S 2 P L t .RIH
k2Z
2/
As before, A A1 C A2 C A3 reflecting the high-high, high-low, and low-high decomposition of the Q˛ˇ -null-form. We will first exclude the contributions by opposing high-high interactions in the null-form, cf. Remark 4.20. Hence, by (4.55) without the hk1 ki2 loss, X X X A1 . 2 k k2Z
.
X
k0 k0 5
2
k
k2Z
.
X
2 jrj 1 Pk0 Q>k0 Q˛ˇ .Pk2 ; Pk3 / L2 t;x X
Pk PQk X
k0 k0 5
2
k
2
Pk k 1 2 kPk Q>k Q˛ˇ .Pk ; Pk / 2 0 0 2 3 L t Lx L t;x X X
k2Z
k0 k0 5
kPk kL1 L2x 2
k0 2
t
kPk2 kS Œk2 kPk3 kS Œk3
2
. k k6S :
For A2 one proceeds similarly, using Lemma 4.24 instead. In fact, due to the hyperbolic nature of 1 ; 3 , X X X A2 . 2 k k2Z
.
X
k2 CO.1/Dk0
2
k
X k2Z
jrj
1
2 Pk0 I c Q˛ˇ .Pk2 ; Pk3 / L2 t;x X kPk kL1 L2x t
k2Z
.
Pk PQk X
k2 CO.1/Dk0
2
k
2
Pk QQ k Q˛ˇ .Pk ; Pk / 2 0 0 2 3 L t;x X X k2 CO.1/Dk0
270
7 Some basic perturbative results
kPk kL1 L2x 2
k3 2
kPk2 kS Œk2 kPk3 kS Œk3
t
2
. k k6S :
This concludes the high-low case A. In the high-high case we write B B1 C B2 C B3 as before. Therefore, B1 .
X
k
2
X
`
X
t
k2Z
.
2k kP` kL1 L2x 2
`>k
XX k2Z
k2 Dk3 CO.1/>` 5
k
`
2 2 kP` kS Œ` 2
`>k
2
PQ` Q>` Q˛ˇ .Pk ; Pk / 2 2 3 L t;x X ` 22
k2 Dk3 CO.1/>` 5
kPk2 kS Œk2 kPk3 kS Œk3
2
. k k6S by Lemma 4.19, whereas
B2 .
X
2
k
X
2k kP` Qj kL2 2
`
X
t;x
k2Z
`>k
`Dk2 CO.1/>k3 5
PQ` QQ ` Q˛ˇ .Pk .
XX k2Z
`>k
k 2
2 kP` kS Œ` 2
`
X
2
2
2
; Pk3 / L2 L2 t
x
k3 2
`Dk2 CO.1/>k3 5
kPk2 kS Œk2 kPk3 kS Œk3
2
. k k6S
by Lemma 4.24 which finishes the analysis of B. We again leave the low-high case to the reader. It remains the bound the contributions by the opposing high-high waves in the inner null-form. Returning to the 1 ; 2 ; 3 notation, we may assume that i D Qki i for i D 2; 3 and that there is an angular separation of the Fourier supports of 1 and 2 , say (since the Fourier supports of 2 ; 3 make a large angle). Hence we may bound the missing contribution to A1 as follows, where we ignore the null-form and replace the outer jrj 1 with a weight by the usual
7.2 Control of wave maps via a fixed L2 -profile
271
convolution logic: X
A1 .
k
2
X
k2Z
X
2
k0
k0 k0 5
h
X c2Dk;k0
Pc
1 Pk0 .Pk2
2 Pk3
2
3 / L2 t;x
i 21 2
k
We now invoke (2.30) to conclude that
X
Pc
c2Dk;k0
2 1 2
3 / L2 t;x
kPc
1 Pk2
2 Pk3 kL 2 1 L
kPc
1 Pk2
2 kL k2L1 L2 2 kPk3
k
. 2k0
X c2Dk;k0
. 2k0 .2
2 P k3
1 Pk0 .Pk2
3k0 2
c2Dk;k0
k
hk0
X c2Dk;k0
.2
3k0 2
hk0
ki
x
k
X
ki
t
12
3 Y
kPki
kPc
t;x
t
2 1 kS Œk kPk2
12
x
k2S Œk2
12
kPk3 kL1 L2x t
k
i kS Œki :
i D1
The loss of hk0 ki here is due to the usual issue of wave-packets which are too thick resulting in the need for Lemma 2.4. Inserting this into the bound on A1 yields A1 .
X X k2Z
k0
2
k0 k 2
hk0
kikPk
1 kS Œk
2
k
2 2 kS k
3 3 kS .
3 Y
k
2 i kS
i D1
as desired. The opposing high-high contributions to the other terms are similar and omitted. We still need to control the contributions from the elliptic terms , leading to higher order nonlinearities. This is again done in the appendix. We can now state the refined version of Lemma 7.6 which gives better control over the linear wave L . As in that lemma are the gauged components of an admissible wave map locally on some time interval ΠT0 ; T1 .
272
7 Some basic perturbative results
Corollary 7.27. Let k kS < C0 , with defined on R2C1 . Given "0 > 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
.j / NL ;
C
.j / L
D0
which satisfies X
Pk
.j /
2 NL S Œk.Ij R2 /
< "0
(7.49)
k2Z
rx;t
.j /
1P L Lt H
1 1
. "0 4 .E C 1/E
(7.50)
where the implied constant in the last inequality is universal and E D k .t /k2 is the conserved energy. In particular,
.j /
.j /
P 1 1 NL S.I R2 / rx;t L H j
by choosing "0 small enough depending on the energy. Proof. We first prove (7.49) and (7.50) by following the strategy of the proof of Lemma 7.6; however, we use Lemma 7.26 instead of Lemma 7.4 when the underlying time interval is small. More precisely, consider the frequency component P0 ˛ . Case 1: The underlying time interval I0 WD . T0 ; T1 / satisfies jI0 j < "1 with an "1 that is to be determined. The main property of this parameter is that it can be chosen to be an absolute constant independently of C0 . The ˛ satisfy the system (1.12)–(1.14). Schematically, this system takes the form @ t P0 j D @j P0 0 C P0 r 1 . 2 / ; j D 1; 2 @ t P0
0
D
2 X
@j P0
j
C P0
r
1
.
2
/
j D1
where the nonlinearity is written schematically. Now define the linear wave P0 to be P0 L;j WD S.t / P0 j .0/; @j P0 0 ; j D 1; 2 P0
L;0
WD S.t / P0
0;
2 X j D1
P0 @j
j .0/
;
L
7.2 Control of wave maps via a fixed L2 -profile
whereas P0 P0 P0
j .t/
NL;˛
WD P0
Thus, for j D 1; 2, Z t Z t P0 @j P0 0 .s/ ds C P0 r j .0/ C
D P0
kP0
P0
˛
L;˛ .
0
L;j .t/ D P0
and similarly for
273
j .0/ C tP0 @j
0,
NL .t/kL2x
1
2
.
/ .s/ ds
0
2 0 .0; / C OL2x .t /
whence for all t 2 I0 , 2
t kP0 .0/kL2x . t 2 kP0 .0/kL2x
Z t
C P0 r 1 . 0 1 C jtj 2 P0 Πr 1 .
2
/ .s; / ds
L2x
2 /
L2t;x
:
In other words, kP0
2 NL kL1 t .I0 ILx /
1 . "21 kP0 .0/kL2x C "12 P0 r
1
.
2
1
2
/ L2 .I0 IL2 / : x
t
As in the proof of Lemma 7.6 one concludes from this that kP0
NL kS Œ0.I0 R2 /
1
. "21 P0 .0/ L2 C "12 P0 r x
.
/ L2 .I0 IL2 / : x
t
Rescaling this bound to general 2k yields the following. Suppose jI j "1 2 Then kPk
NL kS Œk.I0 R2 /
1
. "21 Pk .0/ L2 C "12 2 x
k 2
P0 r
1
2
.
k.
/ L2 .I0 IL2 / : x
t
Now provided I0 Ij where fIj gjMD1 , Ij D Ij .Q"0 ; /, M D M.Q"0 ; C0 / are the intervals constructed in Lemma 7.26, one concludes that X
2 kPk NL k2S Œk.I0 R2 / . "41 .0/ L2 C "Q0 "1 (7.51) x
kWjI0 j"1 2
k
1
where "Q0 is a separate smallness parameter. We now pick "1 WD "04 .1 C E/ 3 4
1
and
"Q0 WD "0 where E D k .t /kL2x is the conserved energy of (for this one needs to remain on the interval on which equals the gauged derivative components of a wave map). This renders the right-hand side of (7.51) less than "0 . As already explained in the proof of Lemma 7.6, we will use this analysis also in the case of large intervals to which we now turn. However, in that case the ˇ estimates obtained here allow one to control the term k ˇŒ T0 ;T0 kS in (2.74) of Section 2.5.
274
7 Some basic perturbative results
Case 2: The underlying time interval I0 D . T0 ; T1 / satisfies jI0 j > "1 where "1 is as in Case 1 (again for the P0 frequencies). Here the analysis of Case 2 of 0 Lemma 7.6 applies verbatim, leading to intervals fIj0 gjMD1 , with M 0 D M 0 ."0 ; C0 / such that X max kPk F˛ k2N Œk.I R2 / < "0 1j M 0
where F D
P
˛
j
k2Z
F˛ stands for the right-hand side of (1.14) as usual.
Now we take the intersections of the intervals Ij and Ik0 which appeared in Cases 1 and 2 above. Denote this collection again by fIj gjMD1 with M D M.Q"0 ; "0 ; C0 /. .j / L
Fix such an Ij . Given k 2 Z, we define Pk
to be the free evolution of .j /
.I /Œt0 where t0 2 Ij is the center of Ij , whereas Pk NL is everything else. By our construction, X
Pk .j / 2 NL S Œk.I R2 / "0 : j
kWjIj j"1 2
k
Combining this with (7.51) this bound implies (7.49). As for the linear wave we note that those k which belong to Case 1 yield .j /
L S Œk.Ij R2 /
Pk
.j / L ,
. kPk k2
with an absolute implicit constant, whereas (2.74) from Section 2.5 yields the bound
Pk .j / . "1 1 kPk k2 : L S Œk.I R2 / j
These estimates imply (7.50).
Remark 7.28. Note that if we a priori work on a time interval Ij of infinite length, the statement of the corollary may be strengthened to
rx;t
.j /
1P L Lt H
1
.E
with universal implied constant. Indeed, in this case, the “time averaging” around the initial data does not cost a large constant. Later we shall need to following corollary which further specifies the Fourier support of L .
7.2 Control of wave maps via a fixed L2 -profile
D Q C M where for some b
Corollary 7.29. Let k kS < C0 . Assume that k M kS C kP.
1;bc
275
Q kS < ı1
1 for some small ı1 . Then there exist intervals fIj gjMD1 as in Corollary 7.27 so that on each Ij one has a decomposition
jIj D
.j / L
C
.j / NL ;
.j / L
D0
.j / .j / .j / .j / .j / .j / .j / where furthermore L D Q L C M L and NL D Q NL C M NL where both Q L .j / .j / .j / and M L are free waves satisfying (7.50) and both Q NL and M NL satisfy (7.49). Furthermore,
.j /
M C P. 1;bc Q .j / . ı1 L L S S
.j /
M C P. 1;bc Q .j / . ı1 NL S NL S
with an absolute implicit constant. Proof. The proof of this statement follows the exact same lines as the proof of the previous corollary. The only difference is that each nonlinearity needs to be split into the contributions made by M and Q , respectively.
8 BMO, Ap , and weighted commutator estimates
In this section we develop some auxiliary tools that will be needed in the implementation of the Bahouri–Gérard theory for wave maps. More specifically, due to the lack of an embedding from energy to L1 in the critical case we need to invoke methods involving BMO and the closely related Ap -classes in order to carry out Steps 1 and 2 of the program delineated in Chapter 1. Lemma 7.19 will play a crucial role here. Moreover, we require a weighted version of the Coifman– Rochberg–Weiss commutator theorem, with the weights belonging to the Ap class. Although it does not seem to be widely known, it is an easy consequence of the standard theory and we sketch the proof for the sake of completeness. The paper [40] contains a more general form of this result. A Calderón–Zygmund kernel here is defined to be any linear operator T bounded on L2 with the additional property that for any f 2 L2 with compact support and all x 62 supp.f /, Z Tf .x/ D K.x; y/f .y/ dy where jK.x; y/j C jx jK.x; y/ jK.x; y/
yj
d
and for some 0 < 1,
jx x 0 j ; jx yjd C jy y 0 j K.x; y 0 /j C ; jx yjd C K.x 0 ; y/j C
8 jx
yj > 2jx
x0j
8 jx
yj > 2jy
y 0 j:
By the Calderón–Zygmund theorem, any such T is also bounded on Lp .Rd / provided 1 < p < 1. Lemma 8.1. Let 1 < p < 1. There exists ı D ı.p/ > 0 with the following property: suppose D 0 C 1 where k 0 kBMO.Rd / < ı and k 1 kL1 .Rd / A. Then
e ŒT; be C.d; A; T; p/kbkBMO (8.1) p!p for any Calderón–Zygmund operator T and b 2 BMO. Moreover, inf ı.p/ > 0 and sup C.d; A; T; p/ < 1
p2I
p2I
278
8 BMO, Ap , and weighted commutator estimates
for any compact I .1; 1/. Proof. Since 1 contributes at most e 2A to the estimate, we can assume that D 0 with small BMO norm. In particular, e 0 2 Ap . We will require the following inequality involving the so-called sharp maximal function M ] f which is defined as Z ˇ ˇ ] 1 ˇf .y/ c ˇ dy .M f /.x/ D sup inf jQj QWx2Q c
Q
where c is a constant. The optimal choice of c is c D fQ WD jQj 1 The estimate then reads (see Theorem 7.10 in [8]) Z Z p .Mf / .x/ w.x/ dx C .M ] f /p .x/ w.x/ dx Rd
R
Q
f .y/ dy.
(8.2)
Rd
for any w 2 A1 with a constant that only depends on the dimension and the constants in (7.26). To avoid trivialities like f D const for which (8.2) fails, one needs to assume Mf 2 Lp0 .Rd / for some 1 p0 p. The proof of (8.1) combines the standard proof of the unweighted Coifman– Rochberg–Weiss bound with the sharp function estimate (8.2). More precisely, fix a cube Q and write ŒT; bf D .b bQ /Tf C T .b bQ / 2Q f C T .b bQ / Rd n2Q f DW AQ C BQ C CQ : To bound M ] .ŒT; bf /, we simply note that for any x 2 Q, and any 1 < s < 1, Z 1 jQj .jAQ .y/j C jBQ .y/j/ dy Q
1 1 C.s; d; T /kbkBMO .M jTf js / s .x/ C .M jf js / s .x/ : Indeed, for A this follows from Hölder’s inequality and the definition of BMO, whereas for B we also invoke the Lq boundedness of T for some 1 < q < s. For CQ we let yQ be the center of Q and estimate for any y 2 Q, Z jCQ .y/ CQ .yQ /j jK.y; z/ K.yQ ; z/k.b bQ /.z/kf .z/j dz Rd n2Q Z jy yQ j C j.b bQ /.z/kf .z/j dz yQ jd C Rd n2Q jz 1
C kbkBMO inf .M jf js / s .x/ x2Q
279
8 BMO, Ap , and weighted commutator estimates
where > 0 is as above. In conclusion, 1 1 M ] .ŒT; bf / C.s; d; T /kbkBMO .M jTf js / s C .M jf js / s :
The lemma follows from (8.2) and the weighted Lp boundedness of M and T . We now apply this to prove the following lemma, which will be important in the implementation of the Bahouri–Gérard decomposition for wave maps. Instead of a general Calderón–Zygmund operator, we restrict ourselves to the subclass of Mikhlin multiplier operators which are of the form Tf D .mfO/_ with m 2 C 3 .R2 n f0g/ and with jD ˛ m./j C.˛/jj
j˛j
;
8 2 R2 n f0g
for all j˛j 3. For simplicity, we also limit ourselves to two dimensions. 1 2 Lemma 8.2. Suppose f'n g1 nD1 ; f n gnD1 lie in the unit-ball of L . Furthermore, assume that 2 k0 c supp.'bn /; supp. jj 2k1 g n / f 2 R W 2 1
for arbitrary k0 < k1 4 and let vn WD e . / 2 'n . Then
j k
Pj v 1 T . n vn / . min.2k1 j ; 2 3 0 /; n 2 provided either j k0 or j k1 . Proof. By Lemma 7.20, for any 1 < p < 1 one has supn supjt j1 Ap .vnt / C.p/. Set R D 2k1 and r D 2k0 . If j > k1 , then
Pj v 1 T . n vn / . 2 j r v 1 T . n vn / n n 2 2 j .2 k n k4 k'n k4 C kr n 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 k n 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 f n 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, k n k2 C k'n k2 C k
n k2
A<1
for all n 1. By Lemma 7.20 one has vn 2 Ap and vn wn 2 Ap for all 1 < p < 1 with Ap constants depending only on A and p. Now fix " > 0 arbitrarily small. Then there is R > 1 so that Z lim sup j'On ./j2 d < "2 : n!1
Œjj
1;
jj>R
281
8 BMO, Ap , and weighted commutator estimates
Fix an R D R."/ with this property. Define '1n WD . ŒR 1 ;R 'bn /_ , '2n WD 'n _ c '1n and 1n WD . ŒR 1 ;R 1n . Then k'2n k2 C k 2n k2 < " n / , 2n WD n for large n whence
.vn wn / 1 T . 2n vn wn / C v 1 T . 2n vn / C.A; T /" n 2 2 as well as k. /
1 2
'2n kBMO < C ". Next, define 1 j D 1; 2: vj n WD exp . / 2 'j n ;
By Lemmas 7.20 and 8.1,
.vn wn / 1 T . 1n vn wn / .v1n wn / 1 T . 1n v1n wn / 2 Z 1
1
.wn v1n v t / 1 ΠT; . / 2 '2n . 1n wn v1n v t / dt 2n 2n 2 0
C.A; T /k. /
1 2
By the same argument,
1
v T . 1n vn / n
'2n kBMO C.A; T /":
v1n1 T . 1n v1n / 2 C.A; T /":
Similarly, set 2n
WD . Œ
1 ;
cn /_ ;
1n
WD
n
where > 1 will be determined later. By assumption, k In particular,
. / 12 2n !0 BMO
2n 2n k2
! 0 as n ! 1.
as n ! 1. Applying Lemma 8.1 as before allows one to remove the weights w2n from (8.3) where 1 j D 1; 2: wj n WD exp . / 2 j n ; Hence, we are reduced to establishing that
.v1n w1n / 1 T . 1n v1n w1n / v 1 T . 1n v1n / C.A; T /" 1n 2
(8.4)
for sufficiently large n. For ease of notation, we shall now drop the subscript 1 1 c from '1n etc. with the understanding that 'bn and n are supported on ŒR ; R and that cn is supported off Œ 1 ; where > 1 is a large number depending on " to be chosen later. Define n;low
WD . .0;
1
cn /_ ;
n;high
WD . Œ;1/ cn /_
282
8 BMO, Ap , and weighted commutator estimates
and write, correspondingly, wn D wn;low wn;high . It is easy to remove wn;high :
.vn wn / 1 T . n vn wn / .vn wn;low / 1 T . n vn wn;low / 2 Z 1
1 t 1
vn wn;low w t
2 n;high . n vn wn;low wn;high 2 dt n;high / ΠT; . / 0
C.A; T / . / C.A; T; R/
1 2
1 2
n;high 4 k n k4
(8.5)
";
t provided is sufficiently large. Here we used that vn wn;low wn;high are A2 weights uniformly in 0 t 1 as well as an interpolation between L2 and BMO to pass to the last line. For the final bound we need " 2 . To remove wn;low we split
T D P<
T
CP
<< T
C P> T
where 2 R " and P< etc. denote Littlewood–Paley projections. Introducing an angular decomposition into finitely many sectors, we may assume that j1 j jj=10 on the support of m. Then for large , and with WD 2 ,
.vn wn;low / 1 P> T . n vn wn;low / 2
1 1
C .vn wn;low / @ P> T .@1 Πn vn wn;low / 1
2
C.A/ k@1 . n vn /k2 C k n @1 . /
1 2
n;low k2
":
For the small frequencies P< T we first recall the following standard fact: with a suitable Schwarz function, P<
gP< f .x/ 2 Z 1Z X 2 .y/yj f .x
.fg/
D
D
j D1 0 2 X
Lj; .
R2 1
y/@j g.x
sy/ dyds
f; @j g/
(8.6)
j D1
where Lj; in the final line denotes a multi-linear expression of the form Z L.f; g/.x/ D
f .x R4
u/g.x
v/ .du; dv/
(8.7)
283
8 BMO, Ap , and weighted commutator estimates
with a measure of mass bounded by some constant (in this case uniformly in all parameters). Using this notation, one has (since kvn 1 k1 C.A; R/)
.vn wn;low /
1
P< T . n vn wn;low / 2
1
C.A; R/ wn;low T .P< . n vn /wn;low / 2 1
C C.A; R/
2 X
w
1 n;low T
Lj; . n vn ; wn;low @j . /
1 2
n;low /
2
j D1
DW In C IIn : To bound In , note that since we may take R 1 , one has P< . n vn / D P< . n .vn 1//. Hence, by the boundedness of T relative to the weight wn;low and Bernstein’s inequality,
kIn k2 P<
D P<
. n vn / 2
Ck n k2 kvn
. n .vn
1// 2 C P<
. n .vn
1/ 1
1k2 "
for 1. Bounding IIn requires more care, as one cannot naively remove the weights as we did in the bound for In . In fact, due to the translation by sy in (8.6) we are in a situation with two different weights, namely wn;low and wn;low . sy/. This means that the usual Ap weight theory applied to IIn does not simply cancel the operator T and the two weights, but rather cancels the outer weight, the operator T and the weight inside T is replaced by an expression of the form 1 wn;low . sy/wn;low ./: Thus, in view of (8.6) and these considerations, IIn is bounded by (using kvn k1 C.A; R/) C.A; R/
2
Z
Z
R2
0
1
2 j .y/j jyj
1 1 sy/wn;low ./r. / 2 n;low . sy/ L2 dyds Z Z 1 ˇ ˇ 1 C.A; R/ 2 k n k2 kr. / 2 n;low k1 2 ˇ .y/ˇ jyj R2 0
wn;low . sy/w 1 ./ 1 dyds n;low L Z C.A; R/ 2 1 2 .1 C jyj/ N .1 C jyj/k.A/ dy " (8.8)
n .
y/wn;low .
R2
284
8 BMO, Ap , and weighted commutator estimates
for some constant k.A/ > 0 provided we choose such that 2 " 1 . To pass to the bound in (8.8), assume first that jyj 1. Then with hn WD 1 . / 2 n;low so that wn;low D e hn , wn;low .x
1 y/wn;low .x/ exp jyjkrhn k1 exp C
1
1
eC " 2
where we used that krhn k1 kr. /
1 2
n;low k1
C
1
:
This implies that on scales 1 , the weight wn;low is essentially constant (up to multiplicative constants). Next, observe that for all cubes ˇ ˇ.hn /Q
ˇ .hn /2` Q ˇ C khn kBMO ` C.A/`;
Hence, partitioning R2 into cubes of side-length
1
one obtains that
ˇ hn .y 0 /ˇ C.A/ log 2 C jy
ˇ ˇhn .y/
8 ` 0:
y 0 j ;
(8.9)
whence sup wn;low .x x
k.A/ 1 y/wn;low .x/ C.A/ 1 C jyj
as claimed. Note that the previous estimates on P<
1
v P< n
T
and P> T also prove that
C v 1 P> T . n vn / ": n 2
T . n vn / 2
Therefore, it remains to prove that
.vn wn;low /
1
T . n vn wn;low /
vn 1 T . n vn / 2 "
where T WD P
<< T
(8.10)
is the operator on intermediate frequencies. Since Tf D .mfO/_ with m 2 C 3 .R2 n f0g/, we conclude that Z P<< 1 Tf .x/ D
K .x
y/f .y/ dy
8 BMO, Ap , and weighted commutator estimates
285
1
with jK .x/j C./.1 C jxj/ 3 . Now, with hn D . / 2 n;low as above, and M denoting the Hardy–Littlewood maximal operator,
.vn wn;low / 1 T . n vn wn;low / v 1 T . n vn / n 2 Z 1
1 t
.vn w t
n;low / ŒT ; hn . n vn wn;low / 2 dt 0 Z 1
1 1 1 t
.vn w t
4 4: / M. v w / C.A; / n n n;low n;low 2 dt C.A; / 0
Here we used that the kernel of ŒT ; hn is of the form K .x; y/.hn .x/ hn .y// and satisfies the bounds, cf. (8.9), ˇ ˇ ˇK .x; y/ hn .x/ hn .y/ ˇ C.A; / min 1 jx yj; jx yj 3 log.2Cjx yj/ ; whence ˇ ˇ ˇŒT ; hn f .x/ˇ C.A; /
1 4
Mf .x/:
Taking sufficiently large (depending on ", R, and A) finishes the proof of (8.3). Lemma 8.2 now implies that vn 1 T . n vn / is 1-oscillatory. The following statement will be an essential technical tool for the Bahouri– Gérard method in the context of wave maps into hyperbolic space. As before, T is a Mikhlin multiplier operator. 2 2 Corollary 8.4. Let ffn g1 nD1 L .R / satisfy supn1 kfn k2 A < 1 and ˚ 1 1 define yn D exp . / 2 fn . Let j WD n;j nD1 be sequences of positive numbers for each 1 j J with the property that n n;j 0 o n;j lim C !1 (8.11) n!1 n;j 0 n;j
for any 1 j ¤ j 0 J . Assume further that fn D
J X
'n;j C !n
j D1 1 2 2 where f'n;j g1 nD1 L .R / is j -oscillatory for each 1 j J , f!n gnD1 is j -singular for every 1 j J , and supn1 k!n kBP 0 < ı. ˚ 1 ˚2;1 1 Then yn 1 T .'n;j yn / nD1 is j -oscillatory, yn 1 T .!n yn / nD1 is j singular for each 1 j J , and
lim sup yn 1 T .!n yn / BP 0 < C.A; T /ı (8.12) n!1
2;1
286
8 BMO, Ap , and weighted commutator estimates
where the constant C.A; T / only depends on A and T . Proof. Define 1 2
vn;j WD exp . /
'n;j ;
wn WD exp . /
1 2
!n
˚ 1 Q so that yn WD wn jJD1 vn;j . By Lemma 8.3, both vn;j1 T .'n;j vn;j / nD1 and ˚ 1 1 yn T .'n;j yn / nD1 are j -oscillatory. Now suppose f n g1 nD1 is an arbitrary j -oscillatory sequence where 1 j J is fixed. Then !n WD yn 1 T .!n yn / satisfies ˝ ˛ h!n ; n i D !n ; yn T . n yn 1 / : By Lemma 8.3, fyn T .
n yn
1 /g1 nD1
is j -oscillatory, whence
lim h!n ;
n!1
ni
D 0:
˚ 1 Therefore, !n nD1 is j -singular for each 1 j J . For the proof of (8.12), we first note that passing to a subsequence if necessary, (8.11) implies that we may assume that n;1 > n;2 > : : : > n;J for all large n, whence for any 1 j J
1
n;j !1 n;j C1
(8.13)
as n ! 1. We also note that J X
k'n;j k22 C k!n k22 A2 C o.1/
j D1
as n ! 1. Now we let m 10 and K 10 be integers (to be determined later) and define e ' n;j WD 'n;j Œ2 e y n WD
J Y
m jrj
exp . /
m n;j 2 1 2
e ' n;j
j D1
e ! n WD !n R2 nSJ
j D1 Œ2
Km jrj
Km n;j 2
287
8 BMO, Ap , and weighted commutator estimates
where the multipliers involving r need to be interpreted on the Fourier side. As in the proof of Lemma 8.3,
lim sup yn 1 T .!n yn / yQn 1 T .e ! n yQn / 2 < C.A; T / ı; (8.14) n!1
provided m is chosen large enough and irrespective of the choice of K 1. We will now fix m so that (8.14) holds. It therefore suffices to show that
lim sup sup Pj ŒyQn 1 T .e ! n yQn / 2 C.A; T / ı (8.15) n!1
j 2Z
provided K is chosen sufficiently large. The idea behind (8.15) is that e ! n behaves like a lacunary series, i.e., each e ! n is the sum of functions whose Fourier supports consist of disjoint blocks which are very strongly separated. In addition, the e ' n;j are Fourier supported on intervals which are well separated from the Fourier support of e ! n . It will turn out that for each j – up to negligible errors as K ! 1 – only one block of frequencies from e ! n (namely the one containing 2j ) contributes to Pj ŒyQn 1 T .e ! n yQn / and, moreover, only those e ' n;j with frequencies much smaller than 2j matter. In this way, we can then essentially pass Pj onto e !n. To establish (8.15), we introduce some more notation: set n
WD
J X
. /
1 2
e ' n;j
j D1
and define ŒT;
.s/ n
iteratively via
ŒT;
.1/ n
WD ŒT;
n ;
ŒT;
.sC1/ n
D ŒT;
n
.s/
;
n
:
Then yQn 1 T .e ! n yQn / D
s X 1 ŒT; n .`/ e !nC `Š `D0 Z 1 1 C .1 t /s yQn t ŒT; sŠ 0
(8.16) .sC1/ .e ! n yQnt / dt: n
(8.17)
Denote the remainder in (8.17) by Rn;s . To bound it in L2 , note that k n k1 C mJA with some absolute constant C . Therefore, placing n , yQn , and yQn 1 in L1 yields for all n 1 kRn;s k2
C mJA s e C mJA .C mJA/sC1 ke ! n k2 e C mJA DW s .s C 1/Š s
288
8 BMO, Ap , and weighted commutator estimates
which clearly goes to zero as s ! 1. In particular, s ı for large s. We now turn PJto the details of the analysis of the main terms in (8.16). First, one has e ! n D j D0 e ! n;j where e ! n;0 WD e ! n Œjrj n;1 2
Km
e ! n;j WD e ! n Œ2Km jrjn;j 2
Km
81j J
;
1
e ! n;J WD e ! n Œ2Km jrjn;J with n large. Then s X 1 ŒT; `Š
.`/ e !n n
D
J X s X 1 ŒT; `Š
n
J X s X 1 ŒT; `Š
. / n;j
.`/
e ! n;j
j D0 `D0
`D0
D
.C/ .`/ e ! n;j n;j
C
(8.18)
j D0 `D0
where we have set, for each 0 j J , . / n;j
WD
X
. /
1 2
.C/ n;j
e ' n;k ;
WD
1kj
X
. /
1 2
e ' n;k :
j
We shall now show that for a given e ! n;j only the small frequency part of n;j , . / i.e., n;j , contributes significantly to the commutators in (8.18) (at least for very large K). To this end write ŒT;
. / n;j
C
.C/ .`/ n;j
D ŒT;
. / .`/ n;j
C
X Œ ŒT;
."1 / n;j ;
."2 / n;j ;
:::;
."` / n;j
"
D Kn;j;` C Rn;j;`
(8.19)
where the sum here runs over `-fold commutators with each "k D ˙, the choice "k D for all 1 k ` being excluded (as it is represented by the first – and main – term on the right-hand side). Next, observe that for each 1 ` s WD
1 .K 2 100
1/m
one has, for each 1 j J and every k 2 Z, Pk Kn;j;` e ! n;j D Pk Kn;j;` Pk
! n;j : 2<
(8.20)
289
8 BMO, Ap , and weighted commutator estimates
In fact, this vanishes unless 2Km Pk
s X 1 ŒT; `Š
.`/ e !n n
D Pk
2 1 n;j
2k 2
KmC2 1 n;j C1 .
J X s X 1 Kn;j;` Pk `Š
Writing
! n;j 2<
(8.21)
j D0 `D0
`D0
C Pk
J X s X 1 Rn;j;` Pk `Š
! n;j 2<
(8.22)
j D0 `D0
it follows from (8.13) that for all sufficiently large n n0 depending on K; m, at most one term in (8.21) can be nonzero for any choice of k 2 Z. Applying the . / decomposition (8.16) and (8.17) with n;j instead of n to (8.21) yields s J X
X 1
sup Pk Kn;j;` Pk `Š k2Z
e ! 2<
j D0 `D0
2
s
X
1
Kn;j;` Pk 2<
. / . /
sup sup e j;n T e j;n Pk 2<
k2Z 0j J
C.A; T /k!n kBP 0
2;1
C s C.A; T /ı:
To pass to the final bound, we note that s ı provided K is chosen suffi. /
ciently large. We also used that the weights e j;n 2 A2 with A2 constant CA uniformly in j; n, cf. Lemma 8.1. As for (8.22), we make the following crude estimate for the `-fold commutator as in (8.19)
Œ ŒT;
."1 / n;j ;
."2 / n;j ;
:::;
."` / ! n;j 2 n;j e C.T /.C mJA/`
It arises by placing one Bernstein’s inequality, ke ! n;j k4 C.2
.C/ n;j
Km
in L4 , all other
n;j C1 /
1 2
."i / n;j
1
.C/ n;j k4
ke ! n;j k2 CA2
1
.C/ ! n;j k4 : n;j k4 ke
in L1 , and e ! n;j in L4 . By Km=2
whereas by interpolation between the L2 and BMO bounds, k
k
2 CA2m=2 n;j C1 ;
1
n;j2 C1 ;
290
8 BMO, Ap , and weighted commutator estimates
whence
Œ ŒT;
."1 / n;j ;
."2 / n;j ;
."` / ! n;j 2 n;j e
:::;
C.T; A/.C mJA/`
1 .1 K/m=2
2
:
Hence, the error resulting from (8.22) can be made as small as we wish by taking K large and we are done. In what follows, we call sequences j RC as in Lemma 8.4 pairwise orthogonal iff they satisfy (8.11). The following auxiliary Lemma 8.5 strengthens 0 the result of Lemma 8.3 by replacing L2 with BP 2;1 , but under slightly different conditions. As before, T is a Mikhlin operator. 1 Lemma 8.5. Suppose f'n g1 nD1 ; f n gnD1 ; f thermore, assume that
1 n gnD1
c supp.'bn /; supp. n / fjj 1g;
lie in the unit-ball of L2 . Fur-
supp. cn / fjj 1g:
Define vn WD exp . /
1 2
'n ;
wn WD exp . /
1 2
n
:
Then given " > 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
n kBP 0
2;1
<ı
(8.25)
where k0 D k0 .T; "/ is some positive integer. Proof. Since (8.23) implies (8.24) it suffices to prove the former. As before, .vn wn /
1
T . n vn wn /
vn 1 T . n vn / Z 1 D .vn wnt / 1 ŒT; . / 0
1 2
t n . n vn wn / dt:
(8.26)
We now estimate the L2 -norm of this expression localized to frequency 2j . First, we consider the case j 0. Then, with yn;t WD vn wnt , and using Bernstein’s
291
8 BMO, Ap , and weighted commutator estimates
inequality, one has the bound
Pj y 1 ŒT; . / 12 n . n yn;t / n;t 2 2 1 j 1
3 2 .2 Pj r yn;t ŒT; . / n . n yn;t / 3 2
2 1 j 1 . 2 3 r yn;t ŒT; . / 2 n . n yn;t / 3 2
2 1 j 1
C 2 3 yn;t ŒT; . / 2 n r. n yn;t / 3 2
2 1 j 1
3 2 C2 yn;t ŒT; r. / n . n yn;t / 3 :
(8.27)
2
Since uniformly in 0 t 1, yn;t ryn;t1 D we can further estimate
r y 1 ŒT; . / 12 n;t
.r. /
1 2
'n C tr. /
1 2
n/
D OL2 .1/;
3 . y 1 ŒT; . / 12 n . n yn;t / n;t 6 2
1 . . / 2 n 12 k n k12 . 1:
n . n yn;t /
To pass to the final bound, the term involving n is estimated via an L2 -BMO interpolation, whereas the n term is controlled by Bernstein’s inequality. The other two terms on the right-hand side of (8.27) are estimated similarly. As for the case j 0, Bernstein’s inequality yields
Pj y 1 ŒT; . / 12 n . n yn;t / n;t 2 j 1 1
. 2 3 Pj yn;t ŒT; . / 2 n . n yn;t / 3 2
j j 1
. 2 3 . / 2 n 6 k n k2 . 2 3 : To obtain (8.23), it suffices to show that for every " > 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 Œ Œjja 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
Œ Œjja 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 ; f n 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 .y j1 /
@j .y k1 / 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 Œ k1n 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 Πk1n 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 ΠQ 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 ΠQ 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 ΠQ 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<
i
k1
i
P
P
kD1;2
kD1;2
4
4
1 @ 1n k k
; 1 @ 1n k k :
Start with the first expression which we write as P<
k1 PŒ
na k0 ;k0 ˛ e
D P<
i
P
kD1;2
h k1 P Œ
4
1@
1n k k
na k0 ;k0 ˛
X
4
1
@j PŒ
k0 ;k0
j D1;2
Πi
X kD1;2
4
1
@j @k k1n e
i
P
kD1;2
4
1@
1n k k
i
:
301
9.2 Step 1: Frequency decomposition of initial data
Using Bernstein’s inequality, we can then estimate
h X
4 1 @j PΠk0 ;k0 P
< k1 PŒ k0 ;k0 ˛na j D1;2
X
Πi
4
1
@j @k k1n e
i
P
kD1;2
1@
4
1n k k
i
L2
kD1;2
. 2 k0
k1
kPŒ
na 1n k0 ;k0 ˛ kL2 k kL2
<"
provided we choose k1 sufficiently large in relation to k0 . The estimate for the second term of the same. Next, consider the “tail term” P is more 1n 1 w˛nA e i kD1;2 4 @k k . That this is an -singular for each 1 a A follows from the preceding via duality. It therefore remains to estimate its k kBP 0 -norm. 2;1 We localize this term to fixed dyadic frequency 2q Pq w˛nA e
i
P
kD1;2
4
1@
1n k k
C Pq w˛nA PŒq
D Pq w˛nA P
i
10 e
P
kD1;2
i
P
kD1;2
1 @ 1n k k
4
C Pq w˛nA P>qC10 e
i
1@
4
1n k k
P
kD1;2
4
1 @ 1n k k
and estimate the three terms on the right separately: First, we have P
nA
Pq w P
1n 1 nA D Pq PŒq 10;qC10 .w˛ /P
nA 10;qC10 .w˛ /kL2
. kw˛nA kBP 0
. ı:
2;1
Next, P
nA
Pq w PŒq 10;qC10 e i kD1;2 4 1 @k k1n 2 ˛ L P
nA i kD1;2 4 D Pq P
D Pq P
1 @ 1n k k
L2
j D1;2
i
X
4
1
@j @k k1n e
i
P
kD1;2
4
1@
1n k k
L2
kD1;2
.2
q
kP
2;1
.ı
where Bernstein’s inequality was used in the last step. The third term in the above Littlewood–Paley trichotomy corresponding to high-high interactions, is treated analogously and omitted.
302
9 The Bahouri–Gérard concentration compactness method
For later reference, it shall be important to construct “partial approximations” of the components ˛n in terms of the ˛na . Specifically, for I f1; 2; : : : ; Ag, we let X Q ˛nI WD Q ˛na : a2I
Then reasoning exactly as in the preceding, and employing the same notation as there, one obtains the following statement. Corollary 9.6. Let ynI WD e
P
kD1;2
P
a2I
4
1@
Q 2na k k
xnI WD
;
X X
4
1
@k ΠQ k1na ynI :
kD1;2 a2I
Then for a 2 I j1na D .ynI /
1
X
4
1
@j @k Q k1na ynI C oL2 .1/:
kD1;2
In particular, we have X a2I
9.3
j1na D
@j xnI C oL2 .1/; ynI
X
j2na D
a2I
@j ynI C oL2 .1/: ynI
Step 2: Frequency localized approximations to the data
Given an essentially singular sequence un with derivatives ˛n , Proposition 9.4 yields a new essentially singular sequence ˛n with the following property: for any A 1 (recall the ˛na are defined inductively) ˛n D
A X
˛na C w˛nA :
aD1
Given ı0 > 0, there exists A 1 so that kw˛nA kBP 0 < ı0 for large n. In what 2;1 follows, we will use smallness parameters 1 "0 ı1 ı0 > 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 <<k n
(9.6)
with an WD log an and a sequence Ln ! 1 which increases very slowly. This can be done since w˛nA0 is an -singular for each 1 a A0 . Thus the frequency .k/
nA0
support of w˛ kn
1 Ln
e
is contained in the annulus
< jj < nkC1
1
e
Ln
;
0n
1
WD 0;
A
.A0 /
n 0
C1 1
WD 1:
9.3 Step 2: Frequency localized approximations to the data
305
Figure 9.1 above is a schematic depiction of the situation A0 D 1 with a unique large atom on the right, but with two smaller atoms on the left which are too large to be included in the Besov error (the three bumpy curves between the atoms). More precisely, w˛nA0 consists of the four small curves between the atoms, and .0/
nA0
w˛
is the sum of the three curves to the left of the big atom together with the nA
.1/
two small atoms, and w˛ 0 the one to the right of the big atom. Note that if we refine the frequency decomposition, i.e., increase A0 to A.k/ .k/
nA0
A0 , then the components w˛
are decomposed into
.k/
nA0
w˛
D
X
najk
˛
C w˛nA
.k/
j
for suitable ajk 2 ŒA0 C 1; A.k/ . In Figure 9.1 one has j D 1; 2 for k D 0 corresponding to the two small atoms to the left of the large one. We may again assume that the ajk are increasing in j and have frequency support with increasing value of jj, for each k. Furthermore, we have X X ˛D0;1;2 j
nak 2 lim sup ˛ j L2 < "0 x
n!1
by asymptotic orthogonality and the choice of A0 . Our first goal, to be dealt with in the following section, is to control the nonlinear evolution of the minimum .0/
nA0
frequency components w˛ energy constraint
. The idea behind this is as follows: Due to the
nA.0/ lim sup w˛ 0 2 Ecrit ; n!1
.0/ nA0
we may subdivide w˛ into finitely many pieces be means of frequency local˚ nA00 2 izations PJ` w˛ 1000Ecrit such that the dyadic intervals J` are disjoint, 1` with [` J` D . 1; .1n /
"0
1 e Ln /,
and furthermore
.0/
PJ w˛nA0 2 < "0 ` L x
Recall that .1n /
1
8`:
is the frequency scale of the first frequency atom ˛n1 . In particnA00
ular, this means that the frequency localized pieces PJ` w˛ 2
We suppress the dependence on n of the intervals J` .
should be treatable
306
9 The Bahouri–Gérard concentration compactness method
via a perturbative argument. More precisely, we shall run an induction in ` on a sequence of approximating maps with Coulomb data essentially (up to errors which can be made arbitrarily small depending on a parameter ı0 , see Lemma 9.8 below) given by .0/
nA PJj w˛ 0
X
i Re
e
P
kD1;2
1@ k
4
P
1j `
.0/ nA0
PJj wk
:
1j `
As always, we face the issue at this point that these gauged components are not necessarily admissible, i.e., they are not given by derivative components of maps R2 ! H2 . In order to apply the perturbative theory we shall need to show that they are close to such admissible data. This in turn follows from Lemma 8.5 provided we chose the intervals J` carefully; for this it is essential that the endpoints of these intervals do not fall onto one of the “small” atoms ˛na . Otherwise, condition (8.25) would be violated. In detail, this is done as follows. Recall that the Jj are chosen to be disjoint and such that .0/
nA0
w˛
D
X
.0/
nA0
PJj w˛
.0/
nA sup PJj w˛ 0 L2 . "0 :
;
x
j
j
On the other hand, upon refining the Bahouri–Gérard frequency decomposition .0/
nA0
applied to w˛
, we can also write .0/
nA0
w˛
D
X
.0/
naj
˛
.0/
C w˛nA :
(9.7)
j 1
.0/ Here A.0/ > 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.ΠT0 ;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 / <
9.4 Step 3: Evolving the lowest-frequency nonatomic part
317
This proposition is the key ingredient in the proof. It asserts that the frequency profile of at time t D 0 is essentially preserved under the evolution up to some frequency leakage, which however is controlled by the size of the underlying Besov error. What allows us to prevent energy of moving from high to low frequencies (which is the main difficulty here) are gains in the high-high-low interactions in the nonlinearities. Without these gains, there could indeed be this kind of energy transfer and the argument would break down. It is essential in Proposition 9.12 that C4 is a constant that does not change throughout the induction, whereas C2 does change. If we accept Proposition 9.12 for now, then it is an easy matter to derive the aforementioned approximate energy conservation. Corollary 9.13. Under the induction hypothesis of Proposition 9.11 and assuming the validity of Proposition 9.12, one has the following: For sufficiently small ı1 (depending on C2 and C4 ) and large n, we have X 2 k˛ kL < "0 1 2 L .I R2 / 1
x
t
˛D0;1;2
where I1 is as above. Proof. (Corollary 9.13) Due to energy conservation for the evolution of we have X 2 k ˛ C ˛ kL 2 D constant:
C ,
x
˛D0;1;2
Similarly, we have X
2 ˛ kL2
k
x
˛D0;1;2
D constant:
The crucial observation now is that k
C
2 kL 2 x
Dk
2 kL 2 x
C
2 kkL 2 x
C 2Re
XZ k2Z
R2
Pk Pk dx
on fixed time slices t D t0 2 I1 , and we can split Z XZ X XZ Pk Pk dx D Pk Pk dx C k2Z
R2
k2[j l
1 Jj
R2
C4 C22 ı1 ,
k2Jl
R2
Pk Pk dx:
Both contributions on the right are . which can be made arbitrarily small by choosing ı1 small enough. To obtain this bound, observe that the induction hypothesis and Proposition 9.12 allow one to transfer Lemma 9.10 to all times in the interval I1 . Cauchy–Schwarz then implies the bound of . C4 C22 ı1 .
318
9 The Bahouri–Gérard concentration compactness method
Corollary 9.13 allows us to keep the energy under control as we inductively pass from I1 to its successor I2 and so forth by restarting the procedure. Indeed, since the number of the “divisibility” intervals is bounded by M."0 ; Ecrit /, we can make ı1 in the corollary so small (depending on this number) and n so large that even the energy of the final is no bigger than 2"0 , say. Even though we will now work on I1 , all arguments carried out below apply to any of the later intervals I2 ; I3 ; : : : as well. Proof of Proposition 9.12. We may reduce ourselves to proving the statement for frequency 20 , i.e., k D 0, by scaling invariance. Recall that we have chosen the intervals Ij in such fashion that (9.16) holds with the stated bounds. In order to obtain the desired estimates on , we distinguish between two cases, depending on the size of the underlying time interval. If it is short, we use the div-curl system. Otherwise we use the wave equation. Case 1: jI1 j < T1 where T1 > 0 is some absolute small constant (to be specified). We shall use the div-curl system linearized around , see (1.12), (1.13), which takes the schematic form @ t D rx C r
1
.
2
/C r
1
. / C r
1
. / C r
1
. 2 / C r
1
. 2 /:
The first linear term rx on the right-hand side is estimated by bootstrap, choosing T1 smaller than some absolute constant. For each of the five nonlinear terms on the right-hand side one needs to consider two cases, depending on whether gets replaced by 1 or 2 . In light of the first part of the proof of Lemma 7.6, it suffices to prove bounds of either the form (all on the set Ij R2 and F any one of the expressions on the right above) 0 X kPk F k k2Z
P LM t H
1 M
2
CL2t HP
3 P 2 CL1 H t
2 21 2
C2 C4 ı1
where the sum is over all those frequencies k such that jIj j < T1 2 k , i.e., such that the re-scaled solution where k becomes 0 falls into Case 1, or else, we have kPk F k
1
PM LM t H
2
CL2t HP
3 P 2 CL1 H t
2
C4 dk :
We consider the frequency mode k D 0 and divide P0 F into pieces which are either substituted into the first or second bound above. (a) The term 1 r 1 . 2 /; we cannot just use Lemma 7.4 of Chapter 7, since smallness there can only be enforced by choosing T1 very small, which is counter
9.4 Step 3: Evolving the lowest-frequency nonatomic part
319
productive in Case 2, when we work on a larger interval. Hence we have to exploit the divisibility of the expression, which forces us to exploit the hidden null-structure. However, we can easily conclude from the proof of Lemma 7.4 that
P0 1 r 1 P< C . 2 / M 2 d0 L t Lx
provided we pick C D C.Ecrit / sufficiently large, and thence
Z t
P0 1 r 1 P< C . 2 / ds 1 2 d0
L t Lx 0
Z t
P0 1 r 1 P< C . 2 / ds 2 2 d0
L t Lx
0
for t 2 ΠT1 ; T1 , and from there
Z t
P0 1 r 1 P<
C.
0
2
/ ds
S Œ0
d0 ;
compare (7.10) (provided T1 < 1, say). Similarly, one checks that the contribution of P0 1 r 1 P>C . 2 / is acceptable, and so we now need to force smallness for P0 1 r 1 PΠC;C . 2 / ; which we do by subdivision into small time intervals (whose number depends on k kS ). First, we observe that choosing C1 large enough depending on C and Ecrit , we can force that
P0 1 r 1 PΠC;C .Q>C / L2 d0 ; 1 t;x
and from here one can again infer that
Z t
P0 1 r 1 PΠC;C .Q>C1
0
/ ds
S Œ0
for t 2 Œ T1 ; T1 , T1 < 1, say. The same applies to P0 Q>C1 1 r 1 PŒ C;C . Hence we may reduce to considering P0 1 r 1 PŒ
C;C .
2
2
/
/:
d0
320
9 The Bahouri–Gérard concentration compactness method
where we automatically assume that D Q
D R
C :
First, substitute the gradient term for either factor , which results in the expression P0 1 r 1 PΠC;C Qj . ; / : Now due to Lemma 4.17 etc that in case of high-low or low-high interactions inside Qj . ; / we can estimate
PΠC;C Qj . ; / 2 . k k2 S L t;x
and one may then pick time intervals Ij with the property that X
I PŒk j
C;kCC Qj .
2 ; / L2 1 t;x
k2Z
which ensures “divisibility”. Thus it remains to deal with the expression P0 1 r
1
PŒ
C;C Qj .P>C2
; P>C2 /
and indeed in light of Lemma 4.17 etc only the case when D 0 needs to be considered. We choose C2 maxfC; C1 g. Note that in this case the inner null-form may have very large modulation (comparable to the frequency of the inputs), in which case we cannot take advantage of the null-structure. The idea then is to use the smoothing effect of integration over time. Specifically, we write schematically P0 1 r
1
PΠC;C Qj .P>C2 ; P>C2 / D P0 1 r 1 PΠC;C @ t .P>C2 jrj 1 P>C2 Rj / P0 1 r 1 PΠC;C @j .P>C2 jrj 1 P>C2 R0 / D P0 @ t 1 r 1 PΠC;C .P>C2 jrj 1 P>C2 Rj / P0 @ t 1 r 1 PΠC;C .P>C2 jrj 1 P>C2 Rj / P0 1 r 1 PΠC;C @j .P>C2 jrj 1 P>C2 R0 / :
(9.21) (9.22) (9.23)
Now it is straightforward to analyze the contribution of each term, keeping in mind our assumptions about hyperbolicity of each input. For the contribution of
321
9.4 Step 3: Evolving the lowest-frequency nonatomic part
(9.21), note that we have Z
t
0
1
PΠC;C .P>C2 jrj 1 P>C2 Rj / ds D P0 1 r 1 PΠC;C .P>C2 jrj 1 P>C2 Rj / .t; / P0 1 r 1 PΠC;C .P>C2 jrj 1 P>C2 Rj / .0; /
P0 @s 1 r
and we can then crudely bound (assuming T1 < 1, say)
Œ
T1 ;T1
P0 Œ1 r
1
1
P0 Œ1 r
C;C .P>C2 jrj
PŒ PŒ
1
C;C .P>C2 jrj
1
P>C2 Rj /.t; / P>C2 Rj /.0; / L2 d0 : t;x
This again suffices for the bootstrapping. Next, for the expression (9.22), we estimate it by
ΠT ;T P0 @ t 1 r 1 PΠC;C .P>C jrj 1 P>C Rj / 2 2 2 1 1 L t;x 1 1 . k@ t 1 kL1 L2x kr PΠC;C P>C2 jrj P>C2 Rj kL1 L2x t
t
d0 : Finally, expression (9.23) is more of the same (due to the hyperbolicity of the inputs) and omitted. We next consider the contribution of the terms arising when the substitute an elliptic term for inside r 1 . 2 /. This leads to an expression of the schematic form P0 "1 r 1 r 1 Πr 1 . 2 / : However, as is easily verified, we have
P0 Pk "1 r 1 .2
1
Pk r
1
.jk1 jCjkj/
Pr Πr 2
1
2
.
ı.jk k1 j/
/
L2t;x
kPk1 "1 kS Œk1 Pr r
1
.
2
/
L2t HP
1 2
;
for suitable > 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
1
.
2
/ ds:
0
The second term here leads to a contribution that is bounded by
. QC ./ L2 P0 ŒQ"r
1
t
.
2
/ L1 L2 sup 2 t
x
jkj
kPk "QkS Œk.R2C1 / :
k2Z
Rt For the first term above, 1 .t / 0 Q
P0 r .2
1
.Pk2 Pk3 1 / LM L2
0 jk2 k3 j
t
x
kPk2 kS Œk2 kPk3 1 kS Œk3 sup 2
0 jk1 j
kPk1 kS Œk1 (9.24)
k1 2Z
for some 0 > 0. This follows by inspecting the proof of Lemma 7.4. If jk2 k3 j > Bj log ı1 j where B is large, one concludes from (9.24) that X
P0 r
1
.Pk2 Pk3 1 / LM L2 t
x
jk2 k3 j>C j log ı1 j
. C4 ı1B0 k kS k1 .0/k2 sup 2 k1 2Z
0 jk1 j
kPk1 kS Œk1 :
324
9 The Bahouri–Gérard concentration compactness method
Replacing P0 by Pk and square summing in k yields a bound of C4 ı1B0 k k2S k1 .0/k2 C4 C2 ı1 for the contribution of this case. This can be done by choosing B large depending on Ecrit , see (9.18). On the other hand, if jk2 k3 j Bj log ı1 j, then we exploit that the Fourier supports of and are essentially disjoint up to small errors (bounded by . ı1 in the S -norm) and exponentially decaying tails. Now we sum (9.24) over this range to obtain X
P0 r 1 .Pk Pk 1 / M 2 2
L t Lx
3
jk2 k3 jBj log ı1 j
X
.
kP0 Πr
1
.Pk2 M Pk3 1 /kLM L2x
(9.25)
t
jk2 k3 jBj log ı1 j
X
C
kP0 Πr
1
.Pk2 Q Pk3 1 /kLM L2x : t
(9.26)
jk2 k3 jBj log ı1 j
For (9.25) one obtains as above (9.25) . k M kS k1 .0/k2 sup 2
0 jk1 j
kPk1 kS Œk1
k1 2Z
with an absolute implicit constant. Replacing P0 with Pk and summing over all scales yields the bound 1
2 . k kS k M kS k1 .0/k2 . "2 4 Ecrit C2 ı1 "0 C2 C4 ı1 1
2 provided we choose "2 4 Ecrit "0 C4 . Next, by the definition of the frequency envelopes ck and dk ,
(9.26) . sup 2
0 jk1 j
kPk1 kS Œk1
k1 2Z
X
2
0 jk2 k3 j
kPk2 Q kS Œk2 kPk3 1 kS Œk3
jk2 k3 jBj log ı1 j
. sup 2
0 jk1 j
kPk1 kS Œk1
k1 2Z
X
2
0 jk2 k3 j
.` 1/
C2 ck2
C4 dk3
k2 k3 jBj log ı1 j
. C2 C4 ı0 sup 2
0 jk1 j
kPk1 kS Œk1 :
k1 2Z .0/
This follows from the fact that ı0 was chosen to control the Besov norm of w˛nA , as well as the fact that the intervals Ji where chosen in such a way that any of the
325
9.4 Step 3: Evolving the lowest-frequency nonatomic part nA
.0/
smaller atoms contained within w˛ 0 are arbitrarily far away from the endpoints of Ji as n ! 1. Rescaling this bound to Pk from P0 and square summing yields a bound of C2 C4 k kS ı0 C2 C4 ı1 by taking ı0 small enough, cf. (9.18). Next, we turn to r 1 . 2 /. Here the smallness comes from “divisibility” again as in case (a). More precisely, reasoning as in (a), we may reduce this expression to the form P0 r 1 PŒ C;C . 2 / where we moreover have D Q
and then
.; x/ .; x
y/ L2 . k k2S ; t;x
which follows from our assumption about the Fourier supports, as well as the fact that both frequencies here are < O.1/. But then we can again force smallness by picking the Ij suitably, such that Z X
2 Pk P
Ij k2Z
L t HP
R2
2
By replacing the output frequency 0 by k and square summing over all frequencies for which the Case 1 condition jIj j < T1 2 k is satisfied, we have then achieved that 0 12 X
Pk r 1 . 2 / 2 C2 C4 ı1 : 3 2 P k
Lt H
2
(c) The term 1 r 1 . 1 / is easy, since it inherits the frequency profile of 1 . More precisely, using the same type of trilinear estimates as in (a) and (b) one obtains
Pk 1 r 1 . 1 / . C4 dk k kL1 L2x k1 kL1 L2x C4 dk S Œk.I1 R2 / t
t
326
9 The Bahouri–Gérard concentration compactness method
using (9.18) and the fact that k1 kL1 L2x 2"0 (taking ı1 small). The other cases t are easier due to the presence of ı1 coming from 2 . (d) The term r 1 . 2 / splits into the terms r 1 .12 /, r 1 .1 2 /, and r 1 .22 /. The last two are easier due to the smallness of 2 . The first one is harder, as it inherits the frequency profile of and therefore needs to be incorporated in 2 . This means that we need to gain the very small ı0 , which is only possible if there are high-high gains in the inner term of r 1 .12 / resulting from 1 . Of course, this requires that we expand this inner expression into a null-form via the usual Hodge decomposition. (i): High-high-low interactions in r 1 . 2 /. This is the following (schematic) type of term: X P0 Pk1 r 1 Pk .Pk2 Pk3 / : k; k1;2;3 ; k k2
It is straightforward to see that we may assume jkj < 3 k2 for some 3 > 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
1
.r
1
C Pk1 r
1
.r
1
Pk2 Q
C Pk1 r
1
.r
1
Pk2 Q
1
C Pk1 r
1
.r
1
Pk2 Q
1
C Pk1 r
1
.r
1
Pk2 Q
C Pk1 r
1
.r
1
Pk2 Q
Pk2 Q
1
. 2 //R Pk3 Q
(9.29)
1
. //R Pk3 Q
(9.30)
. //R Pk3 Q
(9.31)
.
1 1
2
2
//R Pk3 Q
(9.32)
. //R Pk3 Q
. /r
1
Pk3 Q
1
(9.33) . // C : : : (9.34) 2
where : : : denotes the remaining septilinear terms containing mixed --interactions. Again we may substitute 1 everywhere for , the contributions from 2 leading to much smaller contributions. The first two terms on the right are straightforward to estimate: using Bernstein’s inequality, one obtains for (9.27) the bound
P0 Pk r 1 .Pk Q>k 1 Pk 1 / 2 1 2 2 3 L t;x ˚ 1 kP . min kPk1 kL1 L2x ; kPk1 kL1 2 k2 Q>k2 1 kL2t;x kPk3 1 kL1 L x t t t Lx k2 ˚ 1 kP . 2 2 min kPk1 1 kL1 L2x ; kPk1 1 kL1 k2 1 kS Œk2 kPk3 1 kS Œk3 : t Lx t
Keep in mind here we assume k1 D O.1/. Then by an argument similar to the one used to estimate (9.26), replacing the output frequency by 2k and square summing over k D k1 C O.1/ while also summing over jk1 k2 j > Bj log ı1 j, one can bound this contribution by . C2 C42 "20 ı0 , which is enough to incorporate this term into 2 . The second term in the expansion is of course handled identically, and so we now turn to the third term (9.28), which is the most delicate one. The potential difficulty comes when D 0, as the Qj -null-form allows us to pull out one derivative otherwise; indeed, assume first that f; j g D f1; 2g. Then using the identity (and omitting the subscript from for simplicity) R1 1 R2 2
R1 2 R2 1 D @1 Œr
1 1
R2 2
@2 Œr
1 1
R1 2 ;
328
9 The Bahouri–Gérard concentration compactness method
we can estimate (always under the assumption k1 D O.1/ D k)
P0 Pk
1
r
1
Pk .R1 Pk2 Q
R2 Pk2 Q
1
. kPk1 kL1 L2x Pk Œr Pk2 Q
t
x
In order to estimate the right-hand factor, we use the improved Strichartz estimates: We have Pk Œr
1
Pk2 Q
1
Pk Œr
Pc1 Q
c1;2 2Dk2 ; k2 dist.c1 ; c2 /DO.1/
whence
Pk Œr
1
Pk2 Q
.2
Y
t
k2
X
2 kPk3 Q
x
c2Dk3 ;
12
k3
kPkj kS Œkj ;
j D2;3
whence we now have
P0 Pk
1
r
1
Pk .R1 Pk2 Q
R2 Pk2 Q
t;x
. kPk1 kL1 L2x 2
k2 2C
Y
t
kPkj kS Œkj :
j D2;3
From here one can again conclude as in case (b). Hence we now consider the more difficult case where D 0. First, it is straightforward to check that we may reduce the first input Pk1 to modulation < 24 k2 , where for example we may put 4 D 12 . Then we use the schematic representation P0 Pk1 Q< k2 r 1 Pk .R0 Pk2 Q
329
9.4 Step 3: Evolving the lowest-frequency nonatomic part
D P0 @ t Pk1 Q< k2 r 1 Pk Œr 1 Pk2 Q
If one then integrates the transport equation for , the contribution from the above terms is P0 Pk1 Q< k2 r 1 Pk Œr 1 Pk2 Q
0
But under our current assumption k1 D O.1/, k D O.1/, we have the estimate (using Bernstein’s inequality)
P0 Pk Q k2 r 1 Pk Œr 1 Pk Q
P0 Pk1 Q< k2 r 1 Pk Œr 1 Pk2 Q
2
.2
k2
x
kPk1 kL1 L2x kPk2 kL1 L2x kPk3 kL1 L2x t
t
t
and the remaining integral expressions on the right also easily lead to exponential gains in k2 due to the extra r 1 applied to Pk2 Q Bj log ı1 j then allows us to incorporate the contribution of all these source terms into 2 . Note that the cutoff Q< k2 in front of @ t allows us to control the effect 2
of the @ t . We explain here how to deal with the construction of the actual Schwartz extension of " for the contribution of the preceding terms, since this is a bit more complicated than in case (a); thus as at the end of case (a) consider Z 1 .t /
t
.s/P0 0
r
1
."2 / ds
where we have a high-high interaction inside r 1 ."2 / but all other frequencies are O.1/, as discussed in the preceding. In particular, we have " D Pk2;3 " with
330
9 The Bahouri–Gérard concentration compactness method
k2 D k3 C O.1/ Bj log ı1 j. We first observe that we are done provided jI1 j . 2 k2 for some small > 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
N k2
/
and we have " D .R " C /. Of course, inclusion of the cutoff destroys the frequency localization again, but we have 1 Pk2 Q
R "/ D OL2 .2 t;x
N k2 /.
As at the end of case (a), we
observe that if the expression at modulation 2j , j 1, is hit by a time derivative @ t , the definition of S Œ0 gives us a gain of 2 "j , which translates into a gain of 2 "k2 . The remaining terms (9.29)–(9.34) no longer require an integration by parts trick and can be directly placed into L2t;x with the requisite gain in k2 . We treat here the term (9.30) given by Pk1 r 1 r 1 Pk2 Q Bj log ı1 j. The key here is as before the improved Strichartz estimates. Write r
Pk2 Q
1
. 2 /
D Pk2 Q
r
1
X P<0 . 2 / C Pk2 Q
r
1
Ps . 2 / :
s0
We treat here the contribution of the second term on the right, the first being treated in the same vein. Now if s < k2 10, we obtain
Pc Q
X c2Dk2 ;s
r
1
12
2 Ps . / 4 2
L t L1x
k2
.2
3k2 4
2
s k2 2C
2 s kPk2 kS Œk2 kk2L1 L2 : t
x
332
9 The Bahouri–Gérard concentration compactness method
Thus in the case s < k2 10 from Bernstein’s inequality we obtain
Pk r 1 .r 1 Pk Q
Pk r 1 .r 1 Pc Q
. kPk1 kL1 L2x
X
Pc Q
t
c2Dk2 ;s
R Pc2 Q
t
k2
1
R Pc Q
X
t
c2Dk2 ;s
.2
k2
2
3k2 2
22.
s k2 2C /
2 s kPk1
1 kS Œk1 kPk2
x
k2
kS Œk2 kk2L1 L2 : t
x
Summing over 0 < s < k2 results in the bound .2
k2 2C
kPk1
On the other hand, when s k2 kPk2 Q
1
1 kS Œk1 kPk2
kS Œk2 kk2L1 L2 : t
x
10, we simply bound
Ps . 2 //kL4 L1x . 2 t
k1 4
k kS kk2L1 L2 t
x
and from here one estimates the L2t;x -norm of the output as before but without using the improved Strichartz, just the standard L4t L1 x -bound. The remaining terms (9.31) are handled similarly. (ii): High-Low/Low-High interactions within r 1 . 2 / In this case one gains exponentially in the maximum frequency occurring among the two factors , provided this is much larger than 1. In this case one can argue as in case (b) to include this contribution into 2 . (e) The cubic term r 1 . 2 / is easy, and can be treated as in (a) and (b) above. Here the smallness comes simply from the size of . The bootstrap argument for in the small time case is now completed as in the proof of Lemma 7.6, cf. (7.10). Case 2: jI1 j T1 , where T1 > 0 is a small constant depending on Ecrit . Here we have to work with the wave equation satisfied by P0 . We start by recording this equation schematically in its original trilinear form, to which we apply
333
9.4 Step 3: Evolving the lowest-frequency nonatomic part
various Hodge type decompositions as well as localizations in frequency space. The goal is to write the equation in the form of a nonlinear wave equation with a low-frequency magnetic potential term, which we will treat as part of the linear operator. To begin with, we have the schematic equation (here we suppress the fact that really stands for the system of variables f˛ g, ˛ D 0; 1; 2) P0 D rx;t P0 . C /r 1 .Œ C 2 / rx;t P0 . /r 1 . 2 / (9.35) D P0 rx;t Œr
1
.
C P0 rx;t Πr
2
/ C P0 rx;t Πr 1
1
. / C P0 rx;t Œr
2
. / C P0 rx;t Œr
1
1
. /
2
. /:
More precisely, the terms on the right-hand side of (9.35) are exactly those given by (1.14). It is precisely the first term on the last line which causes technical difficulties for the bootstrap argument, and we shall have to include parts of it into the linear operator. However, this will only be made specific once we have localized the terms suitably in frequency space. To begin with, note that we will implement a bootstrap argument in order to deduce bounds on . For this we substitute Schwartz extensions Q˛ for each ˛ on the right-hand side (these extensions agreeing with ˛ on the time interval I1 R2 we are working on), and then solve the inhomogeneous wave equation for ˛ , improving the bounds we used for Q˛ . Denoting the right-hand source term above – with Q˛ instead of ˛ – by FQ˛ , what we really do is solving the problem P0 ˛ D P0 FQ˛ : In order to deduce the S -bounds on P0 ˛ , we split this variable into two parts P0 ˛ D P0 QD ˛ C P0 Q
1
QD P0 FQ˛ Z
P0 Q
t
s/P0 Q
U.t 0
In other words, P0 Q
1
.
2
/C
r C
1
r
. Q / C Q r 1
2
.Q / C Q r
1
. Q / 1
.Q 2 / (9.36)
334
9 The Bahouri–Gérard concentration compactness method
First, we identify the terms which can be included in the right-hand side as source terms since they gain smallness, which is achieved in part by introducing suitable Fourier localizations. To begin with, recall that the basic version of the wave maps equation at the level of the Coulomb gauge is of the schematic form
˛
D i @ˇ Œ
˛ Aˇ
i @ˇ Œ
ˇ A˛
C i @˛ Œ
A :
The estimates of Chapter 5 will be seen to imply that the middle term here can be included entirely in the right-hand side, and the immediately ensuing discussion is only applied to the first and third terms. Split the first term on the right in (9.36) (which is understood to be of the first or third type) into P0 Q
L
C
NL ;
we further decompose (schematically) P0 Q
1
P<
2 D1 . NL /
Due to the smallness of NL and (9.17), it is only the first term on the right which we need to incorporate in part into the linear operator. Of course this requires replacing Q by , which requires some care due to the non-local operator Q
335
9.4 Step 3: Evolving the lowest-frequency nonatomic part
Now we introduce null-structure by performing Hodge decompositions as in Chapter 3, for all the trilinear terms. In particular, the preceding discussion yields that we replace the schematic term P0 Q
F˛3j .P0 Q
D1 I
L;
L/
C P0 Q
D1 I
L;
L/
j D1;3
C
X
P0 F˛3j .Q
D1 I
j D1;3
L;
X
L/
F˛3j .P0 Q
D1 I
L;
L/
D1 I
L;
L/
j D1;3
C
X
P0 F˛3j .QD I P<
D1 I
j D1;3
L;
X
L/
P0 QD F˛3j .Q I P<
j D1;3
C
5 X
P0 Q
D1 I
L;
L /:
kD2
We can now write the wave equation that we use to solve for P0 Q
C
X
C P0 Q
P0 F˛3j .Q
D1 I
j D1;3
L; X
D1 I
L;
L/
L/
F˛3j .P0 Q
D1 I
L;
L/
D1 I
L;
L/
j D1;3
C
X
P0 F˛3j .QD I P< D1 I
j D1;3
L; X
L/
P0 QD F˛3j .Q I P<
j D1;3
C
5 X
P0 Q
C Q ; .
C Q /; .
C Q //
P0 Q
kD2
C P0 Q
P0 Q
D1 I
NL ;
NL ;
NL /
C
L/
C P0 Q
P0 Q
D1 I L;
L;
L/
NL /
336
9 The Bahouri–Gérard concentration compactness method
C P0 Q
(9.37)
The significance of the first term on the right, i.e., the expression X F˛3j .P0 Q
is that it implicitly contains a magnetic potential interaction term, see the discussion at the end of Chapter 3. In order to deduce estimates, we shall re-arrange terms and move the magnetic interaction term contained in the above term 2i@ˇ .P0 Q
Aˇ WD
P<
D1 @j
1
I Qˇj .
L;
L/
(9.38)
to the left, thereby obtaining an equation of the schematic type .P0 Q
(9.39)
The next issue occupying us is the derivation of a priori estimates for this type of equation, at first treating F as a function with good Fourier localization properties and bounded with respect to k kN . 9.4.1 Solving the wave equation with a magnetic potential in the Coulomb gauge For simplicity’s sake, replace .P0 Q D1 .Ecrit /: Then satisfies the bound kkS Œ0 . kF kN Œ0 C k.f; g/kL2x HP
1
with implied constant only depending on Ecrit . Furthermore, there is approximate energy conservation: 2 k@ t .t; /kL 2 C krx .t; /kL2 x x
2 D k@ t .0; /kL 2 C krx .0; /kL2 C c.D1 / C O.kF kN Œ0 k"kS / x x
337
9.4 Step 3: Evolving the lowest-frequency nonatomic part
with c.D1 / ! 0 as D1 ! 1, independently of t. Proof. Recall that 0 ˇ 2, X Aˇ D 4 1 @j P<
D1 I
Rˇ
1 L Rj
2 L
Rˇ
2 L Rj
1 L
j D1;2
and observe that these functions are real-valued and Schwartz for fixed times. The key difficulty comes from the fact that there appears no obvious way to obtain smallness for the linear interaction term 2i @ˇ Aˇ , even when restricting to small time intervals. The easiest way out of this impasse is to use an approximate a priori bound resulting from energy conservation. This will allow us to split the bad interaction term into two, one of which is small due to angular alignment of the inputs, the other of which is controlled due to the a priori bound. Moreover, we note that we may always move parts of the expression 2i @ˇ Aˇ with additional smallness properties, such as extreme frequency discrepancies inside Aˇ or special angular alignments, to the right-hand side, since we gain smallness for them as shown in Chapter 5. More precisely, let us pick a cap size jj D jj.Ecrit /, and write the underlying equation (9.39) in localized form as P C 2i @ˇ P Aˇ D P2 FQ ;
P Œ0 D .P f; P g/:
(9.40)
Here we also assume that D1 above is large enough in relation to jj.Ecrit /. We next pick a cap size j1 j D j1 .Ecrit /j j.Ecrit /j, but such that D1 j1 j 100 , say. Note that a computation similar to (8.19) reveals that kP2 FQ
P2 F kN Œ0 c6 kP kS Œ0
where c6 D c6 .D1 ; Ecrit / can be made arbitrarily small in relation to jj.Ecrit /. Now make the following A priori bound assumption: There exist constants c7 D c7 Ecrit ; jj ; C7 D C7 .Ecrit / kF kN Œ0 C kf kL2x C kgkHP
1
such that
˙
sup P Q<2 log j1 j L1 L2 < c7 kkS Œ0 C C7 :
!…˙2
t!
x!
We first show that this assumption, together with a standard bootstrap procedure, implies the bound of the proposition. Then we establish the a priori bound. Observe that the localization to caps of size jj is important in the first step, while we
338
9 The Bahouri–Gérard concentration compactness method
need to pass to finer caps 1 in order to establish the a priori bound. Thus return to the original equation, which we write in the form D F
X
2i @ˇ P Aˇ ;
Œ0 D .f; g/:
(9.41)
jjDjj.Ecrit /
Decompose the term on the right-hand side into X
F
2i @ˇ P Aˇ
jjDjj.Ecrit /
X
DF
˙ Q 2i @ˇ P Q<2 log j1 j Aˇ
jjDjj.Ecrit /;˙
X
˙ 2i @ˇ P Q<2 log j1 j Aˇ
jjDjj.Ecrit /;˙
X
Q2 log j1 j @ˇ P Aˇ :
jjDjj.Ecrit /
Here we define
X
Aˇ WD
1;2 2Klog jj.Ecrit / ;dist.˙;1;2 /<10jj.Ecrit /
X
I4
1
X
@j Pk1 CO.1/
j D1;2 maxfk1;2;3 gminfk1;2;3 gCC8 .Ecrit / k1 < D1 Rˇ Pk2 ;1 L1 Rj Pk3 ;2 L2 Rˇ Pk3 ;2 L2 Rj Pk2 ;1
X
I4
maxfk1;2;3 g>minfk1;2;3 gCC8 .Ecrit / k1 < D1 Rˇ Pk2 L1 Rj Pk3
1
X
1 L
@j Pk1
j D1;2 2 L
Rˇ Pk2
2 L Rj Pk3
1 L
and furthermore AQˇ WD Aˇ Aˇ . By choosing jj.Ecrit / small enough in relation to Ecrit , and further using Corollary 5.2 as well as Lemma 5.5 as well as their improvements in the small angle case, see Section 5.3, we infer that
X
˙ 2i @ˇ P Q<2 A
log j1 j ˇ ˙
N Œ0
. jjı10 kP kS Œ0 ;
339
9.4 Step 3: Evolving the lowest-frequency nonatomic part
and furthermore, exploiting the alignment of the inputs in the definition of Aˇ as well as Cauchy–Schwarz, we obtain
X
˙ 2i @ˇ P Q<2 A . jj.ı10 / kkS Œ0
log j1 j ˇ N Œ0
jjDjj.Ecrit /;˙
where the implied constant is universal. Next, consider X Q2 log j1 j @ˇ P Aˇ : jjDjj.Ecrit /
Here, we estimate
Q2 log j1 j @ˇ P kL2 L2x kAˇ kL2 L1 < c E ; jj kP kS Œ0 8 crit x t
t
provided D1 is large enough, where we pick c8 .Ecrit ; jj/ small enough in relation to the indicated quantities. Similarly, we obtain
Q>O.1/ Q2 log j j P Aˇ 1 N Œ0 kQO.1/>2 log j1 j P kL2 L2x kAˇ kL1 < c8 Ecrit ; jj kP kS Œ0 : t;x t
P ˇ Finally, consider the most delicate term above, jjDjj.Ecrit /;˙ 2i @ P ˙ Q<2 log j1 j AQˇ . By definition of AQˇ , at least one input (both inputs being free waves) has some angular separation from ˙ for its Fourier support. On the other hand, the frequencies of the inputs are approximately equal to the frequency of the output AQˇ . Now using the “a priori bound assumption” from above, we obtain (with implied constants only depending on EC )
X
ˇ ˙ Q 2i @ P Q<2 log j1 j Aˇ
N Œ0
jjDjj.Ecrit /;˙
.
X
˙ sup kP Q<2 log j1 j kL1 L2x
jjDjj.Ecrit /;˙ !…˙2
X
t!
!
c7 kkS Œ0 C C7 :
jjDjj.Ecrit /;˙
By picking c7 small enough in relation to jj, we can bound the preceding by < c8 kkS Œ0 C C9
340
9 The Bahouri–Gérard concentration compactness method
with C9 D CQ 9 .Ecrit /ŒkF kN Œ0 C kf kL2 C kgkHP 1 . Recalling (9.41) as well as picking c8 small enough, it is now straightforward to deduce the bound kkS Œ0 . kF kN Œ0 C k.f; g/kL2x HP
1
:
We now turn to the proof of the aforementioned a priori bound: In effect, we will first prove a conceptually somewhat simpler standard energy bound where we replace the null-frame energy by the standard energy; this, together with the a priori bound, will in particular imply approximate energy conservation as D1 ! 1, as specified in the proposition. To begin with, pick localizers P , jj D jj.Ecrit ; D1 / with jj ! 0 sufficiently slowly as D1 ! 1, such that if ./ is the corresponding cutoff on the Fourier side, we have X 2 ./ D 1:
Consider the inhomogeneous problem C 2i @ A D F; Œ0 D .f; g/: Under the assumptions of the proposition, we intend to show approximate energy conservation as in the statement of the proposition. We localize this equation as before P C 2i @ P AQ 2i @ P A
D
P 2i @ A C 2i @ P A C P2 F DW FQ
(9.42)
where A is defined as above but with C8 D C8 .Ecrit ; D1 / and C8 ! 1 sufficiently slowly as D1 ! 1. Note that we may arrange that X . log jj/` k P2 F C FQ kN Œ0 ! 0
as D1 ! 1, for any `. Finally, we shall also assume that D P0 Q
2
j@ t P C i AQ0 P j2 C
X j D1;2
j@xj P C i AQj P j2 :
(9.43)
341
9.4 Step 3: Evolving the lowest-frequency nonatomic part
Compute @t
1 2
j@ t P C i AQ0 P j2 C
X 1 j@xj P C i AQj P j2 2
j D1;2
D Re .@ t P C i AQ0 P /.@ t C i AQ0 /2 P X C .@xj P C i AQj P /@ t .@xj P C i AQj P / : j D1;2
The second term on the right satisfies ReŒ.@xj P C i AQj P /@ t .@xj P C i AQj P / D ReŒ.@xj P C i AQj P /.@xj C i AQj /.@ t C i AQ0 /P C ReŒ.@xj P C i AQj P /i.@ t AQj
@xj AQ0 /P
D @xj ReŒ.@xj P C i AQj P /.@ t C i AQ0 /P ReŒ.@xj C i AQj /2 P .@ t C i AQ0 /P C ReŒ.@xj P C i AQj P /i.@ t AQj
@xj AQ0 /P :
In summary, one obtains the following local form of energy conservation: i X h1 X 1 @t j@ t P C i AQ0 P j2 C j@xj P C i AQj P j2 2 2 j D1;2
2 XX
@xj Re .@xj P C i AQj P /.@ t C i AQ0 /P
j D1
X D Re .@ t C i AQ0 /2
X
.@xj C i AQj /2 P .@ t P C i AQ0 P /
j D1;2
C ReŒ.@xj P C i AQj P /i.@ t AQj
@xj AQ0 /P
(9.44)
We furthermore observe that any solution of C 2iA˛ @˛ D F satisfies h i X .@ t C iA0 /2 .@xj C iAj /2 j D1;2
D F C i @ t A0
X j D1;2
X @xj Aj C Aj2 j D1;2
A20 :
342
9 The Bahouri–Gérard concentration compactness method
We now integrate the above relation over a time slice Œ0; t0 R2 , which gives X Z 1 X 1 j@ t P C i AQ0 P j2 C j@xj P C i AQj P j2 .t0 ; x/ dx 2 R2 2 j D1;2 Z X X 1 1 D j@ t P C i AQ0 P j2 C j@xj P C i AQj P j2 .0; x/ dx 2 R2 2 j D1;2 Z X X Re FQ C i.@ t AQ0 C @xj AQj /P Œ0;t0 R2
C.
X
AQj2
j D1;2
AQ20 / P .@ t P C i AQ0 P / dt dx
j D1;2
C
XZ Œ0;t0 R2
C
XZ Œ0;t0 R2
ReŒ.@xj P C i AQj P /i.@ t AQj X
Re Œ.@ t C iA0 /2
@xj AQ0 /P dt dx
.@xj C iAj /2 ; P
j D1;2
.@ t P C i AQ0 P / dt dx: We now estimate the three last integrals, XZ Re .@ t P C i AQ0 P / Œ0;t0 R2
X
FQ C i.@ t AQ0
@xj AQj /P C .
j D1;2
C
XZ
Œ0;t0 R2
X
C
AQ20 /P
dt dx
j D1;2
ReŒ.@xj P C i AQj P /i.@ t AQj XZ
AQj2
@xj AQ0 /P dt dx
Re Œ.@ t C iA0 /2
Œ0;t0 R2
X
.@xj C iAj /2 ; P .@ t P C i AQ0 P / dt dx:
j D1;2
One can classify four types of terms. P R (1) The term Œ0;t0 R2 Re .@ t P C i AQ0 P /FQ dt dx. Here one uses the duality of N and S , Lemma 2.19, as well as the space-time frequency localization of : ˇXZ ˇ X ˇ ˇ Q Q Re .@ t P C i A0 P /F dt dx ˇ . kFQ kN Œ0 kkS ˇ
Œ0;t0 R2
9.4 Step 3: Evolving the lowest-frequency nonatomic part
343
Application of Lemma 2.19 is justified due to our assumptions on the modulation of , which in turn restrict the modulation of FQ to the hyperbolic regime via the equation. P R (2) The terms of the form Œ0;t0 R2 rx;t AQ P rx;t P dt dx. These are controlled due to the angular separation inherent in the definition of AQˇ . Note the schematic identity X rx;t AQˇ D rx;t r 1 Pk1 ŒPk2 L Pk3 L : k1;2 < D1
Here our reductions for AQˇ imply k1 D k2 C O.1/ D k3 C O.1/ (where the implied constant may be quite large depending on Ecrit , D1 ) and furthermore the inputs Pk2;3 L have some angular separation between their Fourier supports and ˙. But from this one infers that ˇZ ˇ ˇ ˇ rx;t P
since one may pair each factor
L
against a factor P .
(3) The terms Z Œ0;t0 R2
A2 P rx;t P dt dx
are easier to handle. Here, one may use that k
kŒPk A2 k
4 L t3 L2x
4 . Ecrit 24 ;
which follows from the usual Strichartz estimates, cf. Lemma 2.17:
r
1
Pk .
2 8 L/ L t3 L4x
.2
k
k
2 L k 16 L t 3 L8x
k
. 28 k
2 L k2
One may then use the L8t L4x -control for to infer that ˇ X ˇˇ Z k ˇ P
Œ0;t0 R2
this bound of course being sub-optimal. R (4) The terms Œ0;t0 R2 Œ.r t;x C iA/2 ; P r t;x dt dx. Here, using the observation that P .fg/ D gP f C .f; rg/ provided f is supported at frequency 1,
344
9 The Bahouri–Gérard concentration compactness method
while g is supported at frequency log jj, and further represents a convolution operator of bounded L1 -mass, we reduce this case to either case (2) or (3) in the immediately preceding. Summation over small k < D1 in (2), (3), (4) now yields the desired smallness provided D1 is large. In view of the preceding, we may conclude that
rx;t .t; / 2 2 D rx;t .0; / 2 2 C O 2 kk2 C kF kN Œ0 kkS Œ0 (9.45) S Œ0 L L x
x
where may be made arbitrarily small by choosing D1 .Ecrit / in the statement of the proposition large enough. Note that we eliminated the magnetic potential 1 . 1. here from the covariant energy by means of the estimate kAˇ kL1 t Lx Almost energy conservation claimed in the proposition follows from this if we assume a priori control over kkS Œ0 . To achieve the latter, all that remains is to establish the “a priori bound” above over one of the null-frame ingredients of k kS Œ0 . This will follows by a very slight modification of the above argument; indeed, the only difference will be that now we are integrating (9.44) over a region A!t;c WD Œ0; t R2 \ ft! > cg for arbitrary c, with ! 2 S 1 being a fixed direction. Recall equation (9.41). Also, recall that we introduced a smaller scale j1 j immediately before the “a priori bound assumption” above. This extra scale now becomes important: We may localize (9.41) further to obtain P D P F
X
P 2i @ P1 A
(9.46)
j1 jDj1 j.Ecrit ;jj/; 1 23
provided D1 is chosen large enough. We further localize this to scale j1 j to obtain for 1 32 ˙ ˙ Q˙ Q P1 Q<2 log j1 j C 2i @ P1 Q<2 log j1 j A D P21 F1
where we construct AQ as in the first part of the proof, but with inputs of angular separation from ˙1 now comparable to j1 j, as well as the quotient of all frequencies involved in this definition. On the other hand, P21 FQ1 incorporates all errors generated, in particular those involving A . In the sequel, we shall omit ˙ the additional localizer Q<2 but keep in mind that P1 has this additional log j1 j localization property. Now we integrate the corresponding divergence identity
345
9.4 Step 3: Evolving the lowest-frequency nonatomic part
(9.44) over A!t;c for fixed c and ! … ˙2. This yields (for fixed 1 ) Z h i X Re .@ t P1 C i AQ0 P1 / .@ t C i AQ0 /2 .@xj C i AQj /2 P1 A! t;c
j D1;2
C ReŒ.@xj P1 C i AQj P1 /i.@ t Aj @xj A0 /P1 dt dx Z h1 i X 1 D j@ t P1 C i AQ0 P1 j2 C j@xj P1 C i AQj P1 j2 dt dx 2 2 0R2 \A! t;c j D1;2
(9.47) h1
Z
2 h1
t R2 \A! t;c
Z C
j@ t P1 C i AQ0 P1 j2 C
X 1 j@x P C i AQj P1 j2 dt dx 2 j 1 i
j D1;2
j@ t P1 C i AQ0 P1 j2 2 X 1 C j@x P C i AQj P1 j2 2 j 1
ft! Dcg\A! t;c
j D1;2
!j ReŒ.@xj P1 C i AQj P1 /.@ t C i AQ0 /P1
i
dx! :
It is the latter integral expression that gives us the additional information we need: Indeed, use the decomposition X j@xj P1 C i AQj P1 j2 j D1;2
ˇ X ˇ2 ˇ X ? ˇ2 .!j @xj P1 Ci !j AQj P1 /ˇ Cˇ Dˇ .!j @xj P1 Ci !j? AQj P1 /ˇ : j D1;2
j D1;2
Recalling that ! … 2, we can conclude that Z ˇ2 ˇ X ˇ ˇ 2 sup sup .!j? @xj P1 C i !j? AQj P1 /ˇ & kP1 kL ˇ 1 2 L c
t
ft! Dcg\A! t;c j D1;2
t!
x!
1 since the magnetic potential is small in L1 t Lx . Here the implicit constant depends on jj, but not j1 j (which we recall was chosen jj); this will be important since we can compensate a loss in this implicit constant by picking j1 j small enough. Next, observe that ˇ2 1 1 ˇˇ X ˇ .!j @xj P1 C i !j AQj P1 /ˇ C j@ t P1 C i AQ0 P1 j2 ˇ 2 2 j D1;2 X !j ReŒ.@xj P1 C i AQj P1 /.@ t C i AQ0 /P1 0; j D1;2
346
9 The Bahouri–Gérard concentration compactness method
and also that, due to the additional localization coming from the (suppressed) ˙ Q<2 applied to P1 , we have log j1 j X
kP1 k2NF Π& kP k2NF Π:
1 32
In order to derive the desired “a priori bound assumption”, we need to estimate the first three integral expressions in (9.47). To begin with, the argument given above for the standard energy conservation implies that 1
X Z 1 32
0R2 \A! t;c
2
j@ t P1 C i AQ0 P1 j2 X 1 j@xj P1 C i AQj P1 j2 dt dx 2
C
j D1;2
1
Z t R2 \A! t;c
2
j@ t P1 C i AQ0 P1 j2 C
X 1 j@xj P1 C i AQj P1 j2 dt dx 2
j D1;2
.
krx;t .0; /k2HP
1
C
kF k2N Œ0
C .D1 /kk2S Œ0 C kF kN Œ0 kkS Œ0 (9.48)
where we have .D1 / ! 0 as D1 ! 1, and the implicit constant is absolute. It remains to control X Z 1 32
A! t;c
h Re .@ t P1 C i AQ0 P1 / .@ t C i AQ0 /2
X
i .@xj C i AQj /2 P1
j D1;2
C ReŒ.@xj P1 C i AQj P1 /i.@ t Aj
@xj A0 /P1 dt dx;
which we do by essentially following the steps in (1), (2), (3) of the standard energy estimate above: proceeding by exact analogy, we need to estimate the following expressions: (10 ): The terms X Z 1 32
A! t;c
Re .@ t P1 C i AQ0 P1 /P21 FQ˙1 dt dx:
347
9.4 Step 3: Evolving the lowest-frequency nonatomic part
˙ Here we recall that we suppress the additional localization operator Q<2 in log j1 j front of . To estimate this term, write it as h X Z i Re .@ t P1 C i AQ0 P1 / A!t;c .t; x/P21 FQ˙1 dt dx 1 32
R2C1
h X Z D Re 1 23
R2C1
h X Z D Re 1 32
R2C1
.@ t P1 C i AQ0 P1 / A!t;c .t; x/P21 FQ˙1 dt dx
i
.@ t P1 C i AQ0 P1 / i ˙ Q˙ dt dx ! .t; x/P2 F P1 Q<2 A 1 log j1 j 1 t;c
where we again exploited the suppressed localization of close to the light cone. Now we use Lemma 2.20, together with Lemma 2.11, as well as Cauchy–Schwarz and the suppressed localization of to bound the preceding by ˇ h X Z ˇ ˇRe 1 32
R2C1
.@ t P1 C i AQ0 P1 / iˇ ˙ ˙ dt dx ˇˇ Q ! .t; x/P2 F P1 Q<2 1 1 log j1 j A t;c . kP 3 kS Œ0 ŒkP 3 kS Œ0 C kP 3 F kN Œ0 2
2
2
where again D .D1 ; j1 j/ can be made small as D1 ; j1 j 1 ! 1. Note that the implicit constant in the preceding inequality depends on jj. (20 ), (30 ), (40 ): The terms X Z 1 23
A! t;c
Q 1 rx;t P1 dxdt; rx;t AP
X Z 1 23
X Z 1 23
A! t;c
A! t;c
AQ2 P1 rx;t P1 dxdt;
.r t;x C iA/2 ; P1 r t;x dt dx:
These are estimated exactly as in (2), (3), (4), exploiting the angular separation of the inherent factors, as well as the fact that we may let D1 ! 1 independently of
348
9 The Bahouri–Gérard concentration compactness method
j1 j. The “a priori bound assumption” now follows from (1’)-(3’) by picking D1 and j1 j 1 large enough such that all terms in the above bounds involving kkS Œ0 come with a factor of at most size c7 .Ecrit ; jj/, the latter as in the “a priori bound assumption” above. This completes the proof of Proposition 9.14. Due to frequency leakage coming from the magnetic term we shall also require energy estimates that take N ! S , or alternatively, preservation of frequency envelope. For the following lemma, we allow more general frequency support of A˛ . Hence consider the following equation3 u C 2i @˛ Œu A˛ D F;
uŒ0 D .f; g/
(9.49)
12 P 2 where F has the property that F D F1 CF2 where kF1 kN WD k2Z kF kN Œk is finite and with F2 controlled by a frequency envelope, i.e., kPk F2 kN Œk ck and fck gk2Z is sufficiently flat (as defined above). Furthermore, .f; g/ D .f1 ; g1 /C.f2 ; g2 / with k.f1 ; g1 /kL2 HP 1 finite and kPk .f2 ; g2 /kL2 HP 1 dk where dk is again a sufficiently flat envelope. Finally, A˛ D
1
X
@j R˛
1 L Rj
2 L
R˛
2 L Rj
1 L
j D1;2
is more general than in the previous proposition. Here L1 and L2 are finite energy free waves (with energy bounded by Ecrit ). Now one has the following result. Lemma 9.15. Let u be a solution of (9.49) with F and f; g as above. Then u D u1 Cu2 Cu3 where ku1 kS . kF1 kN Ck.f1 ; g1 /kL2 HP 1 , and kPk u2 kS Œk . ck , kPk u3 kS Œk . dk . The implied constants only depend on the energy of L1;2 . Proof. We restrict ourselves to Pj u. By scaling j D 0. Now split A˛ D P3 .i / iD1 A˛ where A.1/ ˛ D
X k< C
3
Pk A˛ ;
A.2/ ˛ D
X
Pk A˛
(9.50)
C
Note that here we have to take the derivative outside the product, since otherwise we cannot control high-high interactions. In Step 4 we will revert to the original form of the operator as before, without making a priori modulation or frequency restrictions; this does not lead to problems there since we work with a function u instead of with a different scaling.
9.4 Step 3: Evolving the lowest-frequency nonatomic part
349
The constant C in (9.50) is chosen such that the proof of Proposition 9.14 applies to the low frequency part of A˛ . Then we write P0 u C 2i@˛ P0 u A.1/ ˛ D P0 F
˛ .3/ ˛ .1/ 2iP0 Œ@˛ u A.2/ ˛ C @ u A˛ C L.@ u; rA˛ /
P0 uŒ0 D P0 .f; g/: Here L.; / S stands for the commutator in P0 .uv/ D vP0 u C L.u; rv/. We now divide R D M `D1 I` into disjoint intervals with the property that X
˛ .3/ 2 max P0 Œ@˛ u A.2/ 2 0 jkj kPk ukS Œk ˛ C @ u A˛ N Œ0.I R2 / k L kS `
`
k2Z
where > 0 can be made arbitrarily small, and M D M.Ecrit ; /. To see this, one argues as in several previous instances. First consider A.1/ . In case of angular alignment of the Fourier supports of any two of the inputs, one obtains a gain as shown in Chapter 5. On the other hand, in case of angular separation the desired smallness is achieved by a careful choice of the I` , see Chapter 7. Finally, for A.2/ one uses the high-high-low gains in the trilinear estimates (see the form of the weights w.j1 ; j2 ; j3 / in Chapter 5 when max.j2 ; j3 / > C ). The commutator terms satisfies the bound X
L.@˛ u; rA.1/ / . 2 C k L k2S 2 0 jkj kPk ukS Œk ˛ N Œ0 k2Z .1/
since rA˛ gains a factor of 2 C . We now apply the covariant energy bound of Proposition 9.14 to conclude that (with Pj instead of P0 ) kPj ukS Œj C.Ecrit / kPj .f; g/kL2 HP X 2
1
C . C 2
0 jk j j
C
/
kPk ukS Œk C kPj F kN Œj :
k2Z
The lemma now follows from this estimate provided the frequency envelopes are flat enough compared to 0 .
9.4.2 Controlling the error terms In this section, we complete the proof of Proposition 9.12. This amounts to bounding each of the terms on the right-hand side of (9.37) one by one using the covariant energy estimate of the previous section.
350
9 The Bahouri–Gérard concentration compactness method
P 3j We begin with the first term in (9.37), i.e., j D1;3 F˛ .P0 Q
F˛3j .P0 Q
D1 I
L;
L/
2i @ˇ P0 Q
j D1;3
into the sum of two terms, one of which has controlled frequency envelope and the other small S -norm as in (9.20). By (3.14), this difference equals iP0 Q
D1 Qˇj . L ; ˇ
L/
1
C iP0 Q
L/
L/
D1 Qˇj . L ;
:
L/
Denoting these terms by term11 –term14 , respectively, we now proceed to estimate them by means of Section 5.2. Let us now assume that is of the envelope type, see 1 in (9.19). Then by (5.86) kterm11 kN Œ0 . 2
D1
kP0 ˛ kS Œ0 k
2 L kS
C4 d0
for D1 large depending on Ecrit . The contribution of 2 is estimated similarly. For term12 one uses that by Lemma 5.7
P0 Q
iP0 Q
ˇ
@ P0 Q
1
P< D1 I c Q˛j . L ; L / N Œ0
C P0 Q
L;
L/
N Œ0
: (9.51)
351
9.4 Step 3: Evolving the lowest-frequency nonatomic part
We can bound the first by
ˇ
@ P0 Q
D1 I
c
Q˛j .
L;
L/
N Œ0
kP0 Q kS Œ0 k
2 L kS ;
using Corollary 5.4 as well as choosing the constant in I c large enough, while we can bound
P0 Q
j
1
P<
D1 I @
ˇ
Q˛j .
L;
L/
N Œ0
.2
D1 I
L;
D1
k"QkS Œ0 k
2 L kS ;
having used Lemma 5.7. The third term in (9.37) is the commutator X
P0 F˛3j .Q
D1 I
L;
L/
j D1;3
X
F˛3j .P0 Q
L/
j D1;3
D PQ0 @ˇ ŒQ
D1 I
L;
L/
j D1;3
j D1;3
X
P0 F˛3j .QD I P<
D1 I
L;
L/
X
P0 QD F˛3j .I P<
j D1;3
j D1;3
X
X
D1 I
L;
L/
(9.52) C
P0 QD F˛3j .I P<
D1 I
L;
j D1;3
L/
P0 QD F˛3j .Q I P<
D1 I
L;
L /:
j D1;3
(9.53) First, (9.52) is bounded by 2
D1
kP0 kS Œ0 k
2 L kS ;
by the same commutator logic as before and Lemma 5.1. Second, since D Q on I1 , the length of which is bounded below by an absolute constant by Case 1 above,
352
9 The Bahouri–Gérard concentration compactness method
one obtains that, upon letting IQ1 be a smooth cutoff localizing to an interval containing I1 obtained by adding an interval of length 2
X
3j P Q F . Q I P I ; /
0 D ˛ < D1 L L 2
D 2
to the top of I1
N Œ0.I1 R /
j D1;3
X
. P0 QD IQ1 F˛3j .
Q I P<
D1 I
L;
L/
j D1;3
X
C P0 QD .1
IQ1 /F˛3j .
Q I P<
D1 I
j D1;3
.2
D
kP0 .
C2 . .2
ND
D D1
Q /kL1 L2x kP< t
kP0 .
C2
ND
Q /kS Œ0 k /kP0 .
1
D1 r
.
L1t .I1 IL2x / L;
L/
N Œ0.I1 R2 /
2 1 1 L /kL t Lx
2 L kS
Q /kL1 L2x k t
2 L kS
where we used Bernstein’s inequality in the last line. The fifth term is a collection of quintilinear and higher order terms, and we deal with it in the appendix. The terms six through eight are easy by Lemma 5.1, Lemma 5.5, Lemma 5.7 and (9.17). More precisely, they inherit the frequency profile of Q times a factor 1
of "24 ; this is good enough to bootstrap both 1 and 2 . The ninth term in (9.37) is split as follows, see (3.14): P0 Q
D1 I
L; L/ D i @ P0 Q0 Rˇ Q @j 1 IP>0 Q˛j . L ; L / i @ˇ P0 Q0 Rˇ Q @j 1 IP>0 Qˇj . L ; L / C i @˛ P0 Q
9.4 Step 3: Evolving the lowest-frequency nonatomic part
353
Denote these terms in this order by term91 through term100 . First, we rewrite term91 in terms of the usual trichotomy: i@ˇ P0 Q0 Q˛ I @j 1 P>0 Qˇj . L ; L / (9.54) The first term in (9.54) is rewritten as the sum @ˇ P0 Q
ˇ
@ P0 Q
As usual, replacing the output frequency 0 by k and square summing, this is turned into an S bound, leading to an 2 contribution. If Q D Q2 , then it again suffices to consider Q . In this case, one needs to gain extra smallness by partitioning I1 further; however, the number of intervals needed for this partition only depends on the energy in an absolute way (i.e., not on the stage of the induction). First, we may assume that there is angular separation between the Fourier supports of the two Q L inputs due to the bound X
ˇ
@ P0 Q
.P1 Q L ; P2 Q L / N Œ0 kP0 Q kS Œ0 ;
354
9 The Bahouri–Gérard concentration compactness method
see Section 5.3. Here we used that k L k2S is bounded by the energy in an absolute way, which allows us to chose m0 in the same fashion. On the other hand, the remaining term X
ˇ
@ P0 Q2 m0
is estimated by placing Qˇj .P1 Q L ; P2 Q L / into L2t;x , see the reasoning leading up to (7.16), followed by a decomposition of the interval of integration. Here is important to note that D1 only depends on the energy. For the second term in (9.54) consider first Q1 ; then the frequency envelope of Q1 is inherited by this expression. More precisely, for Pk Q one gains a weight 2 k from Lemma 5.5 which is sufficient for the bootstrap provided k is sufficiently large and negative; if not, then one applies the same divisibility as for the previous terms. the same reasoning applies to Q2 . Finally, for the third term in (9.54) consider first the contribution by Q1 . In that case one has
@˛ P0 Q0 Rˇ Q @ 1 IP>0 Qˇj . j .
L;
0 jk1 k2 j
sup 2
L/
N Œ0
kPk1 Q kS Œk1 kPk2
L kS Œk2
(9.55)
k1 ;k2 >0
which can be made C4 C2 ı1 by choosing ı0 small and n large. On the other hand, if Q D Q2 , then one gains smallness in two ways: if any one of Q , or the two L inputs has large frequency, then ones gains smallness from the weight w in Lemma 5.5. If the three inputs have frequency of size O.1/, then one gains smallness by divisibility as before. Next, we note that term92 is treated in the same fashion as the first term on the right-hand side of (9.54). The terms term93 and term97 are of the high-high type and are estimated exactly as in (9.55), and term94 , term98 are essentially the same as the low-high term on the righ-hand side of (9.54). To bound term95 and term96 one applies the same divisibility considerations as in the high-low case of term91 . Finally, the terms term99 and term100 can be handled similarly, first reducing I c @j 1 P
D1 Qˇj . L ;
L /; I
c
@j 1 P
D1 Q˛j . L ;
L/
to frequency O.1/ via Lemma 5.1, and then using the divisibility property for the term kI c @j 1 PO.1/ Qˇj . L ; L /kXP 0; ";2 etc. 0
355
9.4 Step 3: Evolving the lowest-frequency nonatomic part
The tenth and eleventh terms in (9.37) are essentially the same so it suffices to estimate the former. Since the details are quite similar to the preceding arguments, we will proceed schematically. Beginning with Q D Q1 , we split P0 Q
P0 Q
L;
L ; Q /
L ; Q /
C
C P0 Q
P0 Q
L;
NL ; Q /
NL ; Q /
(9.56)
and furthermore, using Corollary 7.29, P0 Q
L;
D P0 Q
All terms here are going to be placed into the S error 2 since they inherit the frequency envelope of . The trilinear estimates of Lemma 5.5 allows for this, with the required smallness for the terms containing NL is gained by the smallness of k L kS k NL kS . Furthermore, the terms containing M are easy due to the smallness k M kS < C2 ı1 and the bootstrap assumption on Q (one then chooses "0 small enough). The most interesting term here is P0 Q
356
9 The Bahouri–Gérard concentration compactness method
already accomplished in Case 1, we can split P0 Q
1 1
P0 Q
D f0 C g0
T1 ;T1 /Œ0
C4 d0 while for g0 , replacing 0 by general k 2 Z, C4 C2 ı1 . On the other hand, one has Œ
T1 ;T1 /
Œ0
L2 HP
1
k"kS Œ0
by choosing D large enough, and so this also leads to an acceptable contribution This concludes the proof of Proposition 9.12. It is now easy to conclude the proof of Proposition 9.11. More precisely, as indicated in Figure 9.2, one proceeds in the direction of increasing time by passing from I1 to I2 and so on. Writing I2 D Œa2 ; b2 , one introduces the frequency envelope X 1 P 22 2 dQk D 2 jr kj kPr k"1 .a2 ; /k L x
r2Z
and by the preceding argument, using the fact that k".a2 ; 0/kE < "0 , we obtain "jI2 D "1 C "2 with k"2 kS.I2 R2 / . ı1 ; kPk "1 k . dQk with implied constant only depending on Ecrit . But by the preceding step, we also have dQk . dk and we can then again conclude via Corollary 9.13 that the energy of "jI2 < "0 provided n and ı1 1 are sufficiently large. Even though is initially only defined locally, Proposition 7.2 and the k kS -norm bound of Proposition 9.12 imply that exists globally with the bounds stated in Proposition 9.11, see (9.13) and (9.14). Proof of Proposition 9.9. This follows simply by iterating Proposition 9.12, i.e., by passing from J1 to J2 and so forth in Figure 9.2. Even though the constant C2 increases with `, in the end one obtains a bound of the form (9.15). The final statement (9.10) is a consequence of our proof of Proposition 9.12 due to the frequency evacuation of the first Besov error from the atom na . In fact, our estimates are based on control of the frequency envelope which therefore implies (9.10) at all stages of the induction.
357
9.4 Step 3: Evolving the lowest-frequency nonatomic part
We include here two corollaries of the proof just given: Note from the proof of Proposition 9.12 that even if the perturbation Œ0 at time zero is controlled by a frequency envelope ck which is concentrated at much higher frequencies than , nonetheless the evolution of will also involve a component that is no longer well-localized with respect to frequency (which forces us to implement the splitting D 1 C 2 ). The reason for this are interactions of the schematic form such as r t;x Œ r 1 . 2 /, which “inherit” the frequency envelope of . This issue is moot provided there is no distinction between and : Corollary 9.16. Assume that ˛ are the Coulomb components of an admissible wave map from . T0 ; T1 / R2 into H2 , and that we have kPk ˛ kL2 ck for a frequency envelope with cl 2 jk lj ck cl 2 jk lj 8l; k, with > 0 small enough. Then assuming a bound k kS.Œ
T0 ;T1 R2 /
< C0 ;
we can conclude kPk kS.Œ
T0 ;T1 R2 /
< M.C0 /ck :
Proof. One proceeds just as in the preceding proof, but with taking the role of ; thus write schematically P0 D r t;x P0 H r 1 . 2 /L C r t;x P0 H r 1 . 2 /H C r t;x P0 L r 1 . 2 /H where we set fH gL D
X
Pk f P
C1 g;
k2Z
fH gH D
X
Pk f PŒk
C1 :k1 CC1 g;
k2Z
with C1 a constant depending on C0 above. Then arguing exactly as in the preceding proof (in particular, one has to implement the null-form expansion in the nonlinearity) one finds a collection of time intervals Ij , j D 1; 2; : : : ; N.C0 / such that
r t;x P0 H r 1 . 2 /H C r t;x P0 L r 1 . 2 /H N Œ0.I R2 / N Œ0.I R2 / j
j
sup 2
jk lj
l2Z
kPl kS Œl.Ij R2 / ;
provided C1 ; N.C0 / is chosen large enough in relation to C0 . But then using Proposition 9.14, one obtains inductively in j bounds of the form kPk kS Œk.Ij R2 / Mj .C0 /ck
358
9 The Bahouri–Gérard concentration compactness method
via a bootstrap argument on Ij R2 .
Corollary 9.17. Let ˛ be as in the preceding corollary, with k kS.Œ T0 ;T1 R2 / < C0 . Then there exists ı1 D ı1 .C0 / > 0 such that if ˛ C "˛ are Coulomb components of an admissible map wave map with
".0; / 2 < ı1 ; L x
then the wave maps evolution of data ˛ C "˛ exists on Œ T0 ; T1 R2 , and we have k"kS.Œ T0 ;T1 R2 / . ı1 with implied constant depending on C0 . Proof. This is exactly as in the proof of Proposition 9.12, with "1 D 0, and "2 D ".
9.5
Completion of the proofs of Lemma 7.10 and Proposition 7.11
We commence by proving the final assertion of Lemma 7.10. This follows immediately from Corollary 9.16. Next, the proof of Proposition 7.11 follows from Corollary 9.17.
9.6
Step 4: Adding the first large atomic component and preparing the second stage of the Bahouri–Gérard decomposition
Recall from Section 9.2 that we wrote the data ˛n of the essentially singular sequence (at time t D 0) in the form ˛n D
A0 X
˛na C w˛nA0 ;
aD1
where A0 was chosen such that the sum X
na 2 2 "0 : lim sup ˛ L n!1
x
aA0 C1
As before "0 .Ecrit / > 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
C kn1
4
P
1@
k
kD1;2
C oL2 .1/ .0/ nA0 1
wk
4
1@
k
C kn1
.0/ nA0 1
wk
C kn1
C oL2 .1/:
Here the first component satisfies .0/
nA w˛ 0
e
i
P
kD1;2
1@
4
k
.0/ nA0 1
wk
C kn1
.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
C kn1
;
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<n ;jk1 k2 j<n dist.1;2 /&2 n
X
4
1
.0/ .0/ 1nA 2nA @j Pk I .n/ Pk1 ;1 0 Pk2 ;2 j 0
j D1;2 .0/
1nA0
Pk1 ;1 j
.0/
2nA0
; (9.63)
Pk2 ;2
where we have introduced the modulation cutoff X I .n/ WD Pk Q
Here we shall let n ! 1 as n ! 1 sufficiently slowly. The errors thereby generated shall be treatable as perturbative errors. This time we use the full , and not just the free wave part. Our notation is somewhat inconsistent, since we do .0/ not include A0 . Since we keep this parameter fixed throughout this section, this .0/
1nA
omission will be inconsequential. From now on we shall denote 0 D 1n etc. to simplify the notation. We shall tacitly assume that the n allow the usual Hodge type decompositions as in the preceding. Definition 9.18. The covariant wave operator An is defined via An u WD u C 2i @ uAn The fundamental fact about this operator is that solutions obeying An u D 0 preserve the energy in the limit n ! 1. This will allow us to modify the second stage of the Bahouri–Gérard method to the covariant d’Alembertian instead of the “flat” d’Alembertian. We state this rigorously as follows.
362
9 The Bahouri–Gérard concentration compactness method
Lemma 9.19. Assume that u is essentially supported at frequency 1, and that A is essentially supported at frequencies 1. By this we mean that
lim PŒ R;Rc uŒ0 L2 D 0 (9.64) x
R!1
as well as
lim P>
n!1
R
1n S
D0
(9.65)
for any R > 0. If u solves An u D 0; uŒ0 D .@ t u; rx u/ D .u0 ; u1 / 2 L2 L2 then one obtains a global bound (uniformly in the implicit n )
kukSQ .R2C1 / . uŒ0 L2 x
with implied constant depending on Ecrit as well as supn k n kS (which control An ), and we can conclude that
@ t u.t; / 2 2 C rx u.t; / 2 2 D ku0 .t; /k2 2 C u1 .t; / 2 2 C oL2 .1/ L L L L x
x
x
x
as n ! 1, uniformly in t 2 R, provided n ! 1 sufficiently slowly. We have introduced the slightly altered norm kukSQ .R2C1 / D krx ukS.R2C1 / C
X
krx;t Pk Qk uk2 0; 1 ;1 XP
k2Z
21
:
2
We also have lim kPŒ
R!1
R;Rc ukSQ .R2C1 /
D0
Proof. This follows by the same argument that we used to prove Proposition 9.14. In the latter proof, we assumed that the Coulomb potential was defined in terms of free waves. In order to arrive at the present conclusion, one needs to invoke the decomposition from Corollary 7.27 on suitable time intervals. The additional contributions can be handled just as in the proof of Proposition 9.14. We have modified the norm to kkSQ to accommodate the different scaling and to strengthen it in the large modulation regime. This is possible since we do not have a time derivative hitting the term 2i @ uAn , see for example the estimates in [58] in the simple case of large modulation for the output. These considerations easily furnish a bound of the form
kukSQ . uŒ0 L2 x
9.6 Step 4: Adding the first large component
363
with implicit constant depending on k kS . Similarly, one obtains the frequency localized bounds kPk ukSQ . ck where we put ck D
X
2
jl kj
2 kPl uŒ0kL 2
21
x
l2Z
with > 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 @ PΠR;R Q<2R uP< 10R An (9.66) n n PΠR;R Q<2R 2i @ uA C 2i @ PΠR;R Q<2R uP< 10R A PΠR;R 2i @ Q2R uAn C PΠR;R Q2R 2i @ uAn C 2i @ PΠR;R Q<2R uAn PΠR;R 2i @ Q<2R uAn DW FRn :
R;R Q<2R u
D D
But then for fixed R 1 we have
lim FRn NQ D 0
n!1
12 2 where kF kNQ WD k2Z krx Pk F kN1 Œk , and k kN1 Œk is defined as in Definition 2.9 but with k k 1 C"; 1 ";2 replaced by k k 1; 1 ;1 . The latter modiP
XP k
XP k
2
2
fication is again a consequence of the fact that in the large modulation case, one derives a better estimate for the expressions 2iPŒ R;R uAn as there is no outer time derivative, and simple modifications of the proofs in Section 2.3 (see also the proof of the energy estimate in the high modulation case in [58]) yield that this modifcation of k kN suffices to recover k kSQ . Similarly, we have PŒ R;R Q2R u C PŒ R;R Q2R @ uAn D 0; and one checks readily that lim kPŒ
n!1
R;R Q2R
n @ uA kNQ D 0:
But then the argument of the proof of Proposition 9.14 yields that
r t;x PΠR;R Q<2R u.t; / 2 D r t;x PΠR;R Q<2R u.0; / 2 C o.1/ L L and further 8t (where o.1/ indicates the behavior as n ! 1)
lim kr t;x PΠR;R Q<2R u.t; /kL2 D r t;x u.t; / L2 C o.1/ R!1
which gives the desired asymptotic energy conservation.
364
9 The Bahouri–Gérard concentration compactness method
In our applications of Lemma 9.19, (9.64) will hold due to the frequency localization inherent in our construction of the atoms; in other words, u will be 1-oscillatory after rescaling. The other condition (9.65) will hold due to (9.10), at least at the first stage of the construction (i.e., when adding the first atom as we are doing here). For a D 2 etc. we will use the exact same frequency evacuation property which gave rise to (9.10) in the first place.
9.6.1 Dispersion for the covariant wave equation In this section we prove a weak form of dispersion for the initial value problem An u D 0;
uŒ0 WD .f; g/
(9.67)
where An is as in Definition 9.18. For simplicity, we first consider the case where An is defined as in (9.63) but with free waves Ln . We shall assume that .f; g/, whence also u by Lemma 9.15, are essentially supported at frequency 1, see Lemma 9.19. Generally speaking, u depends on n away from the time t0 D 0, but the above limit is uniform in n and holds on any time-slice. We assume that the free waves Ln satisfy
lim P> R Ln L2 D 0 (9.68) n!1
x
for any R > 0. We now claim the following main result of this subsection for the covariant wave equation (9.67). For simplicity, we drop n as a superscript. Proposition 9.20. Let u be a solution of (9.67), with .f; g/ 2 HP 1 L2 . Given
> 0, there exists a decomposition u D u1 C u2 with the following properties: ı u1;2 satisfy the same a priori estimates which were proved for u in Lemma 9.19, ı ku2 kSQ < , ı there exists t0 D t0 . ; f; g; Ecrit / (but t0 does not depend otherwise on L ) such that for jtj > t0 one has that
u1 .t; / 1 < ; (9.69) L x
uniformly for large enough n.
365
9.6 Step 4: Adding the first large component
The proof of this result will be split into several pieces. The idea is to first obtain a “parametrix” for u, which is established by restricting to suitable time intervals (this is done via “divisibility”). Once we have such a parametrix (more precisely, a representation of u as a sum of Volterra iterates starting with the free wave), we can use the dispersion of the wave equation to prove the desired result. First, we follow Tao to establish the following divisibility lemma.5 Lemma 9.21. For any "1 > 0 there exist a partition of R into intervals fIj gjMD1 where M . .Ecrit "1 1 /C for some absolute constant C with the property6 that for any u max k@˛ u A˛ krx 1 N.Ij R2 / "1 kukS : 1j M
Note that the intervals depend on (other than through the energy).
L
(but not on u), but their number does not
Proof. According to the trilinear estimates of Chapter 5, we may assume that there is angular separation between uO and the waves in A˛ . Otherwise there is the desired gain. The amount of angular separation is very small and depends on Ecrit and "1 . We shall now implicitly assume that @˛ u A˛ respects this type of angular separation. Note that we may restrict ourselves to the case of high-low interactions between u and A˛ , since for the other cases, the divisibility follows by using the same argument as in the proof of Lemma 9.15. Also, since by the preceding lemma we have that u is concentrated along frequency 1, we may reduce to considering the zero frequency mode P0 u. By (2.29), X k1 k@˛ P0 u A˛ krx 1 N C.Ecrit ; "1 / 2 2 kP0 u Pk1 L kL2 kPk2 L kS t;x
k1 < C
X C.Ecrit ; "1 / 2 k1 < C
k1
kP0 u Pk1
2 L kL2
t;x
21
k
L kS :
(9.70) Next, by Theorem 1.11 of [52], assuming u to be a free wave, for each k 2 Z there exists a collection Tk of tubes ki of size 1 2k 2k centered along a light-ray 5
6
It appears likely that an alternative approach to the pointwise decay is to use commuting vector fields. This would force us to strengthen the norm assumptions on n even more, however, and so we opted for the present approach. We define kF kr 1 N WD krx F kN . x
366
9 The Bahouri–Gérard concentration compactness method
and aligned with the Fourier support of u such that #Tk .Ecrit " that, where " > 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 k D c uŒ0kL2x : 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 cone with base D . We now show that uQ 1 D u1 C u2 where ku2 kSQ and 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 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/PΠk0 <
Z
t
2i .t; x/
U.t
s/PŒ
ti
k0 <
˛ .Ji ;` @ uQ 1
1/
A˛ .s/ ds:
1 We now show that the second integral splits into a term of small L1 t Lx -norm, and one of small SQ norm. First, consider PŒ< k0 . Then by Bernstein’s inequality, and the energy estimate Z t
.J ;` 1/ U.t s/PŒ< k0 @˛ uQ 1 i A˛ .s/ ds 1
.t; x/ Lx
ti
Z t k0 .2 U.t
s/PŒ<
ti
.2
k0
.J ;` Ecrit uQ 1 i
1/
SQ
k0
˛ .Ji ;` @ uQ 1
C.Ecrit /2
1/
k0
;
A˛ .s/ ds
2 L1 t Lx
(9.77) (9.78)
370
9 The Bahouri–Gérard concentration compactness method
whereas for PŒ>k0 one can essentially (up to tails which are handled by .J ;` 1/ Lemma 7.23, for example) remove the exterior since the interior @˛ uQ 1 i obeys that very localization. In conclusion, the resulting term is placed in u2 . Now consider the main term (9.76). Decompose S 1 into caps of size c.Ecrit ; / which is a small constant. Denote the corresponding decomposition of the double light-cone fjxj jtj C g into angular P sectors by fS g . Associated with the S there is a smooth partition of unity D . Write (9.76) as the sum Z t X ˛ .Ji ;` 1/ 2i .t; x/ U.t s/PŒ k0 <
C 2i
X
Z
t
.t; x/
U.t
s/PŒ
ti
k0 <
.Ji ;` 1/
@˛ uQ 1
A˛ .s/ ds: (9.80)
Here O D jj and the sign is selected according to the decomposition into incoming and outgoing propagator:
U.t / D
1 i t jrj e 2ijrj
e
i t jrj
By Bernstein’s inequality the first term (9.79) satisfies 1 .J ;` 1 C.Ecrit /jj 2 u Q1 i k(9.79)kL1 L x t
1/
S
which can be made small for small . Also note that jt jj ˙ x j & jt j
8 .t; x; /
such that .t; x/ ¤ 0; 2
k0
< jj < 2k0 ;
O D … 2 jj
where the choice of ˙ depends on whether the propagator U is incoming or outgoing. Now we make the inductive assumption (relative to ` and i ) that
P
.Ji ;` 1/
1 u Q .t / O Lx Œ˙62 2 1
C Œjjt j
.J ;` 1/ Q1 i .t; x/ L1 jxjj&jt j u x CN .i; `; Ecrit ; /jtj
N
(9.81)
where the ˙ sign is according to whether the function has space-time Fourier support in the upper or lower half-spaces, i.e., whether > 0 or < 0. Strictly speaking, the cap size here depends on .i; `/ with the size c.Ecrit ; / from above
371
9.6 Step 4: Adding the first large component
being the size at the end of the induction (recall that there are only finitely many choices for these parameters). But for simplicity of notation, we suppress this dependence from the notation. Note that we only have finitely many values of `; i. Now to estimate the second integral term (9.80), we distinguish between a number of cases: first if jsj jtj (where the implicit small constant depends on jj), .J ;` 1/ due to the a priori support conditions satisfied by uQ 1 i which forces jyj < jsj C , we obtain the desired gain in t by integrating by parts with respect to jj in the Fourier integral representation of U . Next, assume that jsj jtj (where the implicit small constant again depends on jj – this will be tacitly understood for the remainder of the proof). Then we first reduce to jjsj jyjj jsj. For this consider the term
2i
X
Z
t
.t; x/
U.t
s/PŒ
ti
k0 <
Œjjsj
jyjj&jsj @
˛ .Ji ;` 1/ uQ 1 A˛ .s/ ds:
Since we assume jsj jt j, the desired gain t N here follows by using the induction hypothesis. Hence we now reduce to estimating
2i
X
Z
t
.t; x/
U.t
s/PŒ
ti
k0 <
˛ .Ji ;` 1/ Q1 A˛ .s/ ds: jyjj jsj @ u
Œjjsj Here we apply a further decomposition Œjjsj
jyjj jsj @
D Œjjsj
˛ .Ji ;` 1/ uQ 1
jyjj jsj
X
.Ji ;` 1/
0 .s; y/@˛ uQ 1
0
D Œjjsj
jyjj jsj
X
0 .s; y/PŒ˙2 O
.Ji ;` 1/
2 0
@˛ uQ 1
0
C Œjjsj
jyjj jsj
X
0 .s; y/PŒ˙2 O
.Ji ;` 1/
.2 0 /c
@˛ uQ 1
:
0
The contribution of the second term here is again rapidly decaying due to the
372
9 The Bahouri–Gérard concentration compactness method
induction assumption. Hence we have now reduced to estimating 2i
X
Z
t
.t; x/
U.t
s/PŒ
ti
Œjjsj
jyjj jsj
k0 <
X
0 .s; y/PŒ˙2 O
.Ji ;` 1/
2 0
@˛ uQ 1
A˛ .s/ ds:
0
Now writing out the free wave parametrix, we see that on the support of the resulting integral in the variables ; y; s, we have that j ˙ jjs C y j jt j; and choosing 0 as well as the implied constant in jjsj we can ensure that
jyjj jsj suitably small,
j ˙ t jj C x j jt j j ˙ jjs C y j on the support of the integrand. Integrations by parts in jj yield the desired rapid decay with respect to jtj. This recovers the first part of the inductive assumption, and the second follows identically, since if jjt j jxjj & jtj, then we necessarily have j ˙ t jj C x j & jtj: The inductive procedure is now completed by means of (9.74) which takes account of the changes in the level i . Recall that we restricted to be a free wave in (9.63). In order to treat the general case, we apply the usual decomposition (9.62). As usual, the smallness of the NL allows one to iterate these terms away. Furthermore, the proof of Proposition 9.20 applies to these terms equally well since we do not rely on any .J ;` 1/ specific structure of the u1 i other than the inductive assumption (9.81), and the formalism of the Volterra iteration by which we represented these solutions. In this way, the same dispersive property may be proved globally, i.e., not just on I1 where (9.62) holds.
9.6.2 The second stage of Bahouri–Gérard, applied to the first large atomic component Recall that we are considering only a D 1. Nevertheless, we keep the parameter “a” in our notation general. We now need to quantify the lack of compactness
9.6 Step 4: Adding the first large component
373
for the functions Q na , V na , na , na , all at time t D 0. We evolve each of these using the covariant wave flow from before and select a number of concentration profiles. The method for this follows exactly the Bahouri–Gérard template, but using Lemma 9.19 instead of standard energy conservation for the free wave flow. In order to define the temporal flow for each component, we need to impose time derivatives at time t D 0. We do this by defining @ t Q na .0; / WD 0na .0; /; @ t na .0; / WD Q 0na ;
@ t V na .0; / WD 0 @ t na .0; / WD 0:
Introduce the following terminology: Definition 9.22. Given data uŒ0 D .u0 ; u1 / at time t D 0, we denote by SAn .t / uŒ0 the solution of An .u/ D 0 with the given data, evaluated at time t. We now describe the important process of extraction of concentration profiles: Consider SAn . na Œ0/, with na Œ0 D . Q 0na ; na /. Following [1] introduce the family VAn . a /, consisting of all functions on V .t; x/ 2 L2t;loc Hx1 \ C 1 L2x such that SAn . na Œ0/ .t C tn ; x C xn / * V .t; x/ 2 as n ! 1 for some sequence f.tn ; xn /g1 nD1 2 R R . Here, the weak limit is in the sense of L2t;loc Hx1 . Observe that such a function V .t; x/ solves V D 0 in the sense of distributions. Thus it makes sense to introduce the quantity ˚ An . a / WD sup E.V /; V 2 VAn . a /
where
Z E.V / WD
R2
jrx;t V j2 dx
We can now state the following lemma that is at the core of the second stage of the Bahouri–Gérard process for wave maps. Recall that a D 1 here. Lemma 9.23. There exists a collection of sequences f.tnab ; xnab /g R R2 , b ab 1, as well as a family of concentration profiles Vab Œ0 WD .Vab .x/; V1 .x// 2 0 2 2 1 2 L .R / HP .R /, with the following properties: introducing the shifted gauge potentials AQnab WD An .t C tnab ; x C xnab /; (9.82) one has
374
9 The Bahouri–Gérard concentration compactness method
ı For any B 1, one can write SAn . na Œ0/ .t; x/ D
B X
SAQnab .Vab Œ0/ .t
tnab ; x
xnab / C WnaB .t; x/:
bD1
Here each function SAQnab .Vab Œ0/ .t the equation An u D 0, and we have
lim
B!1
xnab /, WnaB .t; x/, solves
tnab ; x
lim sup An .W aB / D 0:
(9.83)
n!1
ı One has the divergence relations lim Œjtnab
n!1
0
tnab j C jxnab
0
xnab j D 1
for b ¤ b 0 . ı There is the asymptotic orthogonality relation B X E na Œ0 D E Vab Œ0 C E WnaB .t; / C o.1/: bD1
Here E refers to the standard (flat) energy and the o-term satisfies lim lim sup o.1/ D 0:
B!1 n!1
ı All Vab Œ0, as well as their evolutions SAQnab .Vab Œ0/ and the WnaB are 1oscillatory. Proof. We follow [1]: There is nothing to do provided An . a / D 0. Hence assume this quantity is > 0. Then pick a profile Va1 .t; x/ 2 L2t;loc Hx1 \ C 1 L2x and associated sequence f.tna1 ; xna1 /gn1 such that SAn . na Œ0/ .t C tna1 ; x C xna1 / * Va1 .t; x/ with 1 E.Va1 / > An . a /: 2
375
9.6 Step 4: Adding the first large component
Using the notation of the lemma, consider then SAn . na Œ0/ .t C tna1 ; x C xna1 / SAn SAQna1 .Va1 Œ0/.0 tna1 ; xna1 / .t C tna1 ; x C xna1 / D SAn . na Œ0/ .t C tna1 ; x C xna1 / SAQna1 .Va1 Œ0/ .t; x/: But by our construction, this expression converges weakly to 0. Furthermore, due to Lemma 9.19, we have that E SAQna1 .Va1 Œ0 .0 tna1 ; xna1 // D E Va1 Œ0 C oL2 .1/: Now we repeat the preceding step, but replace na Œ0 by na Œ0 SAQna1 Va1 Œ0 .0 tna1 ; xna1 /: Thus select a sequence f.tna2 ; xna2 /gn1 and a concentration profile Va2 .t; x/ such that 1 E.Va2 / na SAQna1 .Va1 Œ0/.0 tna1 ; xna1 / ; 2 and furthermore SAn na SAQna1 .Va1 Œ0/.0 tna1 ; xna1 / .t C tna2 ; x C xna2 / * Va2 .t; x/: We obtain that necessarily lim jtna1
n!1
tna2 j C jxna1
Furthermore, we claim that E Va1 Œ0 C E na Œ0 SAQna1 .Va1 Œ0/Œ0
xna2 j D 1:
tna1 D E na Œ0 C oL2 .1/:
This follows again just as in the free case, using Lemma 9.19: We need to show that Z rx;t SAQna1 Va1 Œ0 .0 tna1 ; /rx;t na Œ0 SAQna1 Va1 Œ0/.0 tna1 ; dx R2
D oL2 .1/: Due to Lemma 9.19, up to oL2 .1/, the left-hand side equals Z 0
1Z
rx;t SAn SAQna1 .Va1 Œ0/Œ0 tna1 .t C tna1 ; C xna1 / R2 rx;t SAn na Œ0 SAQna1 .Va1 Œ0/Œ0 tna1 .t C tna1 ; C xna1 / dx:
376
9 The Bahouri–Gérard concentration compactness method
But here we can again use that SAn SAQna1 .Va1 Œ0/Œ0
tna1 .t C tna1 ; C xna1 / D Va1 .t; / C oL2 .1/;
provided t 2 Œ0; 1, while by construction SAn na Œ0
SAQna1 .Va1 Œ0/Œ0
tna1 .t C tna1 ; C xna1 / * 0:
The conclusion is that Z 0
1Z
rx;t SAn SAQna1 .Va1 Œ0/Œ0 tna1 .t C tna1 ; C xna1 / R2 rx;t SAn na Œ0 SAQna1 .Va1 Œ0/Œ0 tna1 .t C tna1 ; C xna1 / dx D oL2 .1/;
from which the asymptotic orthogonality follows. All assertions of the lemma now follow by applying the preceding considerations inductively B times. Figure 9.3 depicts various concentration profiles. More precisely, one should view these profiles as being well-localized in physical space centered at their cores in space-time. The figure then shows the approximate support of the wave evolutions of these profiles. Generally speaking, a will always refer to a frequency atom, whereas b refers to the concentration profile generated by a frequency atom. We shall now apply Lemma 9.23 to the covariant evolution of na Œ0, as well as the remaining components na , Q na , V na .
9.6.3 Selecting geometric concentration profiles At this stage, we face the same issue as in Step 1 above: we have a sequence of component functions V˛ab associated with the essentially singular sequence ˛n , but in order to apply the “energy induction hypothesis”, i.e., the assumption that Ecrit is the minimal energy for which uniform control fails, we need to show that the V˛ab can be assembled to form the Coulomb components of actual maps from R2 ! H2 . We now address this task. To begin with, we may assume that .0/ .0/ nA0 ¤ oL2 .1/ since otherwise nA0 is a perturbative error.
377
9.6 Step 4: Adding the first large component
t D t n2
t D t n1
t D t n1
Figure 9.3. The dependence domains of various concentration profiles
To summarize our construction of the concentration profiles: We started with the Hodge decomposition (all at time t D 0) 1na D @1 Q na C @2 V na ; @ t Q na WD 0na ; 2na D @2 Q na @1 V na Q 1na D @1 na C @2 na ; Q 2na D @2 na @1 na :
@ t na WD Q 0na ;
@ t V na D 0 @ t na D 0
378
9 The Bahouri–Gérard concentration compactness method
From here it is immediate that E.
na
/D D
2 X ˛D0 2 X ˛D0
2 k ˛na kL 2 x
D
2 X
2 k@˛ Q na kL 2 C
˛D0 2 X
2 k@˛ na kL 2 C x
x
2 X ˛D0
2 k@˛ V na kL 2
x
2 k@˛ na kL 2: x
˛D0
Now, we evolve each of the Q na etc. in time using the covariant flow, and apply Lemma 9.23. Changing the notation from that lemma, one obtains the decompositions (with AQ as in (9.82)) rx;t Q na D
B X
rx;t SAQnab .VQ1ab Œ0/ .0
t nab ; x
x nab / C rx;t WQ 1naB ;
rx;t SAQnab .VQ2ab Œ0/ .0
t nab ; x
x nab / C rx;t WQ 2naB ;
rx;t SAQnab .V1ab Œ0/ .0
t nab ; x
x nab / C rx;t W1naB ;
rx;t SAQnab .V2ab Œ0/ .0
t nab ; x
x nab / C rx;t W2naB
bD1
rx;t V na D
B X bD1
rx;t
na
D
B X bD1
rx;t
na
D
B X bD1
where the W errors are small in the -sense when B is large, see (9.83). Here we use the same sequences t nab ; x nab for all decompositions, which of course we can by passing to suitable subsequences. Note that we are working with both the and components here, which is needed for the following result. Proposition 9.24. For any 1 b B, and any ı2 > 0, there exists an admissible (derivative components are Schwartz) map from R2 into H2 , with derivative components jnab , j D 1; 2, and a number ı2 nab 2 R, such that ı2
@1 SAQnab .V1ab Œ0/ .0 t nab ; x/ C @2 SAQnab .V2ab Œ0/ .0 t nab ; x/; @2 SAQnab .V1ab Œ0/ .0 t nab ; x/ @1 SAQnab .V2ab Œ0/ .0 t nab ; x/
P P nab nab
i kD1;2 4 1 @k kı i kD1;2 4 1 @k kı nab nab 2 ; 2 e i ı2 nab 1ı e e < ı2 2ı2 2 2 Lx
for large n.
379
9.6 Step 4: Adding the first large component
Proof. Due to the asymptotic orthogonality relation of Lemma 9.23, given ı2 there exists B0 so that for all b > B0 one can simply take the derivative components to equal zero. In other words, it suffices to consider 1 b B0 . Fix a b, we shall pick B larger, if necessary, and also pick n large enough later. For simplicity introduce the notation V3nab WD @1 SAQnab V1ab Œ0 C @2 SAQnab V2ab Œ0 V4nab WD @2 SAQnab V1ab Œ0/
@1 SAQnab .V2ab Œ0
nab and similarly for W3;4 . Note that we here introduce dependence on n again. Thus at time t D 0, we have the identities
1na e
i
P
i
P
kD1;2
4
.0/ nA0 1 1 @ Œw C kna1 k k
4
.0/ nA0 1 1 @ Œw C kna1 k k
D
B X
V3nab .0
t nab ; x
x nab / C W3naB
V4nab .0
t nab ; x
x nab / C W4naB
bD1
2na e
kD1;2
D
B X bD1
where the W ’s satisfy the smallness property (9.83). Then we distinguish between the following two cases: nab (A): V3;4 . that
t nab ; x
x nab / is of temporally bounded type. By this we mean lim inf jt nab j < 1: n!1
By passing to a subsequence, we may then assume that lim sup jt nab j < 1; n!1
or in fact, that limn!1 t nab exists. nab (B): V3;4 . that
t nab ; x
x nab / of temporally unbounded type. By this we mean lim jt nab j D 1:
n!1
Observe that in this latter case, due to Proposition 9.20, we can conclude that nab V3;4 .
as n ! 1.
t nab ; x
x nab / D oL1 .1/ C oL2 .1/
380
9 The Bahouri–Gérard concentration compactness method
We treat these cases separately, commencing with the temporally bounded nab . t nab ; x x nab / can be approximated Case A. We need to show that V3;4 arbitrarily well by the Coulomb components of admissible maps. We shall do this by physical localization: Note that for b 0 ¤ b, we have either 0
lim jt nab j D 1
n!1
or else lim jx nab
n!1
0
x nab j D 1:
We conclude that if nab is a smooth R nab R, R < 1, centered at x , then we
spatial cutoff localizing to a disc of radius have
nab nab 0 0 0 lim R V3;4 .0 t nab ; x x nab / L2 D 0;
n!1
x
using Proposition 9.20. We also claim Lemma 9.25. Choosing B D B.ı2 ; R/ large enough, we obtain (here ı2 > 0 can be prescribed arbitrarily) naB lim sup k nab R W3;4 kL2x ı2 n!1
for all 1 b B0 . Proof. Recall that naB W3;4 D @1;2 W1naB ˙ @2;1 W2naB ; naB where W1;2 solve the covariant wave equation An u D 0 (where as before An .0/
is defined using nA0 ). But then it is straightforward to check that the spacetime Fourier support of u is contained in a small neighborhood of the light cone intersected with the set jj 1, up to arbitrarily small errors. One can then reason exactly as in [1], see Lemma 3.8 in that paper. ab Therefore, given ı2 > 0, we can pick R D R.ı2 ; V1;2 / with the property that
lim sup nab 1na e R
i
P
kD1;2
n!1
4
.0/ nA0 1 @ Œw C k1na k k
2na e i V3ab .0
t nab ; x
x nab /; V4ab .0
;
P
kD1;2
t nab ; x
4
.0/ nA0 1 @ Œw C k1na k k
x nab /
L2x
ı2 :
381
9.6 Step 4: Adding the first large component
We now need to show that the components
na i nab R 1 e
P
kD1;2
4
1@
.0/ nA0 C k1na k Œwk
;
na i nab R 2 e
P
kD1;2
1@
4
.0/ nA0 C k1na k Œwk
are close to the Coulomb components of an admissible map, up to a constant phase shift. To achieve this, we insert the profile decomposition we obtained above for na 1;2 , i.e., write na nab nab R 1 D R
B hX X
@j
b 0 D1 j D1;2 0
SAQnab0 .VQjab Œ0/.0 na nab nab R 2 D R
B hX X
0
t nab ; x
0
x nab / C WQjnaB
i
. 1/j C1 @j C1
b 0 D1 j D1;2
SA
Qnab 0
0 .VQjab Œ0/.0
t
nab 0
;x
x
nab 0
/ C WQjnaB
i
where @j C1 has to be interpreted modulo 2. Now if we choose B large enough (depending on R, chosen further above), and then choose n large enough, we can ensure that
nab na
R
1
nab R
X
@j SAQnab .VQjab Œ0/.0
t nab ; x
x nab / L2 ı2 x
j D1;2
nab na
R
2
nab R
X
. 1/j C1 @j C1
j D1;2
SAQnab .VQjab Œ0/.0
t nab ; x
x nab / L2 ı2 : x
na We continue by approximating the truncated components nab R 1 by the derivative components of an admissible map .Qxnab ; yQ nab / W R2 ! H2 . For this purpose we recall the identity
j1na D .yna /
1
X kD1;2
4
1
@k @j Πk1na yna ;
j D 1; 2:
382
9 The Bahouri–Gérard concentration compactness method
Inserting the above decomposition for the k1na , we obtain j1na D .yna /
1
X
4
1
@1 @j
B h X X b 0 D1
kD1;2 ab 0
ReSAQnab0 .VQjQ Œ0/.0 C .yna /
X
1
t
nab 0
1
4
jQD1;2
i 0 x nab / C ReWQ jQnaB yna
;x
@2 @j
@jQ
B h X X
Q
. 1/j C1 @jQC1
b 0 D1 jQD1;2
kD1;2 0 ReSAQnab0 .VQjQab Œ0/.0
0
t nab ; x
i 0 x nab / C ReWQ jQnaB yna :
Using the frequency localization of all functions involved, and increasing R if necessary (independently of B), we can then achieve that for n large enough
h X X X
nab 1na .yna / 1 4 1 @1 @j nab @jQ
R j R kD1;2 b 0
jQD1;2
i Re SAQnab0 .VQjQ Œ0/.0 t ;x x / C ReWQ jQnaB yna h X X X Q . 1/j C1 @jQC1 C .yna / 1 4 1 @2 @j nab R ab 0
nab 0
kD1;2 b 0
jQD1;2
0 Re SAQnab0 .VQjQab Œ0/.0
From here we infer that
1na lim sup nab R j
nab 0
t
.yna /
nab 0
1
n!1
;x
x
X
4
nab 0
na i
naB Q / C ReWjQ y
L2x
1
1na na @k @j Πnab y
R k
L2x
kD1;2
ı2 :
ı2 :
Now modify y to a function yQ nab by picking numbers R00 ; R0 with R R00 R0 ; to be specified shortly, and setting yQ nab D e
P
kD1;2
nab 4 R0
1@
2na k PΠR00 ;R00 k
;
whence X @j yQ nab D @ nab j R0 4 yQ nab kD1;2
1
@k PŒ
2na R00 ;R00 k
2na D nab C error R0 j
383
9.6 Step 4: Adding the first large component
where we can achieve that kerrorkL2x ı2 by choosing R00 large enough depending on ı2 and the localization of na in frequency space, and then R0 large enough in relation to R00 . Increasing B if necessary and then choosing n large enough, we can then also achieve that
X
nab na j C1 Q ab Œ0 .0 t nab ; x x nab / nab . 1/ @ S V
R0 2
2 0 j C1 nab Q R A jQ Lx
j D1;2
ı2 and then na k nab R 0 2
na nab R 2 kL2x ı2 :
We next show that the expression X .yna / 1 4
1
1na na @k @j nab R k y
kD1;2
is well-approximated by .Qynab /
X
1
4
1
1na nab Q @k @j nab : R k y
kD1;2
To see this, write .yna /
1
X
4
1
1na na @k @j nab y R k
kD1;2 P
De
kD1;2
nab 4 R0
1@
2na k k
X
4
1
P nab 1na kD1;2 R0 4 @k @j nab e R k
1@
2na k k
kD1;2
C error; where we can achieve kerrorkL2x ı2 by choosing R0 large enough in relation to R and the intrinsic Fourier localization properties of na . Split the phase into the product e
P
kD1;2
nab @k 1 k2na R0
De
P
kD1;2
nab @k 1 PΠR0
2na R00 ;R00 k
e
P
kD1;2
P
nab @k 1 PΠR0 nab
1
2na R00 ;R00 c k
:
2na
We need to show that we can eliminate the factor e kD1;2 R0 @k PΠR00 ;R00 c k . Using similar arguments as in Step 1, choosing R00 large enough in relation to R,
384
9 The Bahouri–Gérard concentration compactness method
it is straightforward to show that, with @k 1 WD 1 @k , P
P nab 1 2na X nab 1na
e kD1;2 R0 @k k kD1;2 R0 P. @k 1 @j nab e R k
1;R00 @k
1
k2na
kD1;2
.e P
ı2 .e
P
kD1;2
nab 1 2na kD1;2 R0 @k P>R00 k
X
nab @k 1 P>R00 k2na R0
P
x 2na 00 1;R k
nab 1 kD1;2 R0 @k P.
1/e P nab 1na kD1;2 R0 4 @k 1 @j nab R k e
1/ L2
2na 1P . 1;R00 @k k
L2x
kD1;2
ı2 : We next show that we can also eliminate e proceeding as in the first section, write e
P
kD1;2
nab @k 1 P< R0
2na R00 k
.Qynab /
P
kD1;2
nab @k 1 P< R0
. Indeed,
1
P nab 1 1na nab Q e kD1;2 R0 @k P< @k 1 @j nab R k y
X
2na R00 k
2na R00 k
kD1;2
X nab D lR
nab .l 1/R e
P
kD1;2
nab @k 1 P< R0
2na R00 k
.Qynab /
1
l2 P nab 1 1na nab kD1;2 R0 @k P< Q @k 1 @j nab e y R k
X
2na R00 k
kD1;2
C nab R e X
P
kD1;2
nab @k 1 P< R0
2na R00 k
.Qynab /
1
P nab 1 1na nab Q e kD1;2 R0 @k P< @k 1 @j nab R k y
2na R00 k
:
kD1;2
Here the cutoff nab localizes to a disc of radius lR around x nab . Then pick a lR point plnab in this disc, for each l, and write for fixed l 2 e
P
kD1;2
nab @k 1 P< R0
X
D
e
e
P
kD1;2 P
kD1;2
kD1;2
2na R00 k
.Qynab /
1
P nab 1 1na nab Q @k 1 @j nab e kD1;2 R0 @k P< R k y nab @k 1 P< R0
nab @k 1 P< R0
X kD1;2
@k
1
2na R00 k
1
2na .plnab / R00 k
1na nab Q @j nab R k y
.Qynab / e
e
P
P
2na R00 k
1
kD1;2
@k 1 P< nab R0
2na R00 k
nab 1 2na .plnab / kD1;2 R0 @k P< R00 k
:
385
9.6 Step 4: Adding the first large component
But then we can estimate nab lR
nab .l 1/R
e
e
P
kD1;2
P
kD1;2
nab @k 1 P< R0
nab @k 1 P< R0
2na R00 k
lR 1 D O. 00 /; R
2na .plnab / R00 k
N
and then using the machinery from Step 1 (which yields a l R00 R, we can achieve that
h
e e
P
P
kD1;2
nab @k 1 P< R0
kD1;2
X
nab @k 1 P< R0
2na R00 k
i 1 .Qynab /
2na .plnab / R00 k
h
1
1
1na nab Q @k @j nab P R k y
e
kD1;2
e
P
kD1;2
kD1;2
1
nab @k 1 P< R0
nab @k 1 P< R0
gain), and choosing
2na R00 k
2na .plnab / R00 k
i
L2x
ı2 l
N
:
Similarly, one can eliminate the second instance of e e
P
P
kD1;2
kD1;2
nab @k 1 P< R0
nab @k 1 P< R0
2na R00 k
2na .plnab / R00 k
:
Hence we have now shown that for B and then n large enough, we have that the functions Qj1nab WD .Qynab /
X
1
kD1;2
2nab @j yQ nab 1na nab Qj Q ; ; WD @k 1 @j nab y R k yQ nab
which of course are the derivative components of admissible maps, satisfy the inequalities k Qj1nab
1na nab kL2x ı2 ; R j
k Qj2nab
2na nab kL2x ı2 : R j
Our next task is to approximate the Coulomb components. For this consider Qjnab e
i
P
kD1;2
.0/ nA0
@k 1 Œwk
C k1na
Qjnab D Qj1nab C i Qj2nab :
;
From the preceding, we can arrange that
nab
Q e j
i
P
kD1;2
.0/ nA0
@k 1 Œwk
C k1na na nab R j e
i
P
kD1;2
.0/ nA0
@k 1 Œwk
C k1na L2x
ı2 :
386
9 The Bahouri–Gérard concentration compactness method
We need to show that we can also arrange (i.e., upon choosing B, n large enough) that
nab
Q e
i
j
P
kD1;2
.0/ nA0
@k 1 Œwk
C k1na
Qjnab e
i
for a suitable constant nab . We first remove the phase e the e e
i
i
support P
@k 1 w k
P
nab R ,
of .0/ nA0
i
kD1;2
P
kD1;2
.0/ nA0
@k 1 wk
and
@k 1 Qk1nab i nab e L2x
ı2
: simply pick a point pnab in e
replace
i
P
kD1;2
.0/ nA0
@k 1 wk
by
.pnab /
. Next, we need to show that Qjnab e kD1;2
P
i
P
kD1;2
@k 1 k1na
is close to Qjnab
Q 1nab kD1;2 @k k 1
, up to a constant phase shift. First, pick R1 R such that P
nab i P
1 1na
Q kD1;2 @k k Q nab e i kD1;2 @k 1 k1na 2 ı2 : e nab j R1 j L x
Next, pick
R0
R1 such that
nab nab
Q e R1 j
i
P
kD1;2
@k 1 k1na
Q nab e e i 1nab nab R1 j
i
P
kD1;2
@k 1 PŒ
1na R0 ;R0 k
L2x
ı2
for suitable 1nab . Next, we claim that picking R2 R0 , we can arrange that Q nab e k nab R1 j
i
P
kD1;2
@k 1 PŒ
1na R0 ;R0 k
Q nab e i nab R1 j
P
kD1;2
@k 1 PŒ
nab Q 1nab R0 ;R0 ΠR2 k
kL2x ı2 :
This is a consequence of the fact that (for R2 large and then n sufficiently large)
nab
e
i
P
R1
kD1;2
@k 1 PŒ
nab Q 1nab R0 ;R0 ΠR2 k
k1na
1 L1 ı2 : x
Finally, we claim that Q nab e nab R1 j
i
P
kD1;2
@k 1 PŒ
nab Q 1nab R0 ;R0 ΠR2 k
1 Q 1nab is very close to Qjnab e i kD1;2 @k k , which is what we need to finish case (A). To see this, note that by choosing R2 large enough in relation to R0 , we obtain from Bernstein’s inequality X
X
1
Q 1nab Q 1nab 1 ı2 : 0 ;R0 @k 1 PŒ R0 ;R0 Œ nab @ P Œ R R2 k k k L
P
x
kD1;2
kD1;2
387
9.6 Step 4: Adding the first large component
This immediately implies
nab nab
Q e R1
i
j
P
kD1;2
@k 1 PŒ
nab Q 1nab R0 ;R0 ΠR2 k
Q nab e nab R1 j
i
P
kD1;2
@k 1 PŒ
Q 1nab R0 ;R0 k
L2x
ı2 :
To conclude, picking R0 large enough in relation to R1 allows us to find a phase e i 2nab such that
nab nab
Q e R1
i
j
P
kD1;2
@k 1 PŒ
Q 1nab R0 ;R0 k
Q nab e nab R1 j
i
P
kD1;2
@k 1 Qk1nab i 2nab e L2x
ı2 :
Since we also have, as mentioned before, that
nab nab
Q Qjnab L2 ı2 : R1 j x
Combining all of the preceding steps, we infer the existence of a phase e i nab such that P
nab na i P
1 1na
e kD1;2 @k k Qjnab e i kD1;2 @k 1 Qk1nab e i nab 2 ı2 : j R L x
0 We then obtain for suitable nab
i 0
e nab Q nab e 1;2
i
P
kD1;2
@k 1 Qk1nab i nab
e
ab V3;4 .0
t nab ; x
x nab / L2 ı2 : x
This finally concludes case (A), i.e., the temporally bounded case. (B): Temporally unbounded case. Here we have limn!1 jt nab j D 1, whence using Proposition 9.20, we see that nab V3;4 .0
t nab ; x
x nab / D oL1 .1/ C oL2 .1/
where we recall the notation V3nab D @1 SAQnab .V1ab Œ0/ C @2 SAQnab .V2ab Œ0/ V4nab WD @2 SAQnab .V1ab Œ0/
@1 SAQnab .V2ab Œ0/:
We now make the following Claim: Choosing n large enough, we have
rx;t S Qnab V ab Œ0 2 ı3 2 A L x
388
9 The Bahouri–Gérard concentration compactness method
for any given ı3 > 0. In particular, V2ab Œ0 D 0 for all b of temporally unbounded type. nab .0 t nab ; x x nab / are approximately Thus in Case (B), the components V3;4 given by the gradient of a suitable (complex valued) function. Once the Claim is established, Case (B) will be straightforward to conclude. In order to prove the Claim, we shall use the curl equations satisfied by the components jna . To begin with, pick R large enough such that
PŒ
na i R;R . j /e B X
P
kD1;2
.Vjnab C2 .0
@k 1 kna
t nab ; x
x nab / C WjnaB C2 / L2 ı2 ;
j D 1; 2:
x
bD1
Then using the Littlewood–Paley trichotomy, and choosing R larger if necessary, we can arrange that
PŒ
PŒ
10R;10R B X
na i R;R . j /e
.Vjnab C2 .0
P
kD1;2
t nab ; x
@k 1 kna
x nab / C WjnaB C2 / L2 ı2 ;
j D 1; 2:
x
bD1
Now fix a cutoff nab which localizes to a large annulus of radius jt nab j around x nab and thickness Rn large enough, such that
nab lim sup nab V3;4 .0
t nab ; x
x nab /
n!1
nab V3;4 .0
t nab ; x
x nab / L2 ı2 : x
By removing finitely many “holes” from this annulus and adjusting nab correspondingly, we can ensure that
nab 0 lim nab V3;4 .0
n!1
0
0 x nab / L2 D 0;
t nab ; x
x
b ¤ b 0 ; 1 b 0 B;
for all b 0 of temporally bounded type. We cannot simply arrange that
naB lim nab W3;4 D 0; L2
n!1
0
nab 0 lim nab V3;4 .0
x
n!1
0
0
0
t nab ; x
0 x nab / D 0
nab where V3;4 .0 t nab ; x x nab / is of unbounded type, and it will be more 0 naB nab 0 complicated to disentangle W3;4 and temporally unbounded V3;4 .0 t nab ; x
389
9.6 Step 4: Adding the first large component 0
nab . From the preceding, choosing R and then n large enough, we x nab / from V3;4 P can arrange that (here the sum 0 is over temporally unbounded profiles)
nab
PŒ
nab ŒVj C2 .0 C
PŒ
10R;10R
0 X
na i R;R . j /e
t nab ; x 0
Vjnab C2 .0
P
kD1;2
@k 1 kna
x nab / 0
t nab ; x
0
x nab / C WjnaB C2 L2 ı2 ; x
j D 1; 2:
b¤b 0
Here R only depends on the frequency concentration of na . We now analyze the curl expression 1
r
h @1 nab PΠr
10R;10R 1
PŒ
h @2 nab PŒ
na i R;R . 2 /e 10R;10R
PŒ
P
kD1;2
@k 1 kna
na R;R . 1 /e
i
i
P
kD1;2
@k 1 kna
i :
We shall show that this expression becomes arbitrarily small when n is sufficiently large. Decompose the above expression into r
1
P 1 na @1 nab PΠ10R;10R PΠR;R . 2na /e i kD1;2 @k k P 1 na r 1 @2 nab PΠ10R;10R PΠR;R . 1na /e i kD1;2 @k k P 1 na C r 1 nab PΠ10R;10R PΠR;R .@1 2na @2 1na /e i kD1;2 @k k P 1 na C r 1 nab PΠ10R;10R PΠR;R . 2na /@1 .e i kD1;2 @k k / P 1 na r 1 nab PΠ10R;10R PΠR;R . 1na /@2 .e i kD1;2 @k k / :
For the first two terms, choosing the cutoff nab suitably, it is clear that for n large enough we have
r
1
P 1 na @1 nab PΠ10R;10R PΠR;R . 2na /e i kD1;2 @k k
P 1 na r 1 @2 nab PΠ10R;10R PΠR;R . 1na /e i kD1;2 @k k
L2x
ı2 : For the third term, we use the schematic curl relation @1 2na @2 1na D ". na /2 ". Note that by including a suitable cutoff Q nab having similar characteristics as
390
9 The Bahouri–Gérard concentration compactness method
nab , we obtain
1 nab PΠ10R;10R PΠR;R .@1 2na @2 1na /e
r r 1 nab PΠ10R;10R PΠR;R . Q nab ". na /2 "/e
i
P
i
kD1;2
P
@k 1 kna
kD1;2
@k 1 kna
L2x
ı2 : Now we insert the decomposition jna D
B X
0 VQjnab C2 .0
0
t nab ; x
0 x nab / C WQjnaB C2 ;
j D 1; 2:
b 0 D1
For any chosen B, by picking n large enough, we can achieve that for all temporally bounded b 0 B X
nab
Q
b 0 D1;
VQjnab C2 .0
x nab /
t nab ; x
L2x
b 0 ¤b
and hence we reduce to estimating (where now the sum rally unbounded profiles)
h
1 nab nab PŒ 10R;10R PŒ R;R . Q nab ŒVQ3;4
r 0 X
C
nab 0 naB 2 VQ3;4 C WQ 3;4 /e
P0
i
ı2 ;
only involves tempo-
P
kD1;2
@k 1 kna
i
L2x
b 0 ¤b
:
Now recall from Proposition 9.20 that for all temporally unbounded b 0 nab 0 Q nab VQ3;4 D oL1 .1/ C oL2 .1/:
Hence we obtain
1 nab PŒ
r
10R;10R
PŒ
nab Q nab ŒVQ3;4 R;R .
C
0 X
nab 0 2 VQ3;4 /e
i
P
kD1;2
@k 1 kna
L2x
b 0 ¤b
C r
1
nab PŒ
10R;10R
C
PΠ0 X
b 0 ¤b
nab Q nab ŒVQ3;4 R;R . nab 0 Q naB VQ3;4 W3;4 /e
i
P
kD1;2
@k 1 kna
L2x
ı2
391
9.6 Step 4: Adding the first large component
for n large enough. Finally, consider the term h naB 2 r 1 nab PŒ 10R;10R PŒ R;R . Q nab ŒWQ 3;4 /e
i
P
kD1;2
@k 1 kna
i :
Here we split naB WQ 3;4 D PŒ
Q naB R1 ;R1 c W3;4
C PŒ
Q naB R1 ;R1 W3;4 :
Then if B is chosen large enough in relation to R1 , we obtain both
PŒ R ;R c WQ naB 2 ı2 ; PŒ R ;R WQ naB 1 ı2 : 3;4 3;4 1 1 1 1 L L x
x
Here the first inequality holds of course uniformly in n; B due to the frequency localization. From here we infer that for B and then n large enough, we obtain
P
1 nab nab Q naB 2 i kD1;2 @k 1 kna r P P . Q ΠW /e
2 ı2 : Œ 10R;10R Œ R;R 3;4 Lx
The argument for showing
P 1 na
1 nab PΠ10R;10R PΠR;R . 2na /@1 .e i kD1;2 @k k /
r
P 1 na r 1 nab PΠ10R;10R PΠR;R . 1na /@2 .e i kD1;2 @k k /
L2x
ı2
of course proceeds in identical fashion. Summarizing what we have achieved thus far in Case (B), we have shown that for n large enough, we obtain
P 1 na
1 nab
r @1 PΠ10R;10R PΠR;R . 2na /e i kD1;2 @k k
P 1 na r 1 @2 nab PŒ 10R;10R PŒ R;R . 1na /e i kD1;2 @k k 2 ı2 : Lx
In light of the fact pointed out earlier that (j D 1; 2) h i P 1 na nab PΠ10R;10R PΠR;R . jna /e i kD1;2 @k k is well-approximated by7
nab
0 X nab 0 naB Vj C2 C Vjnab C2 C Wj C2 ; b 0 ¤b
7
Here
P0
again only involves temporally unbounded profiles.
392
9 The Bahouri–Gérard concentration compactness method
we then infer that (recalling the definition of V3;4 , W3;4 )
1 nab
r @1 @2 Œ..SAQnab .V1ab Œ0//.0 t nab ; x x nab / C
0 X
0
0
SAQnab0 .V1ab Œ0//.0
0
t nab ; x
x nab / C W1naB /
b 0 ¤b
@1 Œ.SAQnab .V2ab Œ0//.0 C
0 X
t nab ; x
0
x nab /
0
SAQnab0 .V2ab Œ0//.0
0
t nab ; x
x nab / C W2naB
b 0 ¤b
r C
1
@2 nab @1 Œ..SAQnab .V1ab Œ0//.0
0 X
0
0
SAQnab0 .V1ab Œ0//.0
t nab ; x
t nab ; x
x nab /
0
x nab / C W1naB /
b 0 ¤b
C @2 Œ.SAQnab .V2ab Œ0//.0 C
0 X
t nab ; x
0
0
SAQnab0 .V2ab Œ0//.0
t nab ; x
x nab / 0
x nab / C W2naB
L2x
b 0 ¤b
ı2 : But then choosing the cutoff nab as above and picking n large enough, we conclude (noting cancelations in the preceding expression) that
r
1
C nab
4 SAQnab .V2ab Œ0/ .0 0 X b 0 ¤b
0
SAQnab0 .V2ab Œ0//.0
t nab ; x 0
t nab ; x
x nab / 0
x nab / C nab W2naB
L2x
ı2 :
This inequality, together with the approximate orthogonality of the two summands involved, then gives the smallness of either summand separately: Recall from Lemma 9.23 and its proof that we have (for sufficiently large n) ˇ ˇZ ˇ ˇ rx;t .SAQnab .V2ab Œ0//.0 t nab ; x x nab / rx;t W2naB .0; / dx ˇ ı2 ˇ 2 R ˇZ ˇ rx;t .SAQnab .V2ab Œ0//.0 t nab ; x x nab / ˇ 2 R ˇ 0 ˇ 0 0 rx;t .SAQnab0 .V2ab Œ0//.0 t nab ; x x nab / dx ˇ ı2 ; b 0 ¤ b:
393
9.6 Step 4: Adding the first large component
Now recall the vanishing condition at time t D 0 B X
0
@ t SAQnab0 .V2ab Œ0/.0
0
t nab ; x
0
x nab / C @ t W2naB D 0
b 0 D1
which we used to define the linear covariant evolution of na . Applying the cutoff nab , and choosing n large enough, we infer that
@ t S Qnab .V ab Œ0/.0 t nab ; x x nab / 2 A C nab
0 X
0
@ t SAQnab0 .V2ab Œ0/.0
0
t nab ; x
0 x nab / C nab @ t W2naB L2
x
b 0 ¤b
ı2 : However, this inequality, together with the two preceding ones, implies that
rx;t S Qnab .V ab Œ0/ .0 t nab ; x x nab / 2 C k nab rx;t W naB k 2 2 2 Lx A Lx X
0 0 0 nab ab nab nab
C rx;t SAQnab0 .V2 Œ0/ .0 t ;x x / L2 ı2 : x
b 0 ¤b
Summarizing the state of affairs in Case (B), we have shown thus far that the Claim holds. But this then says that the “diluted concentration profile” given by V3nab D @1 SAQnab .V1ab Œ0/ C @2 SAQnab .V2ab Œ0/ .0 t nab ; x x nab / V4nab WD @2 SAQnab .V1ab Œ0/ @1 SAQnab .V2ab Œ0/ .0 t nab ; x x nab / is given, up to an L2 -error of size ı2 , by the pure gradient term V3nab D @1 SAQnab .V1ab Œ0/.0
t nab ; x
x nab /
V4nab D @2 SAQnab .V1ab Œ0/.0
t nab ; x
x nab /:
We shall now use this to construct a map from R2 ! H2 whose Coulomb derivanab tive components are close to V3;4 . Indeed, picking R large enough and then n sufficiently large depending on R, it is straightforward to check that @j PŒ R;R SAQnab .V1ab Œ0/.0 t nab ; x x nab / D @j PŒ R;R SAQnab .V1ab Œ0/.0 t nab ; x x nab / P i kD1;2 @k 1 @k PŒ R;R SAQ nab .V1ab Œ0/.0 t nab ;x x nab / e C error;
394
9 The Bahouri–Gérard concentration compactness method
where we have kerrorkL2x ı2 . Then we define a map .x; y/ W R2 ! H2 (here we abuse notation heavily, this map of course depends on n; a; b) via x WD RePŒ
R;R
y WD e ImPŒ
SAQnab .V1ab Œ0/.0
R;R
t nab ; x
SAQ nab .V1ab Œ0/.0 t nab ;x
x nab / ; x nab / :
These then satisfy
@ x @j y
j Ci
y y
@j PŒ R;R SAQnab .V1ab Œ0/.0
t
nab
;x
x
nab
/
L2x
ı2 ;
and the associated Coulomb derivative components are the desired approximations. This concludes the proof of Proposition 9.24. Summary thus far, for both (A), (B): We have shown that we have the “covariant Bahouri–Gérard decompositions” 1na e i
P
kD1;2
.0/ nA0
@k 1 Œwk
C k1na
D
B X
V3nab .0
t nab ; x
x nab / C W3nab
V4nab .0
t nab ; x
x nab / C W4nab
bD1
2na e i
.0/ P nA0 1 C k1na kD1;2 @k Œwk
D
B X bD1
where we have V3nab WD @1 SAQnab .V1ab Œ0/ C @2 SAQnab .V2ab Œ0/ V4nab WD @2 SAQnab .V1ab Œ0/
@1 SAQnab .V2ab Œ0/
and similarly for W3;4 . Furthermore, for n large enough and any given ı2 > 0, we can find maps .xı2 nab ; yı2 nab / W R2 ! H2 , with the property that their (spatial) Coulomb derivative components are ı2 close (within the L2 -metric) to constant nab phase shifts of the V3;4 .0 t nab ; x x nab /. We shall now refine the information we have by proving the following Lemma 9.26. Given ı2 > 0, we can pick B and then n large enough such that krx;t W2naB kL2x ı2 :
395
9.6 Step 4: Adding the first large component
Remark 9.27. Recalling the identities W3naB D @1 W1naB C @2 W2naB ; W4naB D @2 W1naB
@1 W2naB :
naB are essentially pure gradient terms, like in We see that this says that W3;4 Case (B).
Proof. (Lemma 9.26) The proof is quite similar to the Case (B) above. Given ı2 > 0, first choose an index B1 such that we have B
X nab lim sup V3;4 .0 n!1
t nab ; x
x nab / L2 ı2 ; x
bDB1
for any B B1 . Further, pick R D R.ı2 / with the property that
lim sup PΠn!1
na i R;Rc . j /e
P
.0/ 1nA0
@k 1 Πk1na Cwk
kD1;2
L2x
ı2 ;
j D 1; 2:
Increasing R if necessary, we can then also achieve that (for n large enough)
PŒ
10R;10R PŒ
na i R;R . 1;2 /e B1 X
P
kD1;2
nab V3;4 .0
.0/ 1nA0
@k 1 Πk1na Cwk
t nab ; x
bD1
x nab /
naB W3;4
L2x
ı2 :
Here we will choose B sufficiently large in relation to B1 ; ı2 . Now pick a cutoff which localizes to the union of large discs covering most of the support (in the nab L2 -sense) of the atoms V3;4 .0 t nab ; x x nab / of bounded type, i.e., for which lim sup jt nab j < 1, 1 b B1 . Of course then depends on a; B1 ; n, but we suppress this dependence here. Picking suitably and then choosing n large enough, we can then ensure that 0B B1 1
hX i X
nab nab nab nab nab nab V3;4 .0 t ;x x / V3;4 .0 t ;x x /
.1 /
bD1
where
P0 B1
bD1
bD1
L2x
ı2 ;
indicates that we only sum over the atoms of “unbounded type”.
396
9 The Bahouri–Gérard concentration compactness method
Summarizing the above steps, we now have
.1
/PŒ 10R;10R PŒ
na i R;R . 1;2 /e
P
.0/ 1nA0
@k 1 Πk1na Cwk
kD1;2
0
B1 X
nab V3;4 .0
t nab ; x
x nab /
naB /W3;4
.1
L2x
bD1
ı2 :
By picking B large enough (recall that we can do so independently of B1 ), we may also assume that k.1
naB /W3;4
naB W3;4 kL2x ı2 :
Here we use Lemma 9.25. Next, we calculate the curl of the Coulomb components, localized as above, and with an extra cutoff .1 /. Thus we want to estimate the expression @2 .1
/PΠ@1 .1
10R;10R PŒ /PŒ
na R;R . 1 /e
10R;10R PŒ
i
P
kD1;2
na R;R . 2 /e
i
.0/ 1nA0
@k 1 Πk1na Cwk
P
kD1;2
.0/ 1nA0
@k 1 Πk1na Cwk
:
This we can estimate as in Case (B): Of course the case when a derivative falls on .1 / is negligible. Then repeating the arguments in Case (B) above, we need to estimate the schematic expression
.1
/PŒ 10R;10R PŒ
R;R .Œ
na 2
/e
i
P
kD1;2
.0/ 1nA0
@k 1 Πk1na Cwk
i :
Here we use the Bahouri–Gérard decomposition of the na , i.e., na 1;2 D
B X
V3;4 .0
t nab ; x
naB x nab / C W3;4 :
bD1
It is clear that we then reduce to estimating 0
.1
/PŒ 10R;10R PŒ
R;R ..1
/ Q
B1 X
V3;4 .0
t nab ; x
x nab /
bD1
C
naB 2 W3;4 /e i
P
kD1;2
.0/ 1nA0
@k 1 Πk1na Cwk
:
397
9.7 Step 5: Invoking the induction hypothesis
But the contribution of the terms V3;4 .0 t nab ; x x nab /, 1 b B1 can be made arbitrarily small by choosing n large enough, while the contribution of naB W3;4 is handled by placing one factor into L2x and the other into L1 x . Summarizing, we have now shown that 0
r
1
@2
B1 X
V3nab .0
t nab ; x
x nab / C W3naB
bD1 0
r
1
@2
B1 X
V4nab .0
t
nab
;x
x
nab
/C
W4naB
bD1
L2x
ı2 :
nab naB But then recalling the defining relations for V3;4 ; W3;4 , we can repeat the argument from part (B) in the preceding proof to conclude that for B and then n large enough, we have krx;t W2naB kL2x ı2 ;
as desired.
Proposition 9.24 together with Lemma 9.26 are key technical tools we shall use in the next section when bounding the wave maps with data 0
w nA0 C na ; where a D 1.
9.7
Step 5: Adding the first large atomic component and invoking the induction hypothesis
In Step 3, we constructed a wave map with data corresponding to the lowest frequency “non-atomic” part, whose Coulomb components are .0/
nA ˚˛ 0
e
i
P
kD1;2
4
1@
.0/ nA0 k ˚k
.0/
D
nA w˛ 0
e
i
P
kD1;2
4
1@
.0/ nA0 k wk
C oL2 .1/:
Our next step now is to prove bounds for the wave map whose Coulomb components are given by n.<1/ ˛
.0/
WD
nA Œw˛ 0
C
˛n1 e i
P
kD1;2
4
1@
.0/ nA0 C kn1 k Œwk
C oL2 .1/;
398
9 The Bahouri–Gérard concentration compactness method
provided we make the following key Energy assumption: All concentration profiles have energy < Ecrit . Thus E.V ab / < Ecrit 8 b:
(9.84)
As before, in order to avoid confusion, we shall denote the superscript 1 here instead by a, it being understood that a D 1. Thus we now intend to prove global bounds for the evolution of the Coulomb data n.
.0/
nA0
WD Œw˛ .0/
nA0
D w˛
e
i
P
C ˛na e
kD1;2
4
i
P
kD1;2
4
1@
.0/ nA0 C kna k Œwk
C oL2 .1/
.0/ nA0 1@ w k k
C
˛na e i
P
kD1;2
4
1@
.0/ nA0 C kna k Œwk
C oL2 .1/:
From the preceding section, we obtain a decomposition of the added term Q ˛na WD ˛na e
i
P
kD1;2
4
.0/ nA0 1 @ Œw C kna k k
as a sum of concentration profiles at time t D 0. Note that this time in principle plays no distinguished role, other than that we are guaranteed existence of the evolution of the wave maps with above data on some small time interval centered at t D 0. Recall the decompositions (for any B 1) 1na e
i
P
kD1;2
.0/ nA0 C kna k Œwk
4
1@
4
.0/ nA0 1 @ Œw C kna k k
D
B X
V3nab .0
t nab ; x
x nab / C W3naB
V4nab .0
t nab ; x
x nab / C W4naB
bD1
2na e
i
P
kD1;2
D
B X bD1
where V3nab WD @1 SAQnab .V1ab Œ0/ C @2 SAQnab .V2ab Œ0/ V4nab WD @2 SAQnab .V1ab Œ0/
@1 SAQnab .V2ab Œ0/
9.7 Step 5: Invoking the induction hypothesis
399
and similarly for W3;4 , while we also have 0na e i D
P
kD1;2
B X
4
1@
.0/ nA0 C kna k Œwk
@ t SAQnab V1ab Œ0 .0
t nab ; x
x nab / C @ t W1naB ;
bD1 B X
@ t SAQnab V2ab Œ0 .0
t nab ; x
x nab / C @ t W2naB D 0:
bD1
These decompositions are understood to hold at time t D 0, of course. Now the fact that for B large enough (and then n large enough) we can arrange that krx;t W2naB kL2x ı2 implies that B
X
@ t S Qnab .V ab Œ0/.0 A
2
t nab ; x
x nab / L2 ı2 : x
bD1
Recall that we have temporally bounded concentration profiles, as well as temporally unbounded ones. Then it is intuitively clear that the evolution of Q na (this is not well-defined strictly speaking, we can only evolve Coulomb components of actual maps; however, we can think of Q na as the difference between the components of maps) will be dominated for a large time interval around t D 0 by the evolution of the temporally bounded concentration profiles, which will exhibit nonlinear behavior, while the temporally unbounded ones will behave like free waves for a long time. In order to make things precise, we introduce a hierarchy of temporal scales, which means we order the times t nab according to whether they are positive or negative and then whether lim .t nab
n!1
0
t nab / D ˙1:
Assume that this way, we arrive at the list of representative time scales, M D M.B/, 0 D t nab1 ; t nab2 ; : : : ; t nabM ; where we have t nabi > 0, say, and lim .t nabj
n!1
t nabj
1
/ D 1;
and furthermore for each b 2 f1; 2; : : : ; Bg, we have t nab D t nabj for some j as above. Note that we have chosen to equate those times here that do not diverge
400
9 The Bahouri–Gérard concentration compactness method
from each other. This can of course be done by passing to a subsequence such that the difference of these times converges, and the redefining the concentration profiles accordingly. We then implement an inductive procedure, controlling the evolution of n.
9.7.1 Proving a priori bounds for the evolution of scale
n.
the lowest time
Here we prove a priori bounds on the (wave map) evolution of the Coulomb comn.
.0/
nA
D w1;20 e C
i
P
B X
kD1;2
4
nab .0 V3;4
1@
.0/ nA0 k wk
t nab ; x
naB x nab / C W3;4 C oL2 .1/
bD1 n.
.0/ nA0
D w0
e
C
i
P
B X
kD1;2
4
1@
.0/ nA0 k wk
@ t SAQnab V1ab Œ0 .0
t nab ; x
x nab / C @ t W1naB :
bD1 naB The “tail ends” W3;4 , @ t W1naB here satisfy the smallness condition naB W3;4 ; @ t W1naB D oL2 .1/ C oL1 .1/
where o./ here is meant in case B; n ! 1. Observe from Lemma 9.26 that we actually have naB W3;4 D @1;2 W1naB C error;
kerrorkL2x ı2 ;
provided we choose B and then n large enough. Furthermore, the proof of Proposition 9.24, case (B), reveals that for concentration profiles which are temporally
401
9.7 Step 5: Invoking the induction hypothesis
unbounded, we have nab V3;4 .0
t nab ; x
x nab / D @1;2 SAQnab V1ab Œ0 .0
t nab ; x
x nab /:
n.
We shall now build the evolution of ˛ as the sum of well-known pieces, namely the evolutions of the atomic profiles, plus an error term, which we will show will remain small. To make things precise, we now use the following construction: We shall use ı2 > 0 as a smallness parameter which will ultimately hinge on intrinsic properties of the concentration profiles as well as the S -bound nA
.0/
on the already constructed low frequency part ˛ 0 , and be specified at the end of the construction. Thinking of ı2 > 0 as fixed for now, we first pick a large cutoff B1 with the property that h lim sup n!1
X
nab V4;3 L2
x
BbB1
X
C
i
@ t SAQnab .V1ab Œ0/ L2 C k@˛ W1naB kL1 CL1 L2x ı2 x
t;x
t
BbB1
for any B B1 . Then we evolve the concentration profiles corresponding to a b 2 f1; 2; : : : ; B1 g as follows: (I): Evolution of temporally bounded concentration profile. Here, by passing to a subsequence, we may assume that t nab converges as n ! 1, and we may then set t nab D 0 by time translation. Also, it is apparent that then V3nab D @1 V1ab .0; // C @2 V2ab .0; / C oL2 .1/ V4nab D @2 V1ab .0; //
@1 V2ab .0; / C oL2 .1/; @ t SAQnab .V1ab Œ0/
D @ t V1ab .0; / C oL2 .1/ are all essentially independent of n. Now according to Proposition 9.24, we can find, for each ı2 > 0, a constant phase ı2 ab and an admissible map from R2 ! H2 whose Coulomb components ˛abı2 satisfy
i
e ı2 ab
abı2 1;2
nab V3;4 ı3 ; L2 x
i
e ı2 ab
abı2 1;2
@ t V1ab .0; / L2 ı3 : x
For the sake of simplicity, we now refer to the Coulomb components of such a map, which we choose for ı3 extremely small (depending on B1 etc. and to be specified later), simply as ˛ab .
402
9 The Bahouri–Gérard concentration compactness method
First, we evolve the components of ˛ab on a large time interval I ab centered at t D 0, using the wave maps flow for the Coulomb components. This yields an a priori bound k ab kS < Cab due to our energy assumption (9.84). Furthermore, due to Corollary 7.27 as well as Remark 7.28, given ı2 > 0, one can then choose time intervals I1 ; I2 ; : : : ; IMab .ı2 / where the final one is of the form Œt1abı2 ; 1/, say, such that ab
jIj D
ab jL
C
ab jNL
with8
ab jNL S.Ij R2 /
ı2 ;
rx;t
ab P jL L1 t H
1
. Ecrit :
ab Here of course jL D 0. Note that the intervals Ij here only depend on a; b as well as the smallness parameter ı2 . By the Huygens principle, one may assume ab that the support of jL is contained in the set jxj jt j C Dab .ı2 / for some (possibly very large number Dab .ı2 /). But then by choosing a much larger time T abı2 , we can arrange that
ab
abı2
; 1/; / L1 CL1 L2 ı2 : Mab L .ŒT t;x
t
x
These considerations reveal that pursuing the wave maps evolution of the components ab long enough, we eventually find that ab
.t; / D oL2 .1/ C oL1 .1/
where o./ is in the sense as jtj ! 1. This conclusion is of critical importance: Note that thus far we have not taken .0/ the low frequency contribution from w nA0 (from Step 3) into account, which starts to play an important role for extremely large times. The above asymptotic description allows us to incorporate this low-frequency effect by adjusting the ab linear evolution of M from flat to covariant. In more precise terms, we now ab L make the following choice of an extension Q ˛ab of the data ˛ab : ı On the interval Œ0; T abı2 , we let Q ab D ab . 8
The implied constant in the second inequality here is universal, independent of ı2 .
403
9.7 Step 5: Invoking the induction hypothesis
ı On the interval ŒT abı2 ; 1/, we let Q ab be the covariant extension of ab ŒT abı2 , i.e., we have An Q ab D 0 .0/
P
1
.0/ 1nA0
. on ŒT abı2 ; 1/, where An is defined with inputs ˚ nA0 e i kD1;2 4 @k ˚k ab Q More precisely, we apply a Hodge decomposition to the data ˛ as in (9.60), (9.61), and evolve these components as in Step 3. In order to avoid a “kink” at the juncture of these two regimes, we define Q ab D .
ŒT abı2 (9.85) where the notation for the second term is schematic, and . 1;T abı2 C10/ .t / smoothly localizes to the indicated interval and satisfies 1;T abı2 C10/ .t /
.
ab
C .1
.
1;T abı2 C10/ /.t /SAn
1;T abı2 C10/ jŒ0;T abı2
ab
D 1:
With these definitions, one can prove the following bound. Proposition 9.28. We have a bound of the form k Q ab kS < C.Cab /; where we recall the assumption k ab kS < Cab from above. Furthermore, denoting by ck , k 2 Z, a frequency envelope controlling the data at time t D 0, i.e., X
2 1 ck D 2 jl kj Pl ab .0; / L2 2 x
l2Z
for sufficiently small a priori constant > 0, one has kPk Q ab kS Œk C.Cab /ck : Proof. The proof of this follows from Proposition 9.14, as well as Lemma 7.23 and its proof. The idea now in Case (I) is to use Q ab as approximate evolution of the data ab globally in time, for n large enough. Thus Q ab j Œ0;T abı2 is the actual wave abı 2 maps flow, while beyond time T , we use the covariant linear evolution.
404
9 The Bahouri–Gérard concentration compactness method
(II): Evolution of temporally unbounded concentration profile. Here we have limn!1 jt nab j D 1, and as before, 1 b B1 , where we have chosen B1 above. In this case, using the argument from Case (B) in the proof of Proposition 9.24 and arguing as at the beginning of the preceding Case (I) (we again write nab instead of nabı2 ), nab D @˛ SAQnab V1ab Œ0 .0 t nab ; x x nab / C error; ˛ D 0; 1; 2; ˛ with kerrorkL2x ı2 . In this case we set Q ˛nab D @˛ S Qnab V1ab Œ0 .t A
t nab ; x
x nab /;
the covariant linear evolution. Of course this becomes inaccurate when t ! t nab and the nonlinear effects start to become relevant, but we recall that we are on the lowest time scale in this subsection, i.e., t t nab2 . Then we have the following bound. Proposition 9.29. There is a bound of the form k Q nab kS < C.Ecrit /: Furthermore, denoting by ck , k 2 Z, a frequency envelope controlling the data at time t D 0, i.e., X
2 1 ck D 2 jl kj Pl nab .0; / L2 2 x
l2Z
for sufficiently small a priori constant > 0, kPk Q ab kS Œk C.Ecrit /ck : naB (III): Evolution of the weakly small error. These are the components W3;4 ; naB @ t W1 . From Lemma 9.26, we know that naB W3;4 D @1;2 W1naB C error
where we can force kerrorkL2x ı2 by choosing B and then n large enough. We then evolve W1naB using the covariant linear evolution, i.e., An W1naB .t; x/ D 0;
W1naB Œ0 D .W1naB ; @ t W1naB /;
naB and then define W3;4 .t; x/ D @1;2 W1naB .t; x/.
9.7 Step 5: Invoking the induction hypothesis
405
We have now defined the evolutions of all the ingredients of Q ˛na . We claim that by choosing ı2 small enough and then B and n large enough, the sum of all these constituents gives the correct evolution of Q ˛na up to a small error. This is clarified the following Core Proposition for Bahouri–Gérard II which ties it all together. Proposition 9.30. There is a cutoff ı2 > 0 sufficiently small, depending on the ab profiles V1;2 Œ0, 1 b B1 , as well as the a priori bound we have established .0/
for nA0 , such that the following holds: picking B1 and then n large enough, we can write (with B1 chosen as above) on Œ0; t nab2 C R2 for C sufficiently large and depending on the Q ˛nab of unbounded type, b D 1; 2; : : : ; B1 , n.
.0/
nA0
D ˚˛ C
B1 X
e
i
P
kD1;2
4
1@
.0/ nA0 k ˚k
.t; x/
Q ˛nab .t; x/ C @˛ W naB1 .t; x/ C ˛ .t; x/; 1
˛ D 0; 1; 2
bD1
where the components Q ˛nab .t; x/, @˛ W1naB1 .t; x/, are constructed as in (I)–(III) above, and with kkS.Œ0;t nab2 C R2 / ı2 : Moreover, kPk kS.Œ0;t nab2 C R2 / is exponentially decaying for frequencies k > log.an / Thus the inequality above implies uniform smallness of .t; x/ for t 2 Œ0; t nab2 C . Remark 9.31. There appears to be circular reasoning in the statement of this result: we need to choose ı2;3 > 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.
the various constituents in the decomposition of ˛ . To begin with, we split the (large) time interval Œ0; t nab2 C into finitely many intervals Œ0; t nab2
1 C D [jMD1 Ij .0/
.0/
nA0
nA0
D ˚˛
where we have a decomposition (with ˛ .0/
nA
.0/
nA
e
i
P
kD1;2
4
1@
.0/ 1nA0 k ˚k
)
.0/
nA0 jIj D jL 0 C jNL0 with, see Corollary 7.27,
nA.0/
0 jNL
.0/
S.Ij
R2 /
rx;t
< "2 ;
nA0 jL
1
P L1 t H
1
2 : . "2 2 Ecrit
We then run a bootstrap argument inductively on each of these intervals, where of course M1 D M1 .Ecrit / is not too large. We shall now work on the interval I1 , say. This enables us to use the covariant energy estimate from Step 3. Clearly, the evolved concentration profiles also interact with ; we then further subdivide the intervals Ij into smaller ones, which by abuse of notation we again label as Ij , such that X X X Q nab jI D Q nab C Q nab : j jL jNL b2f1;2;:::B1 g
b2f1;2;:::B1 g
b2f1;2;:::B1 g
Note that now the number of intervals is of the form M1 D M1 .Ecrit ; ab fV1;2 Œ0gb2f1;2;:::;B1 g /. Furthermore, one has X
Q nab
jNL S.Ij R2 /
< "2
b2f1;2;:::B1 g
while also
X
rx;t Q nab
P jL L1 t H
1
1
2 . "2 2 Ecrit
b2f1;2;:::B1 g
where "2 is a universal constant depending only on Ecrit . The fact that we derived the last inequality with a universal implied constant hinges on the approximate orthogonality of the Q nab for n large enough. One may object at this point that the choice of B1 was dictated by ı2 , and hence may be extremely large, which in turn means that the number M1 of intervals above depends on ı2 and may also become extremely large. The following observation, however, shows that M1 only depends on a fixed number of concentration profiles independent of ı2 :
9.7 Step 5: Invoking the induction hypothesis
407
Important technical observation. Here we note that M1 really only depends on ab Œ0gb2f1;2;:::;B0 g , for some B0 with the property that fV1;2 X
V ab Œ0 2 2 < 0 1;2 L x
bB0
where 0 is the small-energy global well-posedness cutoff. Thus we can make ı2 small without increasing M1 concurrently. To see this, write ˚ ab ˚ ab ˚ ab V1;2 Œ0 b2fB0 ;B0 C1;:::;B1 g D V1;2 Œ0 b21 [ V1;2 Œ0 b22 ; 1 [ 2 D fB0 ; B0 C 1; : : : ; B1 g; ab so that fSAQnab .V1;2 Œ0/.0 t nab ; x x nab /gb21 is the collection of temporally bounded concentration profiles with b 2 fB0 ; B0 C1; : : : ; B1 g. Then the argument that was used for Case (A) in the proof of Proposition 9.24 reveals that we can approximate X nab V3;4 .0 t nab ; x x nab / b22
up to a constant phase shift arbitrarily well by the Coulomb components of an admissible map, and then Proposition 9.14 allows us to evolve the data X nab V3;4 .0 t nab ; x x nab / C oL2 .1/ b22
using the covariant linear flow on Œ0; t nab1 . This leads to bounds that are uniform in B0 ; n only involving 0 . Handling the contribution of X nab V3;4 .0 t nab ; x x nab / C oL2 .1/; b21
i.e., the “tail” of bounded concentration profiles, is more complicated since we may no longer necessarily approximate this sum by Coulomb components of adnab missible maps, but only the individual summands V3;4 .0 t nab ; x x nab /. Thus the correct evolution of this term has to consist of the evolution of the individual ingredients, and one then needs to bound the S -norm of this (very large) sum in terms of an a priori bound, provided n is large enough. In this regard we have the following result. nab Lemma 9.32. For each b 2 1 and t 2 Œ0; t nab1 , denote by V3;4 .t t nab ; x x nab / C oL2 .1/ the (nonlinear) wave maps evolution of the Coulomb components
408
9 The Bahouri–Gérard concentration compactness method
nab .t t nab ; x of an admissible sufficiently good approximation to the data V3;4 x nab /, as in the preceding discussion. Then for n large enough, we have
X nab
V3;4 .t t nab ; x x nab / C oL2 .1/ S Œ0;t nab1 . "0 b21
for a suitable universal implied constant. nab Proof. (Lemma 9.32) For each V3;4 .t t nab ; x x nab /CoL2 .1/, pick an interval Œ0; tQab with the property that we can write nab V3;4 .t t nab ; x x nab / C oL2 .1/ jŒ0;tQab c nab .t t nab ; x x nab / C oL2 .1/ L D V3;4 nab .t t nab ; x x nab / C oL2 .1/ NL C V3;4
where we impose the condition
rx;t V nab .t t nab ; x x nab / C oL2 .1/ 1 P 3;4 L L H t
nab . kV3;4 .0
nab
V .t 3;4
t nab ; x
x nab / C oL2 .1/
x
1
t nab ; x
x nab /kL2x
NL S.Œ0;tQab c R2 /
"0 B1
where the implied constant in the first inequality is universal. That this is possible follows from Corollary 7.27 and Remark 7.28. Choosing n large enough and exploiting essential disjointness of the supports at time t nab1 , we can arrange that X 1
rx;t V nab .t nab1 t nab ; x x nab / C oL2 .1/ 2P 1 2 . "0 3;4 L H x
b21
which then implies (for large enough n)
X nab
V3;4 .t t nab ; x x nab / C oL2 .1/ L S.Œ0;max b21
b21
tQab c R2 /
. "0 :
In order to complete the proof of the lemma, we need to also control
X nab
V3;4 .t t nab ; x x nab / C oL2 .1/ S.Œ0;max : tQab R2 / b21
b21
nab Here we exploit the fact that for n large, the functions V3;4 .t t nab ; x x nab /C oL2 .1/ are supported on disjoint light cones up to small errors with respect to
409
9.7 Step 5: Invoking the induction hypothesis
nab P .t t nab ; x x nab / C oL2 .1/ are S. One then concludes that b21 V3;4 the spatial Coulomb components of a solution to the wave maps problem up to arbitrarily small error (as n ! 1), with energy . "0 at time t D 0. The small energy well-posedness then implies
X nab
V3;4 .t t nab ; x x nab / C oL2 .1/ S.Œ0;tQab R2 / . "0 b21
where the implied constant is universal. Now assume the bound9 kPk kS C5 ı2 :
We show that provided we choose C5 D C5 .Ecrit / large enough, we can bootstrap C5 to C25 , whence we obtain the bound on all of I1 . Then we continue the argument to I2 etc. Note that by choosing ı2 small enough in relation to M1 as well ab as the other a priori data Ecrit , V1;2 Œ0, b D 1; 2; : : : ; B0 , the error term will then remain small. By scaling invariance, it suffices to bootstrap the estimate for P0 . We now bootstrap the bound for P0 . Here we essentially proceed as in step .0/ .0/ (3), the a priori bound for the first non-atomic component nA0 D ˚ nA0 P
1
.0/ nA0
. Thus we distinguish as there between the small time e i kD1;2 4 @k ˚k case, when the div-curl system suffices, and the large time case, when the wave equations are important: we shall work here on the interval I1 containing the initial time slice t D 0. (i): small time case jI1 j < T1 . Here T1 is a sufficiently small absolute constant. Write the equation for , using the div-curl system, schematically as follows: @ t P0 Drx P0 C P0 r
1
.Œ
.0/
nA0
C
B0 X
Q nab C Q a.>B0 / C W naB 2 /
bD1
C P0 Œ
.0/
nA0
C
B0 X
Q nab C Q a.>B0 / C W naB r
1
bD1
.Œ
.0/
nA0
C
B0 X
Q nab C Q a.>B0 / C W naB /
bD1
9
Here stands for the vector with components "˛ , ˛ D 0; 1; 2.
410
9 The Bahouri–Gérard concentration compactness method
C P0 r
1
Œ
.0/
nA0
C
B0 X
Q nab C Q na.>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 Π10C6 ; 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
415
9.7 Step 5: Invoking the induction hypothesis
where D D D.Ecrit / is a sufficiently large constant. Then arguing as in the proof of Proposition 9.12, we obtain two equations A P0 Q
nA0
C
B0 X
Q nab C W naB : L
bD1
The source terms F˛1 are obtained as in Chapter 3, and here we of course linearize around the above expression. Then we re-iterate the estimates in the proof of Proposition 9.12, with replacing 2 and 1 D 0. As in Case 1 above, the only new feature are the source terms coming from nonlinear interactions between the various Q nab , W naB1 . Fortunately, the fact that each of these functions is essentially frequency localized to the same interval, the mechanisms that force smallness reduce as before to either physical separation or dispersive decay. We explain here how to obtain smallness for the trilinear null-form source terms, which we write schematically in the form rx;t Œ1 r 1 Qj .2 ; 3 /, were represents one .0/ P of the functions nA0 , B0 Q nab , W naB . We consider the following cases: bD1
L
(ii.0) Errors due to the gluing construction (9.85). These errors are of the form 00. 0.
.t /
ab
.t /@ t
ab
1;T abı2 C10/
1;T abı2 C10/
00.
1;T abı2 C10/
0.
1;T abı2 C10/
To show the smallness of these, note that rt
rx
D
ab
ab
.t /SAn
.t /@ t SAn
ŒT abı2 ; ab
ŒT abı2 :
solves the schematic div-curl system r
1
.
2
/:
Now since we have ab D oL1 .1/ C oL2 .1/, choosing T abı2 large enough, we see that (with o.1/ in case T ! 1) ŒT;T C10 r t ab rx ab D o M P .1 1 / .1/: Lt H
M
Similarly, by construction, the extensions SAn ab ŒT abı2 also satisfy the (schematic) relations ŒT;T C10 @ t SAn ab ŒT abı2 / rx SAn ab ŒT abı2 / D oL1 L2x .1/; T
416
9 The Bahouri–Gérard concentration compactness method
see Lemma 9.33. But then it easily follows that k 00. k 0.
.t /
ab
.t /@ t
ab
1;T abı2 C10/
1;T abı2 C10/
00.
1;T abı2 C10/
0.
1;T abı2 C10/
.t /SAn
ab
.t /@ t SAn
ŒT abı2 kN ı2 ; ab
ŒT abı2 kN ı2 :
(ii.1) Self-interactions of temporally bounded Q nab . These only occur provided I1 \ Œ0; T abı2 c ¤ ;. Thus assume the latter is the case and consider rx;t Q nab r
1
Qj . Q nab ; Q nab / :
Now the estimates of Section 5.3 imply that we obtain
rx;t Q nab r
1
Qj . Q nab ; Q nab / N.I1 R2 / ı2
provided at least two of the inputs have Fourier support with very close angular alignment, depending on k nab kS , ı2 . Thus we may assume that these inputs have Fourier supports with some amount (albeit very small) of angular separation. Similarly, localizing the Fourier support to frequency 1, say, we may reduce to the expression X
rx;t P0 Pk1 Q nab r
1
Qj .Pk2 Q nab ; Pk3 Q nab / ;
k1;2;3 DO.1/
where the implied constant O.1/ is of course potentially extremely large, depending on k nab kS , ı2 . We may similarly assume that all the modulations present are of size O.1/ at most (which may again be quite large, depending on k nab kS , ı2 ). But then the assumed angular separation between all factors allows us to bound this expression (for fixed frequencies) by
rx;t P0 Pk Q nab r 1
1
Qj .Pk2 Q nab ; Pk3 Q nab / N Œ0
. kPk1 Q nab kS Œk1 r 1 Qj .Pk2 Q nab ; Pk3 Q nab / L2 : t;x
But then the desired smallness follows by interpolating the improved bilinear Strichartz type bound
r
1
Y
Qj .Pk2 Q nab ; Pk3 Q nab / Lp . kPkj Q nab kS Œkj t;x
j D1;2
417
9.7 Step 5: Invoking the induction hypothesis
for some p < 2 following from a result due to Bourgain10 [2] as well as Lemma 2.22, and the smallness bound
1
r Qj .Pk Q nab ; Pk Q nab / 1 1; 2 3 L t;x
which we obtain by letting T abı2 be large enough in relation to ı2 . Replacing 0 by k and square summing over the output frequencies, the desired bound follows easily. (ii.2): Interactions of two different temporally bounded Q nab . Here the mechanism at work is the physical separation of the centers of mass for n large. Thus consider rx;t P0 Pk1 Q nab1 r 1 Qj .Pk2 Q nab2 ; Pk3 Q nab3 / where we have bi ¤ bj for at least one pair i; j . Now if we have I1 Œ0; T abj ı2 for both i; j , then Q nabi;j are essentially supported in disjoint light cones for n large enough. Specifically, due to Lemma 7.22 as well as Lemma 7.23, given ı2 > 0, we may write nabi nabi Q nabi D Q cone C Q cone c nabj nab Q nabj D Q cone C Q conecj
where we have
nabi;j
Q c ı2 cone S
nab while the functions Q cone i;j are supported in disjoint double cones while still satisfying
nabi;j
Q cone . C. Q nab /: S
It is then straightforward to conclude that by choosing n large enough, we may force
rx;t Q nab1 r 1 Qj . Q nab2 ; Q nab3 / ı2 : N If on the other hand we have I1c \ Œ0; T abj ı2 ¤ ; for at least one j , then we use L1 dispersion on this intersection to ensure smallness as before. .0/
(ii.3) Interactions of temporally bounded Q nab and nA0 . We distinguish between I1 Œ0; T abı2 and I1 \ Œ0; T abı2 c ¤ ;. In the former case, where 10
Of course one also has the optimal results due to Wolff and Tao, but those are not really needed here.
418
9 The Bahouri–Gérard concentration compactness method
Q nab is given by the actual wave map propagation, we generate error terms of the form .0/ .0/ rx;t Q nab r 1 Qj . nA0 ; nA0 / : As in case (i.2.a) above, localizing the output to frequency 1, we may reduce to .0/ .0/ the case when r 1 Qj . nA0 ; nA0 / has extremely small frequency. But then one obtains
.0/ .0/
rx;t P0 Q nab r 1 Qj . nA0 ; nA0 / 1 P 1 PΠ5;5 ΠQ nab 1 2 ; L H L L t
x
t
and re-scaling and square summing over the output frequencies, we can force an upper bound ı2 by choosing n large enough. We further generate interaction terms of the form rx;t Œ
.0/
nA0
r
1
Qj . Q nab ;
.0/
nA0
/;
.0/
nA0
rx;t Œ
1
r
Qj . Q nab ; Q nab /:
However, the trilinear estimates in Chapter 5 in addition to the frequency support properties of these inputs reveal that choosing n large enough, we can force
.0/ .0/
rx;t nA0 r 1 Qj . Q nab ; nA0 / ı2 ; N
nA.0/ 1
rx;t 0 r Qj . Q nab ; Q nab / N ı2 : Note that in the second situation above, i.e., I1 \ Œ0; T abı2 c ¤ ;, we essentially no longer generate errors of the form rx;t Q nab r
1
Qj .
.0/
nA0
;
.0/
nA0
/;
(9.86)
as well as similar higher order terms arising from the Hodge expansion of the potential term X
4
1
1nA.0/ 2nA.0/ @j 0 j 0
.0/
2nA0
.0/
1nA0
j
j D1;2
within I1 \ Œ0; T abı2 c since now Q nab is given by the linear covariant evolution. This is made precise as follows: by construction, on the latter intersection, we can write Q 1nab D @1 nab C @2 nab ; Q 2nab D @2 nab
@1 nab ; Q 0nab D @ t nab
where the functions nab ; nab solve the covariant wave equations An nab D An nab D 0:
419
9.7 Step 5: Invoking the induction hypothesis
Here the potential term An is defined as in (9.63), which is to be contrasted with the “true potential” .0/ .0/ X 1nA.0/ 2nA.0/ 1nA 2nA 0 j 0 : An WD 4 1 @j 0 j 0 j D1;2
Consider the terms @˛ nab DW @˛ , ˛ D 0; 1; 2. We make the Claim that the expression X @˛ C i@ˇ @˛ Anˇ C i @˛ @ˇ I c Anˇ C i Pk @ˇ @˛ P
is negligible in that its k kN -norm converges to zero as n ! 1. In light of the decompositions of the nonlinearity in Chapter 3, one then easily concludes that terms of the form (9.86) as well as similar higher order terms are indeed accounted for by the covariant wave evolution An . To see the above Claim, we use the notation fg D fH gH C fH gL C fL gH where we put fH gH D
X
D
X
Pk f PŒk
5;kC5 g;
k
fH gL D
X
Pk f P
5 g;
fL gH
k
Pk f P>kC5 g:
k
Then write i@ˇ Œ@˛ Anˇ D i @ˇ Œ@˛ H .Anˇ /H C i @ˇ Œ@˛ L .Anˇ /H C i @ˇ Œ@˛ H .Anˇ /L : Then the trilinear estimates of Chapter 5 as well as our assumptions on the fre.0/ quency localization of , nA0 imply that ki@ˇ Œ@˛ H .Anˇ /H kN ! 0; ki @ˇ Œ@˛ L .Anˇ /H kN ! 0 as n ! 1, and so it suffices to replace i @ˇ Œ@˛ Anˇ by i @ˇ Œ@˛ H .Anˇ /L . Due to Corollary 5.4, Remark 5.6 and Lemma 5.7 we canP pick a sequence n ! n D nc D 1 sufficiently slowly and such that if we put I k Pk Q
ˇ
i@ @˛ H I nc .An /L ! 0; i @˛ H I n @ˇ .Anˇ /L N ! 0; ˇ N
ˇ
i @ H @˛ .I n An /L ! 0; i @˛ @ˇ H .I nc An /L ! 0: ˇ
N
ˇ
N
420
9 The Bahouri–Gérard concentration compactness method
We then replace i @ˇ Œ@˛ H .Anˇ /L by i @ˇ Œ@˛ H .I n Anˇ /L , up to asymptotically vanishing error. Then write i@ˇ Œ@˛ H .I n Anˇ /L D i Œ@ˇ @˛ H .I n Anˇ /L C i Œ@˛ H @ˇ .I n Anˇ /L D i @˛ Œ@ˇ H .I n Anˇ /L C i Œ@˛ H @ˇ .I n Anˇ /L i Œ@ˇ H @˛ .I n Anˇ /L D i @˛ Œ@ˇ H .Anˇ /L C oN .1/: Proceeding similarly for the remaining terms of (9.87), it then follows that X @˛ C i @ˇ @˛ Anˇ C i @˛ @ˇ I c Anˇ C i Pk @ˇ @˛ P
D @˛ C 2i @ˇ H .Anˇ /L C oN .1/ D @˛ C 2i @ˇ .Anˇ / C oN .1/ where in the last step we have again used (a slight variation11 of) Lemma 5.7. The Claim above follows from this. (ii.4) The remaining interactions the Q nab of bounded or unbounded type, .0/
nA0
, as well as @˛ W naB1 . These offer nothing new: note that both the compoQ nents nab of unbounded type as well as the covariant linear waves W naB1 have extremely small L1 t;x -norm, but enjoy the same frequency localization properties nab as Q ; indeed, for unbounded type Q nab , this follows by choosing the C in the interval we work on Œ0; t nab2 C sufficiently large. Thus any trilinear interactions involving them can be handled as in case (ii.1) in the asymptotic regime. .0/ Also, not that interactions of @˛ W naB1 with nA0 are of schematic type .0/ .0/ rx;t nA0 r 1 Qj .@˛ W naB1 ; nA0 / .0/ rx;t nA0 r 1 Qj .@˛ W naB1 ; @˛ W naB1 / .0/ rx;t @˛ W naB1 r 1 Qj . nA0 ; @˛ W naB1 / ; and hence can be made arbitrarily small with respect to k kN by choosing n large enough. We omit the treatment of the higher order interactions between the Q nab as this offers nothing qualitatively new. Applying the arguments from the proof of Proposition 9.12, we now conclude the proof of Proposition 9.30. 11
Recall that we have stronger estimates for in the regime of large modulations, viz. Lemma 9.19
421
9.8 Completion of proofs n.
Proposition 9.30 allows us to extend the Coulomb components ˛ to the interval Œ0; t nab2 C . But now the profiles Q nab which were temporally bounded with respect to t D 0 become temporally unbounded with respect to the new starting time t nab2 C as n ! 1. Now by repeating the arguments in Section 9.6.3, we see that for those concentration profiles Q nab for which (see the discussion in Section 9.6.3) lim supn!1 jt nab2 t nab j < 1, i.e., they concentrate at time t nab2 or alternatively time t nab2 C , the exact same arguments as in that subsection imply that they can be approximated arbitrarily well in the L2 -sense by Coulomb components of admissible maps (but for this we have to n.
n.
< Ca ˛ S as well as exponential decay of the kPk
9.8
n.
for k logŒ.an /
1 .
Completion of the proof of Proposition 7.15 as well as of Corollary 7.16
Both of these can be deduced by a simpler version of the proof of Proposition 9.30. For Proposition 7.15, one makes the ansatz t0 /.@ t V; V / C ˛ ˛ D @˛ S.0 and performs a bootstrap argument for k˛ kS.. 1;0R2 / for t0 large enough. This is as in the proof of Proposition 9.30 where the free linear evolution of @˛ S.0 t0 /.@ t V; V / replaces one of the temporally unbounded Q nab , say, while all the .0/ other components Q nab , nA0 , @˛ W naB1 vanish. If we pick t0 large enough, all the error terms due to nonlinear self-interactions of @˛ S.t t0 /.@ t V; V / become arbitrarily small due to the reasoning in case (ii.1) of the proof of Proposition 9.30. As there, one then obtains the estimates for via the technique used in the proof of Proposition 9.12. We conclude that for given ı3 > 0, if t0 is chosen large enough, we obtain the a priori bound k˛ kS..
1;0R2 /
ı3 ;
and from here the smoothness of the solution follows, see Proposition 7.3.
422
9 The Bahouri–Gérard concentration compactness method
Next, we prove Corollary 7.16: from Proposition 7.15, we know that we can construct admissible Coulomb components of the form n tn /.@ t V; V / C ˛ ˛ WD @˛ S.t for t 2 . 1; tn
C for some large enough absolute constant C , with lim sup k˛ kS.. n!1
1;tn C R2 /
1:
Now we claim that the functions ˛n .tn 10C; / form a Cauchy sequence in the L2x -sense. To see this, note that for n > m n ˛ .tn
m ˛ .0; /
tm ; / D
C oL2 .1/
as n; m ! 1, whence by Proposition 7.11 one has n ˛ .tn
10C; / D
m ˛ .tm
10C; / C oL2 .1/:
But then also n ˛ .t
C tn ; / D
m ˛ .t
C tm ; / C oL2 .1/;
t 2 . 1; 10C /
again by Proposition 7.11, and furthermore, due to the uniform bounds lim sup k n!1
n ˛ kS.. 1;tn C R2 /
<M <1
for suitable M 2 R, we conclude upon denoting ˛1 .t; / WD lim n
n ˛ .t
C tn ; /
that k ˛1 kS..
1; CQ R2 /
M
for any CQ > 10C , as desired.
9.9
Step 6 of the Bahouri–Gérard process; adding all atoms
In the preceding subsection we derived a priori bounds for the wave maps evolution of the (admissible) Coulomb components .0/
nA .w˛ 0
C
˛n1 /e i
P
kD1;2
4
.0/ nA0 1 @ .w C kn1 / k k
C oL2 .1/
423
9.9 Step 6 of the Bahouri–Gérard process; adding all atoms
under the assumption that either .0/
lim inf kw nA0 kL2x > 0 n!1
or else, applying the second stage Bahouri–Gérard decomposition to the large atom n1 , that all the concentration profiles have energy < Ecrit . We shall henceforth make this assumption. Now we continue the process by extending the data at time t D 0 for the Coulomb components to .0/
nA0
w˛
.1/
nA0
C ˛n1 C w˛
e
i
P
kD1;2
4
1@
.0/ .1/ nA0 nA C kn1 Cwk 0 / k .wk
C oL2 .1/
where we recall that the error term oL2 .1/ is necessary in order to ensure that the data correspond to exact Coulomb components of an admissible map. Denote the wave maps evolution of .0/
nA0
w˛
.1/
nA0
C ˛n1 C w˛
e
i
P
kD1;2
4
1@
.1/ .0/ nA nA0 C kn1 Cwk 0 / k .wk
C oL2 .1/
which is defined at time t D 0, by the same symbol. We state the result: Proposition 9.34. Under the preceding assumptions, the evolution of the preceding Coulomb components exists globally in time. For n large enough, we have an a priori bound
.1/ nA
nA.0/
w˛ 0 C ˛n1 C w˛ 0 e
i
P
kD1;2
4
.1/ .0/ nA nA0 1 @ .w C kn1 Cwk 0 / k k
C oL2 .1/
S.R2C1 /
< 1:
The bound here depends on Ecrit as well as the a priori bounds for the evolution of the concentration profiles extracted by adding n1 . Furthermore, we have the same bounds as in Proposition 9.11 (applied to the union of all Jj ), where the implied constants depend on Ecrit as well as the a priori bounds for the evolution of the concentration profiles extracted by adding n1 . The proof of this is a precise replica of the one given in Step 3. The difference consists in the fact that in the decomposition (see Step 2) .1/
w nA0 D
X j
k
naj C w nA
.1/
424
9 The Bahouri–Gérard concentration compactness method .1/
we now need to ensure that kw nA kBP 0 is small enough depending on both 2;1 Ecrit as well as the a priori bounds for the concentration profiles from Step 4. Next, one extends the data at time t D 0 to .0/
nA .w˛ 0
C
˛n1
.1/
C
nA w˛ 0
C
˛n2 /e i
P
kD1;2
4
.1/ .0/ nA0 nA 1 @ .w C kn1 Cwk 0 C kn2 / k k
C oL2 .1/: Repeating the procedure of Step 4 but with magnetic potential defined in terms of the -evolution of .0/
nA0
.w˛
.1/
nA0
C ˛n1 C w˛
/e
i
P
kD1;2
4
1@
.0/ .1/ nA0 nA C kn1 Cwk 0 / k .wk
C oL2 .1/
one derives the same types of bounds as in Proposition 9.30 and the process continues A0 many times, as we recall from the discussion at the beginning of Step 2. We have finally arrived at the following grand conclusion to this section. Theorem 9.35. Let n be a sequence of gauged derivative components of admissible wave maps un W ΠT0n ; T1n R2 ! H2 . The hypothesis lim k
n!1
n
kS.Œ
T0n ;T1n R2 /
D 1; lim k n!1
n kE
D Ecrit
(9.88)
implies that two possible cases occur: up to rescaling and spatial translations, either we have n 2 ˛ .0; / D V˛ C oL .1/ for some fixed L2 -profile V˛ , or else we have for some sequence t n ! 1 (or t n ! 1) and suitable .@ t V; V / 2 L2 HP 1 , n 2 t n /Œ@ t V; V C oL .1/ ˛ .0; / D @˛ S.0 where S.t/ refers to the standard free wave propagator. In the former case 2 X
2 kV˛ kL 2 D Ecrit ;
˛D0
while in the latter case, one has 2 X ˛D0
2 k@˛ V kL 2 D Ecrit :
(9.89)
9.9 Step 6 of the Bahouri–Gérard process; adding all atoms
425
Note that due to Lemma 7.10, in the first case, there exist T0 > 0, T1 > 0 with the property that sup k n
n
kS.Œ
T0 ;T1 R2 /
< 1;
and we can then define lim
n!1
n ˛ .t; x/
DW ˛1 .t; x/
(9.90)
2 2 where the limit is in the sense of L1 loc .Œ T0 ; T1 I L .R //. Similarly, in the second case, due to Corollary 7.16, we have the corresponding statements on some semi-infinite interval I D . 1; T0 / respectively .T0 ; 1/. We call the maximal such open interval . T0 ; T1 / (respectively . 1; T0 / or .T0 ; 1/) the life-span of the asymptotic object ˛1 .t; x/. Finally, in order to apply the Kenig–Merle type argument, we need the following essential compactness property:
Corollary 9.36. There exist continuous functions xN W I ! R2 and W I ! RC so that the family of functions f.t / 1 ˛1 .t; . x.t N //.t / 1 /g t 2I L2x is precompact. Proof. We may assume that sup 0
k ˛1 kS.Œ0;T2 /R2 / D 1;
(9.91)
see Lemma 7.17. The proof follows [14], [15] and amounts to an argument by contradiction. More precisely, we begin by showing that one can find functions .t/, x.t/ N not necessarily continuous with the desired compactness property. Suppose this fails. Then there exists " > 0 and a sequence of times ftn g I so that
1 inf 1 ˛1 tn ; . x/ N ˛1 .tm ; / 2 " (9.92) 2 >0;x2R N
for any n ¤ m. Necessarily tn ! T1 . Now apply Theorem 9.35 to the sequence f ˛1 .tn ; /g1 nD1 , which satisfies (9.88), but on a shifted time-interval. Note that ˛1 .tn ; / are not admissible in the sense that they are not necessarily given as the Coulomb derivative components of admissible wave maps. However, by approximation by the original sequence ˛n (up to symmetries) one concludes that either for some V˛ 2 L2 .R2 /, 2 ˛1 .tn ; x/ D n 1 V˛ .x xn /n 1 C oL .1/ (9.93)
426
9 The Bahouri–Gérard concentration compactness method
for some sequence n ; xn , or that for some s n ! 1 or sn ! 1, 2 ˛1 .tn ; x/ D n 1 @˛ S. s n /Œ@ t V; V .x xn /n 1 C oL .1/
(9.94)
where V is as in (9.89). Clearly, (9.93) contradicts (9.92). For (9.94), we first 1. Then Proposishow that fsn g1 nD1 has to be bounded. Assume that sn ! tion 7.15 implies for large n that ˛1 exists on Œ0; 1/ R2 and k ˛1 kS.Œ0;1/R2 / < 1 which contradicts our assumption (9.91). If on the other hand sn ! 1, then this implies by the same proposition that sup k ˛1 kS.. n
1;tn /R2 /
< 1:
This again contradicts our assumption (9.91) and we are done. As in [14] one proves by approximation that and xN can be taken to be continuous.
10
The proof of the main theorem
For the purposes of this section, it is sometimes preferable to pass to the extrinsic point of view. Specifically, let S be a compact Riemann surface of the hyperbolic type, i.e., it is uniformized by the hyperbolic plane. Given a covering map W H2 ! S, we obtain a Riemannian structure on S which makes a local isometry. By Nash’s theorem, we may isometrically embed S ,! RN into an ambient Euclidean space. Now denote the compositions U n WD ı un W I R2 ! S defined on I R2 , see the above discussion. We can express these maps in terms of the ambient coordinates. Our first task is to identify an actual map U from I R2 into S ,! RN which in some sense corresponds to the limiting object ˛1 .t; x/. The fact that this can be done follows again from the compactness property of the ˛1 .t; x/. We have the following Proposition 10.1. Under the above assumptions, there exists a subsequence of fU n ; n ; n g which we denote in the same fashion as well as a function U.t; / 2 C 0 .I I HP 1 / \ C 1 .I I L2 / , such that lim U n .t; x/ DW U.t; x/;
n!1
lim rx;t U n .t; x/ D rx;t U.t; x/
n!1
where the former limit is the a.e. pointwise sense and the latter limit is in the L2x sense on fixed time intervals. The map U is a weak wave map (in the distributional sense). Also, the second limit is uniform on compact intervals J I . Finally, the family of functions frx;t U.t; /g t2I L2x is compact up to rescaling and translational symmetries (which may depend on time). Proof. We may assume that for times t 2 I we have n ˛ .t;
/ D ˛1 .t; / C oL2 .1/:
But then it follows that for each such t 2 I , there is a subsequence (depending on t) such that also ˛n .t; / converges in the L2 -sense. To see this, note that ˛n .t; / D . ˛1 .t; / C oL2 .1//e i
1
P2
j D1
jn
428
10 The proof of the main theorem
inherits both the physical L2 -localization coming from ˛1 .t; / as well as the Fourier localization of this profile1 whence it is compact and a subsequence converges as claimed. Picking a dense subset of times fti g1 i D1 I and using the Cantor diagonal argument, one obtains a subsequence which we again denote by n etc. such that n .t ; / converges for each i in the L2 sense. By Corollary 9.36, i it then follows that n .t; / converges in the L2 sense, uniformly on compact subintervals of I . In particular, the limit 1 satisfies 1 2 C 0 .I I L2 .R2 //. We now use this to infer the existence of U.t; x/. First, introduce a global frame fe1;2 g on the pull-back bundle of T S under the wave map U n by projecting down the standard frame fe1 ; e2 g, i.e., ej .t; x/ WD .ej /.un .t; x//. Thus X @˛ U n .t; x/ D ekn .t; x/ ˛k n .t; x/: (10.1) kD1;2
Fix some I 0 I which is compactly contained in I . We now use that the pullback frame is bounded. By the preceding, given " > 0 there exists R so large that
lim sup r t;x U n Œjxj>R L1 .I 0 IL2 .jxj>R// < ": n!1
On the other hand, it is clear that
lim sup r t;x U n kL1 .I 0 IL2 / < 1: n!1
By Rellich’s theorem we now conclude that up to passing to a subsequence, @˛ U n * X˛ in L1 .I 0 I L2 / (in the weak-* sense), as well as U n ! U 2 1 0 P1 2 in L1 loc .I I L / strongly. Necessarily then U 2 L .I ; H .R //, see (10.1) as well as X˛ D @˛ U . One immediately obtains the stronger statement that U 2 C 0 .I; L2 / by integrating in time. One in fact has stronger convergence: first note that @˛ ekn .t; x/ D d. /.d ej / un .t; x/ @˛ un .t; x/ P1 2 which implies that fekn g1 nD1 is compact in H .R /. It now follows from (10.1) and Rellich’s theorem as before that up to a subsequence one has @˛ U n .ti ; / ! @˛ U.ti ; / strongly in L2 . By compactness, one therefore also has strongly in L2 @˛ U n .t; / ! @˛ U.t; / 1
This follows as usual from a Littlewood–Paley trichotomy argument and the energy conservation of the n .
10.1 Some preliminary properties of the limiting profiles
429
uniformly on compact subsets of I . This implies all the convergence and regularity statements of the proposition. The fact that U is a weak wave map follows from this, as well as from [10]. Note that we do not claim that we have uniqueness for the limiting object U , and indeed we only have a well-posedness theory at the level of the ˛ . Thus we cannot purely work at the level of wave maps with compact target S. Nevertheless, the latter will play an important role when ruling out certain pathological behaviors, or also to formulate the conservation laws. For example, we have the following Corollary 10.2. Let U be the weak wave map as in Proposition 10.1. Then one has thePfollowing laws: with j j2 D h; i being the metric on S, R conservation 2 d 2 ı dt ˛D0 R2 j@˛ U.t; x/j dx D 0 R d ı dt U.t; x/; @i U.t; x/i dx D 0 i D 1; 2 R2 h@ t R R d P2 2 ı dt i D1 R2 xi .x=R/h@ t U.t; x/; @i U.t; x/i dx D R2 j@ t U.t; x/j dx C O r.R/ R R d P2 1 2 ı dt ˛D0 R2 xi .x=R/ 2 j@˛ U.t; x/j dx D R2 h@i U; @ t U i dx C O r.R/ where is a fixed bump function which is equal to one on jxj 1 and 2 X
Z r.R/ WD
ŒjxjR ˛D0
j@˛ U.t; x/j2 dx:
Proof. These are standard calculations for smooth wave maps. By Proposition 10.1 one can then pass to the limit. Note that one could alternatively express these in terms of ˛1 . We will now closely follow the arguments in [14].
10.1 Some preliminary properties of the limiting profiles We begin with the following consequence of finite propagation speed. Let I C WD I \ Œ0; 1/ where I is the life-span of ˛1 . Lemma 10.3. Let M > 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 fr Œjxj>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
P
m0
Pk2 ;2
m0 e
n;d
i@
P
1 n;d
m0 e
e
i@
1 n;d
2
2
L t;x
i@
1 n;d
2
2
L t;x
(10.10) C
X
X
k1 >k2 1;2 2Cm0 dist.1 ; 2 /&2
2 k2 Pk1 ;1
n;d
P
m0
Pk2 ;2
m0 e
n;d
i@
P>k2
1 n;d
m0 e
i@
1 n;d
2
2
L t;x
(10.11)
434 C
10 The proof of the main theorem
X
X
k2
k1 >k2 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@
P
i@
m0 e
1 n;d
2
2
L t;x
(10.12) C
X
X
2 k2 Pk1 ;1
k1 >k2 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
P
m0
Pk2 ;2
X
k2
2
k1 >k2 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 @
1 n
i@
1 n
(10.16)
`>k1 CC2 10
C
X
Pk1 Q>k1 CC2 P`
n
P` e
(10.17)
k1 <`k1 CC2 10
C Pk1 Q>k1 CC2 Pk1 CC2 Pk1 n P
i@
1 n
(10.18)
i@
1 n
(10.19)
and then inserts these decompositions into (10.14). For (10.16) one places Pk2 n into L4t L1 x , and its contribution to Pk1 Q>k1 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 P
. 2 kQ>k1 CC2 . 2k2 2
k1 CC2 2
10 Pk1
kPk1
n
n
1 n
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 P
. n C r
1
5 Q>k1 CC2 10
. n n //e
i@
1 n
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 apply1 n 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
@ P
5 Q>k1 CC2 10 .r
1
.
n
n
/e
i@
1 n
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 P
. n C r
1
5 Q>k2 CC2 10
. n n //e
i@
1 n
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 P
1 n . n n //e i @ / L2 t;x
n 1 k@ t Pk2 CC2 10 Pk2 L1 t;x 1 n .r 1 . n n /e i @ / L2 L1
.r
. kPk1 n kL1 L2x kQk2 CC2 t
5 Q>k2 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@
1 n
/ Pk2 ;2 . n e
i@
1 n
/
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@
1 n
3 X i1 ;i2 D1
/ Pk2 ;2 . n e Pk1 ;1 . n Ai1 e
i@
i@
1 n
/
1 n
/ Pk2 ;2 . n Bi2 e
i@
1 n
/ (10.24)
438
10 The proof of the main theorem
where A1 D P
A2 D Pk1
m00 10 ;
A3 D Pk1 CC2
m00 10
and similarly for Bi . Here C2 is large depending on Ecrit . If i2 D 3, then one estimates
Pk
1
n
1 n
. n Ai1 e i @ / Pk2 ;2 . n Bi2 e i @ X X 2 jk1 mj kPm n kL1 L2x .
1 ;1
/ L2
t;x
t
m
`k2 CC2
X
`
2
kPc1 n kL4 L1 kPc2 Πn e x
1 n
i@
t
kL4 L1 x t
c1 ;c2 2D`;k2 ` dist.c1 ;c2 /.2k2
.2
C2 10
2
k2 2
X
jk1 mj
2
kPm n kL1 L2x
X
t
m
.` k2 /
2
n
kP`
kS Œ`
2
`>k2
with an implicit constant which is allowed to depend on the energy. Therefore, this is placed in gk1 ;k2 . The case where i1 D 3 is similar. Next, suppose that i1 D 1 and i2 D 1. Then the cap localization passes on to the n and due to (10.23) one places the resulting expression into fk1 ;k2 C gk1 ;k2 . We are left with three cases: i1 D 1; i2 D 2, and i1 D 2; i2 D 1, and i1 D i2 D 2. Next, observe that we may assume that Pk1 ;1 . n Ai1 e
i@
1 n
Pk2 ;2 . n Bi2 e
i@
1 n
/ D Pk1 ;1 .P>k1
C2
n
Ai1 e
i@
1 n
/ D Pk2 ;2 .P>k2
C2
n
Bi2 e
i@
1 n
/
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
P
Pk2 ;2 .P>k2 D Pk1 ;1 .P>k1
C2
n
m00 10 e C2
P
Pk2 ;2 P>k2
n
i@
Pk2
m00 10 e C2
n
1 n
/
m00 10
Pk2
1 n
i@
1 n
/
/
m00 10
1
Πn e
i@
1 n
:
439
10.1 Some preliminary properties of the limiting profiles
Now we distinguish two more cases: either the exponential in the second factor 0 has frequency < 2k2 m0 20 or not. In the former case, one obtains Pk1 ;1 .P>k1
C2
n
P
Pk2 ;2 P>k2
C2
n
D Pk1 ;1 P>k1
C2
Pk2 ;2 .P>k2
C2
m00 10 e n
Pk2
P
n
Pk2
1 n
i@
/ 1
m00 10
i@
m00 10 e
Πn e
i@
1 n
/
m00 10
1
Πn P
m00 20 e
i@
1 n
:
Now perform a cap decomposition of the first and second n factors inside the Pk2 ;2 term. Observe that due to the fact that the frequencies of these factors are approximately 2k2 at least one of them has to have angular separation with the 0 cap 1 from the first factor by an amount comparable to 2 m0 . We may therefore place this expression into fk1 ;k2 C gk1 ;k2 in view of (10.23). If, on the other hand, 0 the exponential in the second factor has frequency > 2k2 m0 20 , then one writes D Pk1 ;1 .P>k1
C2
Pk2 ;2 P>k2 D Pk1 ;1 .P>k1
n
P
C2
C2
n
n
m00 10 e
Pk2
P
i@
1 n
/
m00 10
m00 10 e
i@
1 n
1
Πn Pk2
1 n
i@
m00 20 e
/
Pk2 ;2 P>k2 C2 n Pk2 m00 10
Pk
1 ;1
.P>k1
C2
n
P
m00 10 e
i@
1 n
/ L1 L2 . kPk1 n kL1 L2x x
t
t
followed by the estimate
Pk
2 ;2
P>k2
C2
n
Pk2
m00 10
Πn Pk2 k2
. 2 Pk2 ;2 P>k2
C2
n
Pk2
1
m00 20 @
1
m00 10
1
n Pk2
m00 20 @
Πn e
1
Πn e
i@
1 n
L2 L1 x
t
i@
1 n
L2 L2 t
x
440
10 The proof of the main theorem
. 2k2 Pk2 ;2 P>k2 C2 n Pk2 m00 10
i@
1 n
L2 L2
x
t
(10.25) X
C 2k2
Pk
2 ;2
P>k2
C2
n
Pk2
m00 10
Πn e
i@
1
jk k 0 jm00 CC4 kk2 m00 C4
Pk n Pk 0 @
1
1 n
L2 L2 : (10.26) t
x
Note that we may reduce (10.26) to (with possibly very large O.1/ but only depending on the energy)
2k2 Pk2 ;2 Pk2 CO.1/ n @
1
Pk2 CO.1/ Pk2 CO.1/ n Pk2 CO.1/ @ 1 1 n ŒPk2 CO.1/ n e i @ L2 L2 t
(10.27)
x
since the extremely large frequencies give a gain of a smallness factor whence that case can be place entirely into the bootstrap term gk1 ;k2 . We chose C4 here so large that the entire expression (10.25) is placed in the bootstrap term gk1 ;k2 . To see this, one estimates kPk2 CO.1/ n kL6 kP
n
Pk CO.1/ Πe i @ 1 n 6 6 2 L t Lx X . 2 k2 kPk2 CO.1/ n kL6 kP` n kL6 L6x
(10.25) . 2
k2
t;x
t
`
.2
k2
Pk CO.1/ Œ n e i @ 1 n 6 6 2 L t Lx X ` 2 2 kP` n kS Œ`
kPk2 CO.1/ n kL6
t;x
`
Pk .2
C4 2
2
k2 2
X
2
1 4 j`
k2 j
kP`
n
2 CO.1/
kS Œ`
3
Πn e
i@
1 n
L6 L6 t
x
:
`
Second, with each S0 WD
P
j
Pj Qj CC3 and S1 WD
P
j
Pj Q>j CC3 where C3
441
10.1 Some preliminary properties of the limiting profiles
is a large constant depending only on the energy, (10.27) .
X
2
k2
kPk2 CO.1/ Si1 n kL6 kPk2 CO.1/ Si2 n kL6 L6x t;x
t
i1 ;i2 ;i3 D0;1
kPk2 CO.1/ Si3 n kL6 L6x : t
Now note the following: X X 2 . 2 2 k kPk QkCC3 n kL 6 t;x
k2Z
k
2 kPk QkCC30 n;d kL 6
t;x
k2Z
.
X
k
2
kPk
k2Z
.
X
kPk
n;d 2 kL6 t;x
n;d 2 kS Œk
Dk
n;d 2 kS
. M 2;
k2Z
see above. On the other hand, the elliptic piece satisfies kPk Q>kCC3 n kL6 . 2 t;x
C3 10
k
22
X
2
1 4 j`
kj
kP`
n
kS Œ`
`2Z
via the same arguments we used in the elliptic case earlier in this proof. The remaining cases i1 D 2; i2 D 1, i1;2 D 2, are treated similarly. This now concludes the proof of (10.5), and therefore of the proposition. Next, we formulate the analogue of Proposition 4.11 in our context. Proposition 10.7. Let I C D Œ0; 1/ and assume that .t / > 0 > 0 for all t 0. Then for i D 1; 2, Z Z h@i U; @ t U i dx D Re i1 N 01 dx D 0 R2
R2
for all times in I C . Proof. In view of Proposition 10.6 we may also assume that I a contradiction, assume that Z Re 11 N 01 dx D > 0:
D . 1; 0. For
R2
As in [14] one now obtains the following statements, cf. (4.10) and (4.11) in [14]:
442
10 The proof of the main theorem
ı Given " > 0 there exists R0 ."/ > 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
Œjxj<1 logŒ
yn;d;R n;d;R y0
.t0 ;/
where Œjxj>M 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<1
log yn;d;R .x/ dx1 dx2 :
Also, denote by hn;d;R .t; / the above expressions with t0 replaced by t . As in the proof of Lemma 10.3, one then checks that for these data we have Z
d e.hn;d;R /.t0 ; / dx < Ecrit 2 where e is the energy density, provided we choose R large enough, " and d small enough, and then n large enough. Now consider the wave maps evolution of the data H n;d;R .t0 ; / WD hn;d;R .t0 ; /; @ t hn;d;R .t0 ; / : Our energy induction hypothesis implies that this evolution is defined globally in time, and upon denoting the corresponding Coulomb derivative components by n;d;R ;˛ ;
we obtain a global bound k
n;d;R ;˛ kS.R2C1 /
.Ecrit ; d; / < 1:
444
10 The proof of the main theorem
Denote the time evolution of the data H n;d;R .t0 ; / by H n;d;R .t; /, and the corresponding derivative components (not in the Coulomb Gauge) by n;d;R ;˛ :
We now undo the Lorentz transformation Ld , i.e., consider hn;d;
d;R
.t; / WD hn;d;R .t; / ı L
d:
The argument in the proof of the preceding proposition then yields that we also can conclude that the Coulomb derivative components of hn;d; d;R .t; /, which n;d; d;R , also satisfy a bound of the form we denote by ;˛
n;d; d;R
2C1 0 .Ecrit ; d; / < 1: ;˛ S.R / Furthermore, denoting the standard derivative components of hn;d; n;d; d;R , by finite propagation speed we have ;˛ n;d; ;˛
d;R
.0; x/ D ˛n;R .0; x/;
d;R .t; /
by
˛ D 0; 1; 2;
1 provided jxj < 10 , say, where ˛n;R .0; / are the standard derivative components of un .Rt; Rx/ at time t D 0. To conclude the proof of the proposition, we note that by the convergence of the ˛n at time t D 0 in the L2 -sense, picking R large enough and then also n large enough, we may arrange that (for suitable constants
nm 2 R)
n;d; d;R
.0; / e i nm ˛m .0; / L2 "1 ; m n ;˛ x
0 .Ecrit ;
where "1 is as in Proposition 7.11, with A D d; /. But this then yields the contradiction lim sup k ˛m kS.R2C1 / < 1 m!1
and we are done.
10.2 Rigidity I: Harmonic maps and reduction to the self-similar case As in [14] one now has the following rigidity theorem. Proposition 10.8. With f ˛1 g2˛D0 as above, and with life-span . T0 ; T1 / one cannot have T1 or T0 finite. Moreover, if .t / 0 > 0 for all t 2 R, one necessarily has ˛1 D 0 for ˛ D 0; 1; 2.
10.2 Rigidity I: Harmonic maps and reduction to the self-similar case
445
The proof of it will follow from a sequence of lemmas, and only be completed after Proposition 10.17. We begin with the case where T1 D 1 and .t / 0 > 0 on Œ0; 1/. Assuming that ˛1 do not all vanish, the logic then is to extract a nonconstant harmonic map of finite energy into the compact Riemann surface S, leading to a contradiction. The following lemma is the analogue of Lemma 5.4 in [14]. While the statement is identical with that in [14], its proof is slightly different and invokes in a crucial way the geometry of the target. In the statement, we use a function ! R0 ./, defined as follows: by compactness, for every " > 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 I such that sn C .sn / 1 I Jn for each n. Therefore, (10.31) implies that Z Z ˇ ˇ2 ˇ ˇ .t; x/ dxdt D 0: 0 I
R2
This implies that 0 .t; / D 0 for all t 2 I . 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
where we assumed that T1 D 1 as we may. Recall also that in this case supp ˛1 .t; / B.0; 1 t /; 0 < t < 1; see Lemma 10.5. Next, we prove an upper bound on .t / which places us in the self-similar context. Lemma 10.11. Assuming that T1 D 1 there exists a constant C1 .K/ such that C1 .K/ .t /; 1 t
0 < t < 1:
Proof. Suppose this fails. Let z.t / WD
2 Z X j D1
xj j1 .t; x/ N 01 .t; x/ dx;
0 < t < 1:
10.2 Rigidity I: Harmonic maps and reduction to the self-similar case
449
Note that z.t/ ! 0 as t ! 1. Moreover, by Corollary 9.36 one has Z ˇ 1 ˇ ˇ .t; x/ˇ2 dx: z 0 .t / D 0 Hence, 1Z
Z z.t / D
t
ˇ 1 ˇ ˇ .s; x/ˇ2 dxds: 0
We now distinguish two cases: Either there exists ˛ > 0 such that 1Z
Z t
ˇ 1 ˇ ˇ .s; x/ˇ2 dxds ˛.1 0
t /;
0
(10.33)
or not, i.e., there exists a sequence Jn D .tn ; 1/ with tn ! 1 such that Z Z ˇ 1 ˇ ˇ .s; x/ˇ2 dxds ! 0 as n ! 1: jJn j 1 0
Jn
(10.34)
If the first alternative (10.33) holds, then one is lead to a contradiction as in [14]. On the other hand, we will now reduce the second alternative (10.34) to the existence of a nontrivial harmonic map into S by a similar argument as in the proof of Lemma 10.9, see also Struwe [50]. By the Vitali argument from above, one selects intervals Jn0 WD .sn .sn / 1 ; sn C .sn / 1 / with sn 2 Jn such that Z Z ˇ ˇ 1 ˇ .s; x/ˇ2 dxds ! 0 as n ! 1: jJn0 j 1 0 Jn0
Now one uses compactness as in the proof of Lemma 10.9 to conclude that there exists a limiting wave map ˛ on some nonempty interval I with 0 D 0 on I . Therefore, leads to a a harmonic map U of energy Ecrit into S, which gives the desired contradiction. This now allows us to reduce to the exactly self-similar case. Corollary 10.12. If T1 D 1, then the set ˚ .1
t / ˛1 .t; .1
t /x/ W 0 < t < 1;
˛ D 0; 1; 2
is compact in L2 . Proof. This is as in Proposition 5.7 of [14].
450
10 The proof of the main theorem
10.3 Rigidity II: The self-similar case We now turn to the last step in the Kenig–Merle program (modulo the issue of removing the assumption .t / > 0 for infinite times) which consists of excluding the possibility of self-similar blow-up. As in [31], [32] we set yD
x 1
t
;
sD
log.1
t /;
0
and W .y; s; 0/ WD U.x; t / D U.e
s
y; 1
e
s
0s<1
/;
where U is a weak wave map as constructed in Proposition 10.1. By construction, rs;y W is supported in fjyj 1g. Next, for ı > 0, introduce yD
x ; t Cı
1
sD
log.1
W .y; s; ı/ WD U.e
s
t C ı/;
y; 1 C ı
Then we have that W .y; s; ı/ is defined for 0 s < ˚ supp @˛ W .; ı/ jyj 1
0 < t < 1;
e
s
/:
(10.35)
log ı and ı :
The W solve the equation in the distributional sense @2s W D
1 div rW
.y rW /y @s W
2y r@s W A.W / .@s C y ry /W; ry W
(10.36)
where the nonlinearity stands for the second fundamental form on the Riemann surface S relative to its embedding into RN . We now state the following properties of W . Henceforth, j j when applied to derivatives of W will denote the metric on S and W D W .; ı/. Lemma 10.13. For ı > 0 fixed, ı supp.@ ıg ˛ D 0; 1; 2 ˛ W .; ı// fjyj 1 R ı .jry W j2 C j@s W j2 / dy C P2 R ı j@˛ W .s; y/j2 j log.1 jyj2 /j dy C j log ıj R P˛D0 1 1 2 2 jyj2 / 2 dy C ı 2 . ı ˛D0 j@˛ W .s; y/j .1 Proof. By direct calculation.
451
10.3 Rigidity II: The self-similar case
As in [14] one now introduces a Lyapunov functional Z 1 EQ W .s/ WD j@s W j2 C jry W j2 jy ry W j2 .1 2 D
jyj2 /
1 2
dy:
This quantity satisfies Proposition 10.14. For 0 < s1 < s2 < log. 1ı /, the following identities hold: R R j@s W j2 Q .s2 // E.W Q .s1 // D s2 (1) E.W 2 3=2 dy ds, s1
(2)
lim
s!log. 1ı /
D .1 jyj /
Q .s// Ecrit . E.W
Proof. This is proved as in [31], see Lemma 2.1 there. The difference is of course that we have a different equation, namely (10.36). However, the point is that the second fundamental form is perpendicular to @s W and ry W whence it drops out of the calculation needed for the first identity. The second property is verified as in [14]. As a corollary, one now has the following: Lemma 10.15. For each ı > 0 there exists sNı 2 . j log2 ıj ; j log ıj/ such that 1
Z
sNı Cj log ıj 2
j@s W j2
Z
sNı
D
.1
jyj2 /
dyds
3 2
Ecrit 1
:
j log ıj 2
Proof. By Proposition 10.14, Z 0
j log ıj Z D
j@s W j2 .1
3
jyj2 / 2
dyds Ecrit ;
whence the claim. The goal is now to a stationary solution of ˛ D 0; 1; 2, .1
obtain a limit W as ı ! 0 and to show that W is (10.36). To this end, select ıj ! 0 such that for each tNıj / ˛1 tNıj ; .1
tNıj /x ! ˛ .x/
strongly in L2 , see Corollary 10.12. In fact, we may arrange also that .1 C ıj
tNıj / ˛1 tNıj ; .1 C ıj
tNıj /x ! ˛ .x/
(10.37)
452
10 The proof of the main theorem
in L2 . Now consider the evolution on the level of the with data given by the lefthand side of (10.37), see Section 7.2. By our perturbation theory of Section 7.2 j we conclude from (10.37) that these evolutions ˛ .t; x/ exist on some fixed life-span, and moreover, ˛j .t; x/ D .1 C ıj tNıj / ˛1 tNıj C .1 C ıj tNıj /t; .1 C ıj tNıj /x on that life-span Œ0; T / where we may assume that T < 1. Note that on account of this identity, ˚ supp ˛j .t; / jyj
1 1 C ıj
tNıj tNıj
t <1
t
for each ˛ D 0; 1; 2 and 0 < t < T . Now note that by the construction in the proof of Proposition 10.1 we may arrange that the weak wave maps U j j associated with ˛ and U associated with ˛1 satisfy U j .t; x/ D U tNıj C .1 C ıj tNıj /t; .1 C ıj tNıj /x : Note that for fixed times t 2 .0; T / one has that f@˛ U j .t; /g form a compact set in L2 whence the argument in the proof of Proposition 10.1 implies that up to passing to a subsequence @˛ U j .t; / ! @˛ U .t; / strongly in L2 uniformly on compact subintervals of time. Moreover, U is a weak wave map and satisfies the conservation laws. Next, we switch to the .s; y/ variables. Define Wj .y; s/ WD U tNıj C .1 C ıj tNıj /t; .1 C ıj tNıj /x with the same relation between .s; y/ and .t; x/ as above. Similarly, define W .y; s/ D U .t; x/: Then by the preceding, uniformly in 0 s log.1 ˛ D 0; 1; 2, @˛ Wj .; s/ ! @˛ W .; s/
T =2/ DW TQ and for
in the strong L2 sense. Moreover, with W as in (10.35), one has with sNıj D log.1 C ıj tNıj /, Wj .y; s/ D W .y; sNıj C s; ıj /
10.3 Rigidity II: The self-similar case
453
@˛ W .y; sNıj C s; ıj / ! @˛ W .; s/
(10.38)
and therefore also strongly in L2 uniformly in 0 s TQ . Moreover, W is a solution of (10.36) and ˚ supp @˛ W .s; / jyj 1 as well as trace W .s; / D const where trace is the L2 -trace. Lemma 10.16. Let W be as above. Then, W .y; s/ D W .y/ and W 6 const : Proof. With S D Z 0
S
log.1
TQ / and j large one has
ˇ2 ˇ2 Z ˇˇ Z S Z ˇˇ @s W .y; s ıj C s; ıj /ˇ @s W .y; s/ˇ dyds 3=2 dyds lim 3=2 D 1 D j !1 0 jyj2 1 jyj2
by (10.38). The right-hand side is bounded by Z
S Cs ıj
lim j !1 s ıj
ˇ2 Z ˇˇ @s W .y; s; ıj /ˇ j log ıj j 3=2 dyds . j lim !1 D 1 jyj2
1=2
D 0;
by Lemma 10.15. This shows that W .y; s/ D W .y/ as claimed. The fact that W 6 const follows as in [14]. In other words, we have now obtained a stationary, nonconstant, distributional solution to (10.36) with finite energy (relative to the y variable) (as well as finite /). The following proposition now leads to the desired contradiction. Q E.W Proposition 10.17. Let W be a distributional stationary solution to (10.36) of finite energy Z jrW .y/j2 dy < 1: D
Then W D const. This thus contradicts the preceding construction of W and completes the proof of Proposition 10.8.
454
10 The proof of the main theorem
Proof. We follow the argument of Shatah–Struwe, see [42]: first, W is a weakly harmonic map from D ! S where D is equipped with the hyperbolic metric d2 2 C d! 2 .1 2 /2 1 2 where .; !/ are polar coordinates on D. This means that q ! W 1 2 W C p ? TW S: 1 2 Note that by Helein’s theorem, p this holds in the classical sense in the interior. Integrating by parts against 1 2 W implies that Z Z d 2 2 2 jW! j2 d! D 0 .1 /jW j d! d S 1 S1 and thus
Z
2
.1 S1
2
Z
/jW j2 d!
S1
jW! j2 d! D C0 :
Setting D 0 one concludes that C0 D 0 and sending ! 1 along a suitable subsequence j implies that Z ˇ ˇ ˇW .j !/ˇ2 d! ! 0: lim ! j !1 S 1
On the other hand, by the trace theorem, sup 1 <<1 kW .!/k C kW
cludes via change of variables
./ D exp
Z
1
1
HP 2 .S 1 /
2
Since clearly also sup 1 <<1 kW .!/kL2 .S 1 / < 2 interpolation that trace.W / D const as the L2 trace
k H 1 .D/ .
1, one conon S 1 . The
du p u 1 u2
provides a conformal equivalence between the hyperbolic disk and the disk D with the Euclidean metric. In fact, d 2 C 2 d! 2 D
2
.1
2 /
d2 2 2 C d! : .1 2 /2 1 2
By the conformal invariance of the Dirichlet energy in two dimensions, it follows that v.; !/ WD W .; !/ is a weakly harmonic map D ! S with the Euclidean
455
10.3 Rigidity II: The self-similar case
disk D. Moreover, one checks that v has finite HP 1 energy relative to the .; !/coordinates and that trace.v/ D const in this setting as well. By a result of N And then the result of Lemaire [27] gives Qing [37], it follows that v is C 1 on D. the desired conclusion that W D const. The only remaining case is to show that .t / does not approach zero along some subsequence. This case is handled as in [14] or [29]. We follow the argument [14] essential verbatim. Lemma 10.18. Let ˛1 be the limiting object as above and suppose that T1 D 1. Then .t/ > 0 > 0 for all t 0. Proof. Suppose this fails. Then there exist tn ! 1 so that .tn / ! 0; in fact, one may assume even that .tn /
inf .t /:
t 2Œ0;tn
From Corollary 9.36 one has ˛n WD .tn /
1
˛1 tn ; x.t N n / .tn /
1
! ˛
strongly in L2 . Then E. / D Ecrit and we may assume that the life-span . T0 ; T1 / of ˛ has the property that T0 < 1. Otherwise one obtains a con tradiction from Proposition 10.8. Now define ˛n . ; x/ and ˛ . ; x/ to be the evolutions of ˛n and ˛ . By the perturbation theory of Section 7.2 we conclude n that lim infn!1 T0 . ˛ / D 1 and ˛n . ; x/ ! ˛ . ; x/ 2 in L1 loc .. 1; 0 L /. By uniqueness of the -evolutions
˛n . ; x/ D .tn / for all 0 tn C
. .tn /
1
˛1 tn C .tn /
We claim that n WD
1
; x
x.t N n / .tn /
1
tn .tn / satisfies
lim. n / D 1 n
so that for all 2 . 1; 0, for n large, 0 tn C .t n / tn . In fact, if 0 < 1, then x x.t / n ˛n .x; n / D . n / 1 ˛1 ;0 .tn /
n !
456
10 The proof of the main theorem
would converge to ˛ .x; 0 / in L2 , with .tn / ! 0, which contradicts ˛ ¤ 0. We now make the further claim that k ˛ kS. 1;0/ D C1. Otherwise, by the perturbation theory of Section 7.2 for n large, T0 . ˛n / D 1 and k ˛n kS. 1;0/ M , uniformly in n, which contradicts our assumption that k ˛1 kS.0;C1/ D C1. This is on account of Corollary 7.14, since for every interval Œ0; Q , one may find Œ Q1 ; 0 with the property that the map ! tn C .t n / takes the latter interval into the former. Now fix 2 . 1; 0. Then for n sufficiently large, tn C .t n / 0 and .tn C .t n / / is defined. Let .tn C
/ .tn /
1
˛1
/ .tn / ; tn / .tn /
x.tn C
x
.tn C
.tn / 1 n x e D n . / ˛ C
e x n . / e n . /
;
2 K;
with e n . / D
.tn C
Now, since n 1 f
/ .tn /
.tn /
x xn n
1; e x n . / D x.tn C
/ .tn /
x.tn / : e n . /
(10.39)
! f strongly in L2 with either n ! 0 or C1,
n!1
or jxn j ! 1 implies that f 0, we see that we can assume, after passing to a subsequence, that e n . / ! e . /, 1 e . / < 1 and e x n . / ! e x . / 2 R2 . This implies that e x . / 1 x e . / ˛ ; 2 K: e . /
Hence, by Proposition 10.7 and 10.8, ˛ D 0, which is a contradiction.
Proof of Theorem 1.1. We first address global existence and regularity and the global control of the S -norms. In fact, instead of (1.2) we of course require the stronger k ˛ kS K.Ecrit / from which (1.2) then follows by standard Littlewood–Paley calculus and the Strichartz component of the S -norm. Assume that this strengthened assertion of the theorem fails. Recall that Ecrit was defined as the smallest energy with the property that there exists an essentially singular sequence of admissible maps
457
10.3 Rigidity II: The self-similar case
at energy Ecrit . In other words, there exists a sequence fun g1 nD1 of admissible wave maps . T0n ; T1n / R2 ! H2 with associated gauged derivative components f ˛n g1 nD1 and such that ı E.un / ! Ecrit ı max˛D0;1;2 k ˛n kS.. T0n ;T1n /R2 / ! 1 as n ! 1. The Bahouri–Gérard decomposition of Chapter 9 together with the Kenig–Merle argument of this section now lead to a contradiction whence such an essentially singular sequence cannot exist. This now gives the result, at least up to the scattering statement. As for the latter, we argue as follows. It suffices to carry this out for H2 . Then by applying Lemma 7.6 we may represent the gauged derivative components for any ı > 0 in the form D
.ı/ L
C
on a time interval of the form .T0 ; 1/ where k
.ı/ NL .ı/ 2 NL kL1 t Lx
< ı and
wave. The scattering for the free wave is automatic, and the iterated away.
.ı/ NL
.ı/ L
is a free
error can be
11
Appendix
11.1 Completing the proof of Lemma 7.6 We need to show, see (7.13), that there exist time intervals Ij , j D 1; 2; : : : ; M1 , with M1 only depending on k kS ; "0 , with the property that max
1j M1
X
kP` F˛ . /k2N Œ`.I
2 j R /
< "0 C06 :
(11.1)
`2Z
Here we need to verify this for F˛ of at least quintic degree. In fact, the verification of this is more or less the same for all the higher order terms, and we explain it in detail for a quintic term of first type. From the discussion at the end of Chapter 6 we see that we may assume the expression to be reduced. Thus consider for example the expression h rx;t Pk0
0r
1
Pr1 Pk1
1r
1
2r
Pr2 Pk2
1
Pr3 Qk .Pk3
3 ; Pk4
i :
4/
From Lemma 6.1 we infer that
h
rx;t Pk Pk0 Pk2
2r
0r 1
1
Pr1 Pk1
Pr3 Qk .Pk3
1r
1
Pr2
3 ; Pk4
i
4/
N Œk
.2
ık0
kPk0
0 kS Œk0 :
It then follows upon square summing over all k 2 Z that the contribution from those expressions with k0 k in the sense that k0 k > 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 Q
460
11 Appendix
Without loss of generality write this as h rx;t Pk Pk0
0r
1
Pr1 Pk1
1r
Pk2
1
Pr2 1
2r
Pr3 Qk .Pk3 Q>k3 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 Q
0r
1
Pr1 Pk1
Pk2
2r
1
1r
1
Pr2
Pr3 Qk .Pk3 Q>k3 CC
3 ; Pk4 Q
i
4/
;
but where now kj D ri C O.1/ for all i; j , and the output inherits the modulation from the large modulation term Pk3 Q>k3 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 Q
i
2 //
4
N Œk0
< "0 :
461
11.1 Completing a proof
Hence we have now reduced to establishing “divisibility” for the space-time frequency reduced expression (with kj D ri C O.1/ for all i; j ) h rx;t Pk Pk0
1
0r
Pr1 Pk1 2r
Pk2
1
1r
1
Pr2
Pr3 Qk .Pk3 Q
3 ; Pk4 Q
i / : 4
But since we may estimate this by
h
rx;t Pk Pk0
0r
.Pk2 . kPk0
2r
1
Pr1 Pk1 1
0 kS Œk0 kPk1
kPk2
kr 2 kL4 L1 x t
1r
1
Pr2
Pr3 Qk .Pk3 Q
3 ; Pk4 Q
i
4/
N Œk0
1 kL4 L1 x t
1
Pr3 Qk .Pk3 Q
3 ; Pk4 Q
: 4 /kL2 L1 x t
Then use the bound
r
1
Pr3 Qk .Pk3 Q
. Pr3 Qk .Pk3 Q
which follows from Lemma 4. 16, as well as Bernstein’s inequality and our assumptions on the frequencies/modulations. But then again using the “divisibility” of the space L2t;x , we may pick time intervals fIj g as before such that h X
r P
x;t Ij k Pk0
0r
1
Pr1 Pk1
1r
1
Pr2
k0 2Z
Pk2
2r
1
Pr3 Qk .Pk3 Q
3 ; Pk4 Q
i
2 /
4
N Œk0
< "0
for all Ij . This furnishes the proof of claim (11.1) for the first type of quintilinear null-form. The remaining short error terms of either first or second type are treated similarly. For the higher order errors of long type (see the discussion at the end of Chapter 6 for the terminology), the claim follows from Proposition 6.5 as well as the divisibility of L8t;x .
462
11 Appendix
11.2 Completing the proof of Lemma 7.9. Recall the setup in the proof of Lemma 7.9: we have a frequency envelope ck controlling the data at time tj . We then make the bootstrapping assumption kPk kS Œk.Ij R2 / A.C0 /ck : The time intervals Ij have been chosen such that we have a clean separation .j / L
jIj D
C
.j / NL
where we X
Pk
.j /
2 NL S Œk.Ij R2 /
< "0
k2Z
rx;t
.j /
1P L Lt H
1
1
. k k3S "0 M
for large M , say M D 100. We need to check that by refining each Ij if necessary into finitely many subintervals Jj i such that we have
Pk F 2lC1 . / ˛ N Œk.J
2 j i R /
ck
where now l D 2; 3; 4; 5. We outline the argument for the quintic errors of first type, the remaining ones following a similar pattern. Thus consider the expression X
h rx;t Pk Pk0 r
1
Pr1 Pk1 r
1
Pr2 Pk2 r
1
Pr3 Qk .Pk3 ; Pk4 /
i :
kj ;ri
By picking M large enough, it is clear that the only contribution that matters is when we replace each factor Pkj , j D 1; 2; 3; 4, by Pkj L . However, we note here in passing that one can also handle interactions of L and N L terms with at least factors L present by means of the type of “divisibility” argument to follow. Hence consider now X
h rx;t Pk Pk0 r
1
Pr1 Pk1
Lr
1
Pr2
kj ;ri
P k2
Lr
1
Pr3 Qk .Pk3
L ; Pk4
i
L/
:
463
11.3 Completion of a proof
Due to Proposition 6.1, it is clear that we obtain the desired bound
X h
rx;t Pk Pk0 r
1
Pr1 Pk1
Lr
1
Pr2
kj ;ri
Pk2
Lr
1
Pr3 Qk .Pk3
i
L/
L ; Pk4
N Œk
ck
provided either jk0 kj 1, and similarly we may assume that kj D ri C O.1/ for j D 1; 2; 3; 4, i D 1; 2; 3. Thus we now reduce to estimating the expression where the summation is reduced to k0 D k C O.1/; kj D ri C O.1/ for j D 1; 2; 3; 4, i D 1; 2; 3. But in this case, the same type of divisibility argument used in the immediately preceding proof reveals that we may pick intervals Jj i whose number depends only on Ecrit and which are independent of k such that X
Pr Qk .Pk 3 3
L ; Pk4
2
L/
r3 Dk3 CO.1/Dk4 CO.1/
L2t HP
1 2
1
and then the same estimates as in the preceding proof reveal that
X h
rx;t Pk Pk0 r
1
Pr1 Pk1
Lr
1
Pr2
kj ;ri
Pk2
Lr
1
Pr3 Qk .Pk3
L ; Pk4
i
/
L
N Œk
ck ;
as desired. The argument for the remaining error terms is similar.
11.3 Completion of the proof of Lemma 7.26 To complete the proof, we need to show that the contributions of the -factors when implementing the Hodge decomposition for the factors of jrj 1 . 2 / in X
Pk
jrj
1
.
2
2 / 2
L t HP
k2Z
1 2
is also controllable in terms of k kS . Using the schematic relation D jrj
1
jrj
1
.
2
/;
464
11 Appendix
we need to bound X
Pk
1
jrj
1
.jrj
1
Πjrj
2
.
2 / / 2
L t HP
k2Z
X
Pk jrj
1
jrj
1
1
Πjrj
2
.
/jrj
1
1
Πjrj
2
.
1 2
2 / 2
L t HP
k2Z
1 2
:
We deal with the first expression, the second being treated along similar lines. Thus consider Pk jrj 1 .jrj 1 Œ jrj 1 . 2 / / D Pk PŒk 10;kC10 jrj 1 .jrj 1 Œ jrj 1 . 2 / / C Pk P>kC10 jrj 1 .jrj 1 Œ jrj 1 . 2 / / C Pk P
10;kC10
X
D
1
jrj
.jrj
Pk PŒk
1
Πjrj
1
10;kC10
2
.
/ /
jrj
1
Pr .jrj
1
1
Πjrj
2
.
/ / :
r
Now assume the most delicate case, in which we have a high-high-low scenario inside the expression jrj
1
with respect to the factors jrj jrj
1
Pr jrj
D
1
Πjrj X
1
.
2
Pr .jrj 1Œ
/
jrj
1
Πjrj 1.
1
2 /,
2
.
/ /
. Thus in this case we can write
jrj
1
Pr jrj
1
Pr1 Πjrj
1
.
2
jrj
1
Pr jrj
1
Pr1 Πjrj
1
P
/Pr2
r1 Dr2 CO.1/>rCO.1/
X
D
2
/Pr2
r1 Dr2 CO.1/>rCO.1/
C
X r1 Dr2 CO.1/>rCO.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 P
L2t L1 x
r1 Dr2 CO.1/>rCO.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
/Pc2
r1
2 2 kPr1 kS Œr1 kPr2 kS Œr2 kP
2
L2t L1 x
/kL1 t;x
r1
2 2 kPr1 kS Œr1 kPr2 kS Œr2 k k2E :
Hence we obtain the bound
Pk PŒk
X
10;kC10
jrj
1
Pr
r1 Dr2 CO.1/>rCO.1/ 1
.jrj X
.
2
r1 k 2
2.1
1
Pr1 Πjrj
"/.r r1 /
2
P
/Pr2 /
L2t HP
r1
2 2 kPŒk
1 2
kL1 L2x
10;kC10
t
r1 Dr2 CO.1/>rCO.1/
kPr1 kS Œr1 kPr2 kS Œr2 k k2E : If we now square this expression and sum over k 2 Z, it is straightforward to check that one obtains the upper bound . k k6S k k4E : Next, consider the contribution of the expression X
jrj
1
Pr jrj
1
1
Pr1 Πjrj
Pr .
2
/Pr2
r1 Dr2 CO.1/>rCO.1/
D
X
X
jrj
1
Pr jrj
1
Pr1 Πjrj
1
PrQ .
2
/Pr2
r1 Dr2 CO.1/>rCO.1/ rQ r
D
X
X
X
r1 Dr2 CO.1/>rCO.1/ rQ r c1;2 2Dr1 ;rQ
jrj
r1 dist.c1 ;
jrj
1
1
Pr
c2 /.2r
Pc1 Πjrj
1
PrQ .
2
/Pc2
:
466
11 Appendix
Now for fixed r; r; Q r1;2 , we can estimate, using as before the improved Strichartz estimates as well as Bernstein’s inequality
X
jrj 1 Pr jrj 1 Pc1 Πjrj 1 PrQ . 2 /Pc2 2 1
c1;2 2Dr1 ;rQ
. 2.1
r1 dist.c1 ;
L t Lx
c2 /.2r
X
"/r c1;2 2Dr1 ;rQ
Pr
r1 dist.c1 ;
c2
jrj r1
. 2 2 2.1
/.Q r r1 /
2
1
Y
.1 /Q r
/.2r
Pc1 Πjrj
1
PrQ .
2
/Pc2
L2t L1C x
r
1
kPrj kS Œrj k k2E . 2 2 2. 2
"/.r rQ /
j D1;2
Y
kPrj kS Œrj k k2E
j D1;2
and from here the estimate continues as before. The remaining frequency interactions inside jrj 1 Pr jrj 1 Πjrj 1 . 2 / are handled similarly and omitted. Next, consider the case of high-high interactions, i.e., Pk P>kC10 jrj 1 .jrj 1 jrj 1 . 2 / / X D Pk Pk1 jrj 1 Pr .jrj 1 jrj
1
.
2
/ / :
k1 DrCO.1/>kC10
We shall again consider the most delicate case when there are high-high interactions within rj 1 Pr jrj 1 jrj 1 . 2 / X D rj 1 Pr jrj 1 Pr1 jrj 1 . 2 / Pr2 : r1 Dr2 CO.1/>rCO.1/
But then arguing just as above one obtains the bound
rj
1
Pr .jrj
1
Pr1 Πjrj
1
.
.2
2
/Pr2 / L2 L2C t
x
. 21 "/r . 12 "/.r r1 /
2
Y j D1;2
kPrj kS Œrj k k2E ;
467
11.4 Completion of a proof
and from here one obtains
X
Pk Pk1 jrj
1
Pr .jrj
1
Πjrj
1
.
2
/ /
k1 DrCO.1/>kC10
.
1
X
2. 2
"/.k r/ . 12 "/.r r1 /
2
L2t HP
1 2
kPk1 kS Œk1
k1 DrCO.1/>kC10
Y
kPrj kS Œrj k k2E :
j D1;2
Squaring and summing over k again results in the same bound as before. The case of low-high interactions, i.e., Pk .P
10
jrj
1
.jrj
1
Πjrj
1
.
2
/ //;
is more of the same and omitted.
11.4 Completion of the proof of Proposition 9.12, Part I Here we show how to deal with the higher order terms encountered in the decomposition (9.37), i.e., the fifth term there. We shall again explain the method for the quintilinear terms of first type, the remaining higher order terms being treated similarly. Thus consider the expression h i rx;t Pk Pk0 0 r 1 Pr1 Pk1 1 r 1 Pr2 Pk2 2 r 1 Pr3 Qk .Pk3 3 ; Pk4 4 / : We use the letter here to imply either a -factor or one of 1;2 , the the setup in the proof of Proposition 9.12. Now we distinguish between a number of cases: (1) At least one factor of both 1 and 2 is present. In this case, the entire expression contributes to 2 , as follows from Proposition 6.1. Indeed, we can sum over all kj ; rj and then square sum over k 2 Z and bound the entire expression by . k2 kS k1 kS where the implied constant only depends on Ecrit . By choosing "0 , which controls k1 kS , small enough, we can bootstrap. (2) Only 1 factors in addition to -factors. First, assume that there are at least two 1 factors. If one of them is 0 , then the output inherits the frequency envelope of
468
11 Appendix
1 from Proposition 6.1, and the smallness follows from the presence of the extra factor 1 . If the first factor 0 is a , then we need to show that the expression contributes to 2 . But this again follows from Proposition 6.1, essentially as in Case (1) (d) of the proof of Proposition 9.12. Next, assume that there is only one 1 factor present. If this factor is not 0 , then the expression contributes to 2 , following the same reasoning as in Case (1), (b). Thus assume now that we have 0 D 1 , which is the expression rx;t Pk Pk0 1 r 1 Pr1 Pk1 r 1 Pr2 .Pk2 r 1 Pr3 Qk .Pk3 ; Pk4 / : Recall from the proof of Proposition 6.1 that here really stands for L or L, but we suppress this here. What matters is that k kS depends on Ecrit N in a universal way independent of the stage of the iteration in the proof. As usual we may reduce to kj D ri C O.1/, j D 1; 2; 3; 4, i D 1; 2; 3, and k0 D k C O.1/ > 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 j 1;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 ;jrj 1
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/ l 1
Pk1 Q
10 1 r
1
Pr Qj I c .Pk2 Ql ; Pk3 QlCO.1/ / :
471
11.5 Competion of proofs
We can estimate this by (using Bernstein’s inequality)
rx;t P0 QD Pk Q
D rx;t P0 QŒD;lCO.1/ Pk1 Q
t;x
Dj lCO.1/
kPk1 Q
2 10 1 kL1 t Lx
C4 d0 : The case when 1 has comparable modulation is of course similar. Hence we may assume that if one of the inputs Pk2;3 is of elliptic type, the output inherits its modulation. In order to obtain smallness in this case, we can form example use divisibility of L2t;x by applying suitable cutoffs Ij for which X 2 k Ij R Pk2 Q k2 kL 1: 2 k2 2Z
t;x
Next, assume that both inputs Pk2 ;3 of the null-form are of hyperbolic type. Then using the bilinear estimates of Chapter 4, we can estimate
X
rx;t P0 QD
k1;2;3 CO.1/DrDO.1/
Pk1 1 r 1 Pr Qj I c .Pk2 Q
X
rx;t P0 QD k1;2;3 CO.1/DrDO.1/
Pr Qj I c .Pk2 Q
. kPk1 1 kL1 L2x Qj I c .Pk2 Q
1
t
XP 0
/
1 2 C"; 1 ";2
L2t;x
:
In this case, smallness is again forced by subdividing into suitable time intervals Ij with the property that
I Qj I c .Pk Q
This completes treatment of (11.2). Next we turn to (11.3). The same reasoning as for (11.2) shows that we may assume all frequencies k1;2;3 ; r (which we introduce in the same fashion as before) to be of size O.1/. Now a technical issue
472
11 Appendix
arises when the operator R D R0 . Indeed, in this case, it may happen that the output inherits the modulation of the first input Pk1 R 1 , and the remaining inputs necessarily need to be placed into the energy space which is “not divisible”. However, this problem is somewhat artificial, since of course the Hodge decomposition for the temporal components becomes counterproductive for in the large modulation (elliptic) case. Thus for the expression X rx;t P0 QD R0 Pk1 Q 1 1 r 1 Pr Qj I.Pk2 ; Pk3 / ; k1;2;3 CO.1/DrDO.1/
it is best to re-combine it with the term X rx;t P0 QD R0 Pk1 Q 1 2 r
1
Pr Qj I.Pk2 ; Pk3 / ;
k1;2;3 CO.1/DrDO.1/
as well as the “elliptic” error 0 coming from 0 D R0 C 0 and replace it by Pk1 Q 1 D Pk1 Q 1 1 C Pk1 Q 1 2 . Unfortunately, we encounter here the technical issue that the inputs 1;2 , on the right-hand side are really Schwartz extensions of the actual components beyond the time interval I we work on, and hence do not exactly satisfy the div-curl system. The way around this is to work on a slightly smaller time interval IQ obtained by removing small intervals I1;2 from the endpoints of I with I1;2 of length T1 with T1 as in case 1 of the roof of Proposition 9.12. When we restrict the source terms to I , we may invoke the div-curl system for extremely elliptic (i.e., difference of modulation and frequency very large) terms up to negligible errors. This allows us to obtain bootstrapped bounds for 1;2 on IQ, and at the endpoints, we can re-iterate the argument of Case 1. Then the 1;2 on the full interval I can be re-assembled from these pieces via partition of unity with respect to time. The preceding discussion reveals that we may as well suppress the operator R . But once this is done, the divisibility argument used for (11.2) may be repeated to give the desired smallness upon suitably restricting the time intervals. (1b): The argument for rx;t P0 QD Œ1 r 1 Qj . ; / is exactly the same, one square sums over the output frequencies instead. (2): Next we consider the schematically written terms of type rx;t P0 QD Œ r 1 Qj .; /. Again these split into two sub-types:
473
11.5 Competion of proofs
(2a): Terms of type rx;t P0 QD Πr 1 Qj .1 ; /. These contribute to 2 , and indeed apart from the fact that one uses trilinear estimates from Chapter 5 for elliptic outputs, the smallness follows formally just as in Case 1 (b) (of the proof of Proposition 9.12). (2b): Terms of type rx;t P0 QD Πr 1 Qj .2 ; /. Here one encounters again the issue with the terms containing R0 2 and of extremely large modulation. As in (1a) above this is handled by undoing the Hodge decomposition for these terms by restricting to a smaller time interval, up to negligible errors. This, as well as arguments as in (1) above, allow one to reduce to an expression of the form rx;t P0 QD Pk1 r 1 Qj .Pk2 2 ; Pk3 / where all inputs are of hyperbolic type (up to large constants depending on the energy alone). But then the smallness can be forced by reducing to frequencyseparated inputs Pk1;3 (via the estimates of Section 5.3. But then divisibility is obtained by grouping the inputs Pk1;3 together and placing their product into L2t;x . (3) The remaining trilinear null-forms with elliptic output are easier to handle, since they contain at least two factors of type 1;2 , and hence the smallness follows simply by the smallness assumptions on these factors (bootstrap assumptions), as well as the trilinear estimates of Chapter 5. We omit the details. The higher order contributions from the QD
5 hX
F˛2iC1 .
C /
i D2
are estimated in a similar vein and omitted.
QD
5 X i D2
i F˛2iC1 . /
474
11 Appendix Table 11.1. Table of notations
Notation
Meaning
Instance
u x; y e1 ; e2 j ˛
wave map from R2C1 into H2 components of u in standard coordinates standard orthonormal frame for T H2 derivative components of u with respect to e1;2 Coulomb derivative components Components of an essentially singular sequence of wave maps elliptic term in the Hodge decomposition of
(1.1) (1.4) before (1.4) (1.5), (1.6)–(1.9) (1.10) After Proposition 9.1 (1.16)
˛; n ˛,
j ˛ ˛n ,
xn ; yn
ˇ
ˇ
Ecrit t! , x! k kS Œk; k kS Œk k kXP p;q;r k k kN Œ k kN Œk Pk ; Qj ; Qj˙ Rk;
minimal blow up energy null-frame coordinates null-frame component of the frequency localized norm k kS Œk Norm used to control the frequency localized Coulomb components Pk homogeneous Besov X s;b -type norm null-frame component of the norms k kN Œk used to control the nonlinear source terms Norm used to control the nonlinear source terms Frequency and modulation cutoffs Rectangular slabs in Fourier space
(1.23) (2.5) (2.12) (2.17) (2.1) (2.11) Definition 2.9 Before (2.1) (2.1)
475
11.5 Competion of proofs Table 11.2. Table of notations, continued
Notation
Meaning
Instance
˛na
Frequency atoms of the Bahouri–Gérard frequency decomposition of the ˛n weakly small error in Bahouri–Gérard frequency decomposition
Lemma 9.5
Lowest frequency nonatomic derivative components
Lemma 9.7 Prop. 9.9
ck ; dk
Lowest frequency nonatomic derivative Coulomb components various frequency envelopes
Aˇ
various versions of the Coulomb potential
An SAn uŒ0
Covariant wave operator Covariant wave propagation associated with An applied to data uŒ0 Concentration profiles obtained in second stage of covariant Bahouri–Gérard Weakly small error of profile decomposition in second stage of covariant Bahouri–Gérard
w˛nA
Lemma 9.5
.0/
nA0
˚˛
.0/
nA0
˛
.l/
Vab WnaB
Prop. 9.11, Prop. 9.12 After Prop. 9.14 Def. 9.18 Def. 9.22 Lemma 9.23 Lemma 9.23
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Index
adapted to k, 25 adding all atoms, 22, 422 admissible wave map, 6, 14, 20, 23, 245, 293, 425, 430 algebra estimate for S Œk, 90 angular alignment, 108, 180, 238 angular localization, 100 angular separation, 231, 243, 325, 343, 353, 365, 416 Ap -class, 250 approximate energy conservation, 22, 310, 315, 336, 340 approximating maps, 21, 310, 359, 442 approximations by smooth wave maps, 244 asymptotic orthogonality relation, 374, 379 atomic space, 10, 29, 30 atoms, 15, 21, 30–39, 80, 261, 294, 303, 305, 395, 422 auxiliary map, 308 Bahouri–Gérard decomposition, 3, 11, 13–19, 293, 358, 397 Bahouri–Gérard method, 293 basic perturbative results, 223 Bernstein’s inequality, 27 Besov error, 21, 303, 317, 356 Besov norm, 307, 324 bilinear estimates, 73, 78, 90, 93, 204 blow-up, 2, 17, 448, 450 blow-up criterion, 180, 223 BMO, 20, 250, 277 bootstrap assumption, 22, 239, 337, 409, 421, 462 boundary regularity version, 18 Bourgain, Jean, 417 break down in finite time, 239
Calderón–Zygmund operators, 251, 277 characteristic frames, 9 Christ–Kiselev argument, 53 Coifman–Rochberg–Weiss commutator estimate, 20, 277 compactness property, 13–18, 23, 425, 427 concentration cores, 22, 376 concentration profiles, 12–16, 22, 373–376, 399, 400, 423 conformal invariance, 454 conserved energy, 2, 317 Coulomb components, 15, 65, 223, 244, 294, 397, 407, 423 Coulomb derivative components, 4, 15, 19, 65, 245, 294, 443, 444 Coulomb gauge, 4, 5, 13, 63, 65, 245, 336 Coulomb wave map propagation, 246, 248, 402 Coulomb wave maps, 13, 20, 22, 248, 397 covariant dispersion, 364 covariant energy density, 340 covariant evolution, 16, 22, 376, 404, 410 covariant wave equation, 13 covariant wave operator, 15, 22, 361 critical solution, 2, 11, 16 critical theory, 2 curvature, 5, 18 dependence domains, 377 derivative components, 224 dispersive estimates, 25, 36, 415 disposable multipliers, 39, 232 distributional stationary solution, 453 div-curl system, 5, 20, 65, 228, 240, 264, 318, 409, 472 divergence relations, 374
482
Index
divisibility, 21, 315, 318, 325, 354, 365, 460, 468 "-oscillatory, 280 "-singular, 280 elliptic, 37 embedding lemma, 250 energy build-up, 310 energy estimate, 42 energy induction hypothesis, 376, 443 energy transfer, 315, 317 errors of first type, 70 errors of second type, 70 essential Fourier support, 310 essentially singular, 14, 294, 302, 358, 376 expansion graph, 69 extraction of concentration profiles, 373 extrinsic formulation, 1 frequency atoms, 14, 15, 20–22, 294, 305 frequency envelope, 7, 224, 239, 356, 357 frequency evacuation, 356 frequency localization procedure, 14 functional framework, 25 geometric concentration profiles, 15, 22, 376 geometric decomposition, 14, 22, 299 geometric lemma for cones, 73 global frame, 428 Hardy–Littlewood maximal operator, 251 harmonic map, 1, 2, 18, 444, 454 Helein’s theorem, 18, 454 hierarchy of temporal scales, 399 high-high cascades, 20 high-high-low interactions, 36, 211, 317, 326, 349 high-low-low interactions, 9, 13, 115, 180, 181, 231 higher order error terms, 66, 68, 197, 461, 468 Hodge decomposition, 6, 65, 217, 240, 266, 326, 377, 463
Huygens principle, 369, 402 hyperbolic part of phase space, 37 hyperbolic plane, 3 hyperbolic Riemann surface, 3 hyperbolic S -waves, 148 improved bilinear L2 bound, 79 improved bilinear Strichartz bound, 416 improved Strichartz, 28, 328, 331, 332 improved trilinear estimates, 180 infinite Duhamel expansion, 367 integrability conditions, 5 interaction terms, 2, 71, 410, 418 intermediate wave maps, 359 intrinsic formulation, 1 John–Nirenberg inequality, 251 Kenig–Merle method, 13, 16–18, 23, 425 Klainerman–Machedon, 2, 224, 243 lacunary series, 287 Lagrangian, 1 large data problem, 3 Lemaire’s theorem, 18, 455 life-span, 246–249, 431 Littlewood–Paley projection, 25 Littlewood–Paley trichotomy, 66, 201, 258, 260, 266, 301, 353, 388, 428, 437 local form of energy conservation, 10, 341 logarithmic divergences, 9, 258 Lorentz transform, 77, 431, 435, 443, 444 loss of compactness, 12 Lyapunov functional, 451 magnetic potential, 13, 310, 333, 344, 415 magnetic potential interaction term, 71, 336 maximal interval of existence, 16, 223, 244, 425 Metivier–Schochet scale selection, 297 microlocal gauge, 2 Mikhlin multiplier operator, 279, 285, 290 mild null-structure, 197
Index Mockenhaupt’s square function, 76 modulation, 25 modulation balanced, 76 non-atomic, 14, 22, 309, 397, 409 non-atomic components, 294 non-concentration property, 249 nonconstant harmonic map, 445 nonlinear self-interactions, 410 null-form, 2, 20, 101, 111 null-form bounds, 101, 111 null-frame space, 11, 28, 31, 261, 344 null-structure, 2, 6, 65–72, 197 opposing .CC/ waves, 75 opposing waves, 96 orthonormal frame, 4 outgoing edge, 217 parallel interactions, 11 parametrix, 355, 365 perturbative theory, 223 plane-wave decomposition, 9 Poincaré’s inequality, 251, 430 profile decomposition, 294 pull-back bundle, 1, 428 quintilinear null-form, 208, 461 quintilinear term, 67–72, 197–216 radiation term, 410 reduced expressions, 217, 459 reduction to the hyperbolic case, 116 Rellich’s theorem, 428 reverse Hölder inequality, 251 Riesz projection, 240 Riesz transform, 6, 231 rigidity, 18, 444 scattering problem, 247 scatters at infinity, 223 second fundamental form, 1, 450 second stage of Bahouri–Gérard decomposition, 358, 423 self-similar blow-up, 2, 18, 444–453
483
septilinear, 69, 326, 327 sharp improved Strichartz endpoint, 219 sharp maximal function, 278 Shatah–Struwe, 12, 18, 454 single limiting profile, 17 small atoms, 21, 305–307, 325 small data theory, 8 small energy wave maps, 2, 8, 249, 252, 264 smooth wave map, 5 splitting the wave map, 225 square function, 32, 35, 40, 76, 89 stability result, 244 standard frame, 428 stationary phase, 243 Strichartz estimate, 6, 9, 20, 27, 28, 41, 225, 343, 433, 434 Strichartz norm, 6, 11, 12, 31, 40, 52, 53, 204, 221 Struwe, Michael, 2, 449 subcritical theory, 2, 8, 9 Tao’s divisibility lemma, 365 Tao’s trilinear estimate, 116 Tataru, Daniel, 2, 9, 25, 27, 28 temporally bounded type, 379 temporally unbounded concentration profile, 404 temporally unbounded type, 387, 388 time localizations, 57 transport equation, 329 trilinear estimates, 115 trilinear nonlinearities, 11 trilinear null-form, 11, 66, 116, 148, 181, 188, 197, 231, 415, 469 vanishing moment condition, 431 Vitali covering lemma, 446, 449 Volterra iteration, 365, 369, 372 wave maps with L2 data, 245 wave-packet atoms, 10, 11, 36, 37, 43, 45, 46, 49, 52, 53, 195, 221, 261 wave-packet decomposition, 96
484 wave-packets, 33, 36, 232, 271 weak wave map, 429 weakly harmonic map, 18 weighted commutator estimates, 277 Whitney decomposition, 55
Index
