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1 UC Bereley UC Bereley Previously Published Wors Title Local wellposedness of Chern-Simons-Schrödinger Permalin Journal International Mathematics Research Notices, 014(3) ISSN Authors Liu, B Smith, P Tataru, D Publication Date DOI /imrn/rnt161 Peer reviewed escholarship.org Powered by the California Digital Library University of California

2 LOCAL WELLPOSEDNESS OF CHERN-SIMONS-SCHRÖDINGER BAOPING LIU, PAUL SMITH, AND DANIEL TATARU arxiv: v1 [math.ap] 6 Dec 01 Abstract. In this article we consider the initial value problem for the Chern-Simons-Schrödinger model in two space dimensions. This is a covariant NLS type problem which is L critical. For this equation we introduce a so-called heat gauge, and prove that, with respect to this gauge, the problem is locally well-posed for initial data which is small in H s, s > Introduction 1. Gauge selection 3. Reductions using the heat gauge 4 4. Function spaces 7 5. The linear part of N(φ,A) 0 6. Bilinear estimates 1 7. Estimates on A, B 7 8. Cubic terms Quintic terms Conclusion 39 References 39 Contents 1. Introduction The two dimensional Chern-Simons-Schrödinger system is a nonrelativistic quantum model describing the dynamics of a large number of particles in the plane, which interact both directly and via a self-generated electromagnetic field. The variables we use to describe the dynamics are the scalar field φ describing the particle system, and the electromagnetic potential A, which can be viewed as a one-form on R +1. The associated covariant differentiation operators are defined in terms of the electromagnetic potential A as D α := α +ia α. (1.1) With this notation, the Lagrangian for this system is L(A,φ) = 1 Im( φd t φ)+ D x φ g R +1 φ 4 dxdt+ 1 A da (1.) R +1 Although the electromagnetic potential A appears explicitly in the Lagrangian, it is easy to see that locally L(A, φ) only depends upon the electromagnetic field F = da. Precisely, the Lagrangian is invariant with respect to the transformations for compactly supported real-valued functions θ(t, x). φ e iθ φ A A+dθ (1.3) The second author was supported by NSF grant DMS The third author was supported by NSF grant DMS and by the Simons Foundation. 1

3 Computing the Euler-Lagrange equations for the above Lagrangian, one obtains a covariant NLS equation for φ, coupled with equations giving the electromagnetic field in terms of φ, as follows: D t φ = id l D l φ+ig φ φ t A 1 1 A t = J (1.4) t A A t = J 1 1 A A 1 = 1 φ where we use J i to denote J i := Im( φd i φ) Regarding indices, we use α = 0 for the time variable t and α = 1, for the spatial variables x 1,x. When we wish to exclude the time variable in a certain expression, we switch from Gree indices to Roman. Repeated indices are assumed to be summed. We discuss initial conditions in. The system (1.4) is a basic model of Chern-Simons dynamics [14, 5, 6, 13]. For further physical motivation for studying (1.4), see [15, 17, ]. The system (1.4) has the gauge invariance (1.3). It is also Galilean-invariant and has conserved charge M(φ) := φ dx R and energy E(φ) := 1 D x φ g R φ 4 dx. As the scaling symmetry φ(t,x) λφ(λ t,λx), φ 0 (x) λφ 0 (λx); λ > 0, preserves the charge of the initial data M(φ 0 ), L x is the critical space for equation (1.4). Local wellposedness in H is established in []. Also given are conditions ensuring finite-time blowup. With a regularization argument, [] demonstrates global existence (but not uniqueness) in H 1 for small L data. Our goal in this paper is to establish local wellposedness for (1.4) in spaces over the full subcritical range H s with s > 0. However, in order to state the result we need to first remove the gauge freedom by choosing a suitable gauge. This is done in the next section, which ends with our main result.. Gauge selection In order to interpret the Chern-Simons-Schrödinger system (1.4) as a well-defined time evolution, we need to impose a suitable gauge condition which eliminates the gauge freedom described in (1.3). One can relate the gauge fixing problem here to the similar difficulty occurring in the study of wave and Schrödinger maps. Both wave maps and Schrödinger maps are geometric evolution equations, and in such settings the function φ taes values not in C, but rather more generally in some (suitable) manifold M. A gauged system arises when considering evolution equations at the level of the (pullbac of the) tangent bundle φ TM, where φ denotes the pullbac. A classical gauge choice is the Coulomb gauge, which is derived by imposing the constraint A x = 0. In low dimension, however, the Coulomb gauge has unfavorable high high low interactions.

4 To overcome this difficulty in the d = setting of wave maps into hyperbolic space, Tao [19] introduced the caloric gauge as an alternative. See [1] for an application of the caloric gauge to large data wave maps in d = and [1] for an application to small data Schrödinger maps in d =. We refer the reader to [0, Chapter 6] for a lengthier discussion and a comparison of various gauges. Unfortunately, the direct analogue of the caloric gauge for the Chern-Simons-Schrödinger system does not result in any improvement over the Coulomb gauge. Instead, in this article we adopt from [4] a different variation of the Coulomb gauge called the parabolic gauge. We shall also refer to it as the heat gauge. The defining condition of the heat gauge is A x = A t (.1) Differentiating in the x 1 and x directions the second and third equations (respectively) in (1.4) yields { t 1 A 1 1 A t = 1 Im( φd φ) t A A t = Im( φd 1 φ) Adding these, we get t ( A x ) A t = 1 J + J 1, which, in view of the heat gauge condition (.1), implies that A t evolves according to the nonlinear heat equation ( t )A t = 1 J + J 1 (.) Similarly, we obtain (coupled) parabolic evolution equations for A 1 and A : { ( t )A 1 = J 1 φ ( t )A = J φ (.3) We still retain the freedom to impose initial conditions for the parabolic equations for A in (.) and (.3), in any way that is consistent with the last equation in (1.4). We impose A t (0) = A x (0) = 0 To see that such a choice is consistent with (1.4), observe that A x (0) = 0 coupled with the fourth equation of (1.4) yields the system { 1 A 1 (t = 0)+ A (t = 0) = 0 1 A (t = 0) A 1 (t = 0) = 1 φ 0 (.4), which in turn implies { A 1 (t = 0) = 1 φ 0 A (t = 0) = 1 1 φ 0 (.5) Substituting (.5) into (.3) yields { t A 1 (t = 0) = Im( φd φ) t A (t = 0) = Im( φd 1 φ), which is exactly what we obtain directly from the second and third equations of (1.4) at t = 0 with the choice A t (t = 0) 0. 3

5 So having imposed an additional equation in order to fix a gauge, we study the initial value problem for the system D t φ = id l D l φ+ig φ φ t A 1 1 A t = Im( φd φ) t A A t = Im( φd 1 φ) (.6) 1 A A 1 = 1 φ A t = A x with initial data Our main result is the following. φ(0,x) = φ 0 (x) A t (0,x) = 0 A 1 (0,x) = 1 1 φ 0 (x) A (0,x) = φ 0 (x) Theorem.1. For any small initial data φ 0 H s (R ), s > 0, the equation (.6) with initial data (.7) has solution φ(t,x) C([0,1],H s (R )), which is the unique uniform limit of smooth solutions. In addition, φ 0 φ is Lipschitz continuous from H s (R ) to C([0,1],H s (R )). We remar that ideally one would lie to have global well-posedness for small data in L. Unfortunately, in our arguments we encounter logarithmic divergencies at nearly every step with respect to the L setting, maing it impossible to achieve this goal. Another interesting remar is that while the initial system (1.4) is time reversible, the parabolic evolutions added by our gauge choice remove the time reversibility. One may possibly view this as a disadvantage of our gauge choice. Our result is proved via a fixed point argument in a topology X s, defined later, which is stronger than the C([0,1],H s (R )) topology. Thus we directly obtain uniqueness in X s, as well as Lipschitz dependence on the initial data with respect to the X s topology. (.7) 3. Reductions using the heat gauge Let f denote the space-time Fourier transform f(τ,ξ) := e i(tτ+x ξ) f(t,x)dtdx We define H 1 as the Fourier multiplier H 1 f := 1 (π) 3 1 iτ + ξ ei(tτ+x ξ) f(τ,ξ)dτdξ (3.1) Applied to initial data, it taes the form H 1 (f(x)δ t=0 ) = 1 {t 0} e t f(x) We define H 1 similarly: H 1 f := 1 (π) 3 1 e i(tτ+x ξ) f(τ,ξ)dτdξ (3.) (iτ + ξ ) 1 4

6 Here we use the principal square root of the complex-valued function iτ+ ξ by taing the positive real axis as the branch cut. As the above symbol is still holomorphic for τ in the lower half-space, it follows that its ernel is also supported in t 0. In what follows all these operators are applied only to functions supported on positive time intervals. Using (.), we can rewrite A t as A t = H 1 ((Q 1 ( φ,φ))) H 1 ( 1 (A φ ))+H 1 ( (A 1 φ )), (3.3) where Q 1 (φ, φ) := Im( 1 φ φ φ 1 φ). Similarly, by (.3) and initial condition (.5), we can rewrite A x as follows: Here A 1 = H 1 A 1 (0) H 1 [Re( φ φ)+im( φ φ)] H 1 (A φ ) A = H 1 A (0)+H 1 [Re( φ 1 φ)+im( φ 1 φ)]+h 1 (A 1 φ ) A 1 (0) = 1 1 φ 0 and A (0) = φ 0 Our strategy will be to use the contraction principle in the equations (3.4) in order to bound A 1 and A in terms of φ, and then to use (3.3) to estimate A t. The contraction principle is not applied directly to A 1 and A, but instead to These functions solve the system B 1 = H 1 (A φ ), B = H 1 (A 1 φ ), B 1 = H 1 ((H 1 A (0)+H 1 [Re( φ 1 φ)+im( φ 1 φ)]) φ )+H 1 (B φ ) B = H 1 ((H 1 A 1 (0) H 1 [Re( φ φ)+im( φ φ)]) φ ) H 1 (B 1 φ ) We observe here that the first components of A 1 and A depend only on the initial data; therefore they effectively act almost as stationary electromagnetic potentials for the linear Schrödinger equation. The difficulty is that even if φ 0 is localized, both of these components have only x 1 decay at infinity, which in general would mae them nonperturbative long range potentials. Fortunately A 1 (0) and A (0) are not independent, and taing into account their interrelation will allow us to still treat their effects in the Schrödinger equation as perturbative. However, this ends up causing considerable aggravation in the construction of our function spaces. Now we turn our attention to the first equation in (1.4), which we expand using (1.1) as (3.4) (3.5) (i t + )φ = N(φ,A) := ia l l φ i l A l φ+a t φ+a x φ g φ φ (3.6) where A t, A 1 and A are given by (3.3) and (3.4). Our plan is to solve this equation perturbatively. However, in order for this to wor, we need to use (3.3) and (3.4) to expand the A s in the nonlinearity and replace them by B s. Even this expansion is not sufficient in the case of the first term in N(φ,A), for which we need to expand once more. Eventually this leads to an expression of the form 7 N(φ,A) = Lφ+N 3,1 +N 3, +N 3,3 +N 5,1 +N 5, + where the terms above are as follows: 1. L contains the linear terms in φ, which arise from the first term in N and the first term in the expansion of A. It has the form Lφ = iq 1 (C,φ), C = H 1 1 φ 0 (3.7) 5 j=0 E j

7 where Q 1 (C,φ) = 1 C φ C 1 φ As mentioned before, this term significantly affects our function spaces constructions.. The terms N 3,1, N 3, and N 3,3 are the cubic terms in φ, described as follows: N 3,1 = H 1 ( φ 1 φ) φ H 1 ( φ φ) 1 φ, (3.8) originates from the first term in N and the second term in A. The null structure in N 3,1 is crucial in our estimates. N 3, = H 1 (Q 1 ( φ,φ))φ, (3.9) originates from the second term in N and the second term in A, and also from the third term in A and the first term in A t. This also exhibits a null structure that we tae advantage of. is the contribution of the last term in N. N 3,3 = φ φ (3.10) 3. The terms N 5,1, N 5, and N 5, are the quintic terms in φ, described as follows: N 5,1 = H 1 (H 1 ( φ φ) φ ) φ (3.11) occurs in the reexpansion of A 1 and A in the first term in N. N 5, = H 1 ( φ φ)h 1 ( φ φ)φ (3.1) occurs in the fourth term in N, corresponding to the second term in A 1,A. N 5,3 = H 1 (H 1 ( φ φ) φ )φ (3.13) occurs in the third term in N, corresponding to the second term in A 1,A arising in the A t expression. In all these terms it is neither important which spatial derivatives are applied nor where the bar goes in φ φ. While they loo somewhat different, in the proofs of the multilinear estimates they turn out to be essentially equivalent by a duality argument. 4. The error terms are those which can be estimated in a relatively simpler manner. We begin with the multilinear terms containing the data for A x, namely from the reexpansion of the first term in N, from the second and third terms in N, from the fourth term in N, E 1 = H 1 (H 1 A x (0) φ ) φ (3.14) E = H 1 (H 1 A x (0) φ )φ (3.15) E 3 = H 1 A x (0)H 1 ( φ φ)φ (3.16) E 4 = (H 1 A x (0)) φ+h 1 A x (0)Bφ+B φ. (3.17) Finally we conclude with the remaining terms involving B, from the reexpansion of the first term in N, from the second and third term in N, E 5 = H 1 (B φ ) φ (3.18) E 6 = H 1 (B φ )φ (3.19) E 7 = H 1 ( φ φ)bφ (3.0) 6

8 from the fourth term in N. 4. Function spaces In this section we define function spaces as in [16, 9], but with some suitable adaptations to the problem at hand. Spaces similar to those in [16, 9] have been used to obtain critical results in different problems [8, 10]. We refer the reader to [8, ] for detailed proofs of the basic properties of U p,v p spaces. For a unit vector e S 1, we denote by H e its orthogonal complement in R with the induced measure. Define the lateral spaces L p,q e with norms ( ( f L p,q e = with the usual modifications when p = or q =. R f(xe+x,t) q dx dt H e R )p q dx )1 p Define the operator P N,e by the Fourier multiplier ξ ψ N (ξ e), where ψ N has symbol ψ N (ξ) given by (4.8). The following smoothing estimate plays an important role in our analysis. Lemma 4.1 (Local smoothing [11, 1]). Let f L (R ), N Z,N 1, and e S 1. Then e it P N,e f L, e Also recall the well-nown Strichartz estimates. N 1 f L (4.1) Lemma 4. (Strichartz estimates [18, 0]). Let (q, r) be any admissible pair of exponents, i.e. 1 q + 1 r = 1 and (q,r) (, ). Then we have the homogeneous Strichartz estimate e it f L q t Lr x(r R ) f L x(r ) (4.) 4.1. U p and V p spaces. Throughout this section let H be a separable Hilbert space over C. Let Z be the set of finite partitions t 0 < t 1 <... < t K of the real line. If t K = and v : R H, then we adopt the convention that v(t K ) := 0. Let χ I : R R denote the (sharp) characteristic function of a set I R. Definition 4.3. Let 1 p <. For any {t } K =0 Z and {φ } K 1 =0 H with K 1 =0 φ p H = 1, we call the function a : R H defined by a = K χ [t 1,t )φ 1 =1 a U p -atom. We define the atomic space U p (R,H) as the set of all functions u : R H admitting a representation u = λ j a j for U p -atoms a j, {λ j } l 1 j=1 and endow it with the norm u U p := inf λ j : u = j=1 λ j a j, λ j C, a j a U p -atom j=1 7 (4.3)

9 Remar 4.4. The spaces U p (R,H) are Banach spaces and we observe that U p (R,H) L (R;H). Every u U p (R,H) is right-continuous. On occasion the space U p is defined in a restricted fashion by requiring that t 0 >. Then u tends to 0 as t. The only difference between the two definitions is in whether or not one adds constant functions to U p. Definition 4.5. Let 1 p <, We define V p (R,H) as the space of all functions v : R H such that ( K ) 1 p v V p := sup {t } K =0 Z v(t ) v(t 1 ) p H (4.4) is finite. =1 Remar 4.6. WerequirethattwoV p functionsbeequaliftheyareequalinthesenseofdistributions. Since such functions have at most countably many discontinuous points in time, we adopt the convention that all V p functions are right continuous, i.e., we assume we always wor with the unique right-continuous representative from the equivalence class. The spaces V p (R,H) are Banach spaces and satisfy U p (R,H) V p (R,H) U q (R,H) L (R;H), p < q (4.5) We denote by DU p the space of distributional derivatives of U p functions. Then we have the following very useful duality property: Lemma 4.7. The following duality holds (DU p ) = V p, 1 p < (4.6) with respect to a duality relation that extends the standard L duality. We refer the reader to [8] for a more detailed discussion. We also record a useful interpolation property of the spaces U p and V p (cf. [8, Proposition.0]). Lemma 4.8. Let q 1,q >, E be a Banach space and T : U q 1 U q E a bounded bilinear operator with T(u 1,u ) E C j=1 u j U q j. In addition, assume that there exists C (0,C] such that the estimate T(u 1,u ) E C j=1 u j U holds true. Then T satisfies the estimate T(u 1,u ) E C (ln C +1) u j C V, u j V, j = 1,. Proof. The proof is the same as that in [10, Lemma.4]. For fixed u, let T 1 u := T(u,u ). Then we have that T 1 u E D 1 u U q 1 and T 1 u E D 1 u U. Here D 1 = C u U q,d 1 = C u U. From the fact that u U q j u U and [8, Proposition.0], we obtain j=1 T(u 1,u ) E = T 1 u 1 E C (ln C C +1) u 1 V u U (4.7) 8

10 Then we can repeat the argument by fixing u 1, using estimate (4.7), and T(u 1,u ) E C u j U q j C u 1 V u U q j j=1 Let ψ : R [0,1] be a smooth even function compactly supported in [,] and equal to 1 on [ 1,1]. For dyadic integers N 1, set ( ξ ) ( ξ ) ψ N (ξ) = ψ ψ, for N and ψ 1 (ξ) = ψ( ξ ). (4.8) N N For each such N 1, define the frequency localization operator P N : L (R ) L (R ) as the Fourier multiplier with symbol ψ N. Moreover, let P N := 1 M N P M. We set u N := P N u for short. We now introduce U p,v p -type spaces that are adapted to the linear Schrödinger flow. Definition 4.9. For s R, let U p Hs (resp. V p Hs ) be the space of all functions u : R H s (R ) such that t e it u(t) is in U p (R,H s ) (resp. V p (R,H s )), with respective norms u U p Hs = e it u U p (R,H s ), u V p Hs = e it u V p (R,H s ) (4.9) Remar The embeddings in Remar 4.6 and Lemma 4.8 naturally extend to the spaces U p Hs and V p Hs. To clarify the roles of the U,V spaces, we introduce the X0,b -type spaces defined via the norms ( and u X 0,1,1 = ϑ u Ẋs, 1, = sup ϑ ( For these spaces we have the embeddings [16] ũ(τ,ξ) τ ξ dξdτ τ ξ =ϑ ũ(τ,ξ) τ ξ dξdτ τ ξ =ϑ Ẋ 0,1,1 U L V L Ẋ0,1, (4.10) From these inclusions we can conclude that the U and V norms are equivalent when restricted in modulation to a single dyadic scale. Another straightforward consequence of the definitions (see for instance [8, Proposition.19]) is that one can extend the local smoothing estimate and Strichartz estimates to general U p functions: e it P N,e f L, N 1 f e U (4.11) e it f L q t Lr x(r R ) f U p (4.1) Here (q,r) is any admissible pair of exponents and p := min(q,r). Finally, in the case of free solutions for the Schrödinger equation we can easily do orthogonal frequency decompositions. For the U and V functions we have the following partial substitute: 9 )1 )1

11 Lemma Let 1 = R P R(D) be a locally finite partition of unity in frequency, with uniformy bounded symbols. Then we have the dual bounds P R u U u U respectively R R P R f R V R f R V The proof is straightforward and is left for the reader. 4.. Lateral U p and V p spaces. Unfortunately the above function spaces are insufficient for closing the multilinear estimates in our problem. Instead we also need to define the lateral U p and V p spaces. Given a unit vector e S 1, we consider orthonormal coordinates (ξ e,ξ e ) with ξ e = ξ e. Then we define the Fourier region In this region we have A e = {(τ,ξ) R ; ξ e > 1 4 ξ, τ ξ < 1 3 ξ } and therefore can factor the symbol of the Schrödinger operator: τ ξ e 1 64 ( τ +ξ e ) (4.13) ξ τ = (ξ e + τ ξ e )(ξ e τ ξ e ) ξ (ξ e τ ξ e ) Hence instead of considering the forward Schro dinger evolution we can wor with the lateral flow e il e, L e = i t e for functions frequency localized in the region (4.13). We denote the corresponding U p and V p function spaces by U p e and V p e, respectively. Now we are ready to define the nonlinear component of our function spaces, namely U, and V,. For that we need some multipliers, denoted by P e, adapted to the regions A e. The space U, is given by U, = U + D 1 Σ e P e U e (4.14) Inother wordsit can bethought of as anatomic space wheretheatoms arenormalized U functions and normalized U e functions. The space V, is given by the norm φ V, = φ V +sup D 1 P e φ V e (4.15) e By U,,, respectively V, we denote the corresponding spaces of functions which are localized at frequency. The main properties of these spaces are summarized in the following Proposition 4.1. The spaces U, and V, defined above have the following properties: a) Inclusion: U, V, (4.16) 10

12 b) Duality: [(i t )U, ] = V, (4.17) c) Truncation. For any time interval I we have χ I : U, V, (4.18) We postpone the proof of this result for later in the section. Part (a) is a special case of (c) when I = R. Part (b) is a direct consequence of the duality result in Lemma 4.7. Part (c) is proved in Lemma The l spatial structure. We need one additional structural layer to overlay on top of the U and V structure, which has to do with the fact that we are seeing to solve the problem locally in time. Thus all the estimates we will have to prove apply to functions which are localized in time to a compact interval. Within such an interval, waves at frequency travel a distance of O( ), with rapidly decreasing tails farther out. Thus if we partition the space into sized squares, the interaction of separated squares is negligible. This leads us to introduce a local in time partition of unity 1 = χ m (x,t), χm (x,t) = χ 0(t)χ( x m) m Z and corresponding norms φ l U, = m Z χ m (x,t)φ U, We similarly define the l V, and l DU, norms. To relate these norms with the previous ones we have the following: Proposition a) For all φ U, localized at frequency we have φ l U, φ U, (4.19) b) For all φ localized at frequency and with compact support in time we have φ V, φ l V, (4.0) 4.4. The angular spaces. Theabove spaces sufficeinorderto treat thenonlinear partof N(φ,A). However, for the linear part L we need an entirely different type of structure. To set the notation, we denote the angular derivative centered at x 0 by / x0 = (x x 0 ) x We also set x x 0 = ( +(x x 0 ) ) 1 Let σ > 0 be a fixed constant. For x 0 R and Z we define the space X x 0,σ with norm φ x X 0,σ as well as the smaller space X x 0,σ, We further define X σ = φ X x 0,σ, X x0,σ x 0 R = (1 σ) x x 0 1 σ / σ φ L (4.1) with norm = φ(0) L + (i t )φ X x 0,σ,, X σ, = X x0,σ, x 0 R 11, X σ, = x 0 R X x0,σ, (4.), (4.3)

13 where the first space is the dual of the second. Thesespaces areused for frequency solutions to the Schrödinger equation. Theirmain properties are stated in the following Proposition The spaces defined above, restricted to frequency functions, have the following properties: a) Solvability: b) Moving centers: c) Nesting: φ X σ, φ(0) L + (i t )φ X σ, φ X σ + φ L L φ X σ, φ X σ 1 (4.4) (4.5) φ σ X, σ < σ 1 (4.6) In this result is a scaling parameter and can be set to 0. Part (a) is a straightforward consequence of the definitions. However, part (b) is far less trivial, and requires two separate estimates. First, for fixed x 0 we need to show that which is done in Lemma φ X x 0,σ Secondly, for x 1 x 0 we need to show that φ X x 1,σ + φ L L φ X x 0,σ, φ X x 0,σ + φ X x 0,σ, (4.7) (4.8) This is achieved in Lemma Part (c) follows from the similar property for fixed x 0, which is straightforward in view of the frequency localization Dyadic norms and the main function spaces. Finally, we are ready to set up the global function spaces where we solve the Chern-Simons-Schrödinger problem. For the solutions at frequency we use two spaces. The stronger norm X represents the space where the solutions actually lie and is given by X = l U, +X σ, (4.9) Here 0 < σ < 1 is a fixed constant. However, this is a sum type space and so multilinear estimates would be quite cumbersome, with many cases. Furthermore, the above space is not stable with respect to time truncations. Instead we also introduce a weaer topology X = l V, X σ (4.30) For the inhomogeneous term in the equation we have the space Y which has X as its dual, The main result concerning our function spaces is in the following Y = l DU, +X σ, (4.31) Theorem 1. The following properties are valid for frequency functions: a) Linear estimate: φ X φ(0) L + (i t )φ Y (4.3) 1

14 b) Weaer norm: c) Time truncation: d) Duality: φ X φ X (4.33) χ I φ X φ X, I R (4.34) Y = X (4.35) Part (a) is a direct consequence of the preceding three propositions. The estimate in (b) is a special case of (c). Part (c) also follows in part from the two preceding propositions. However, we still need to address the cross embeddings, χ I l U, X σ respectively χ I X σ, l V, The truncation in the first embedding can be harmlessly dropped as it is bounded on X σ. It remains to show the embedding l U, X σ (4.36) which is proved in Lemma The truncation in the second embedding can also be dropped. To see this recall that X σ, L L. This allows us to freely replace arbitrary functions u X σ, by solutions to the homogeneous equation outside I. But then χ I u u U and we can use (4.16). Once χ I is dropped, using again X σ, L L, the problem reduces to which follows from (4.36) by duality. X σ, l, DV In this article we wor with H s initial data. Correspondingly, we define the spaces X s, and X s for solutions, respectively Y s for the nonlinearity, by φ X s, = 0 s P φ X (4.37) φ X s = 0 s P φ X (4.38) where P 0 includes all frequencies less than 1. f Y s = 0 s P f Y (4.39) 4.6. The nonlinearity N(φ, A). Here we turn our attention to the nonlinear equation (i t )φ = N(φ,A), φ(0) = φ 0, A = A(φ), where A(φ) is obtained by solving (3.3)-(3.4). We see to solve this equation for positive t and locally in time; therefore we can harmlessly insert a cutoff function χ = χ [0,1] in time and solve instead the modified equation (i t )φ = N(χφ,A), φ(0) = φ 0, A = A(χφ) (4.40) Any global solution to this modified equation will solve the original equation in the time interval [0,1]. 13

15 We use the H s version of the linear estimate (4.3) to solve this equation in the space X s, using the contraction principle. Thus we need to show that we have a small Lipschitz constant for the map X s, φ N(χφ,A) Y s, A = A(χφ) We subdivide this problem into two completely different problems, which correspond to the decomposition of N(χφ,A) into a linear and a nonlinear part. To estimate the linear part L(χφ) we will use the H s version of the embedding (4.34) and select only the X σ,s part of the X s norm, neglecting the l structure. Then we can drop the cutoff χ, and it remains to prove the bound This is achieved in Section 5, Proposition 5.1. Lφ X σ,s, φ X σ,s φ(0) H s (4.41) To estimate the nonlinear part Nl(χφ,A) = N(χφ,A) Lχψ we see to prove X s, φ Nl(χφ,A) Y s, A = A(χφ) Retaining only the U - V part of our function spaces, it suffices consider the map l V,,s φ Nl(φ,A) l DU,,s, A = A(φ) for φ localized in time. By duality, this translates to Lipschitz continuity of the form l V,,s l V,, s (φ,ψ) Nl(φ,A) ψ dxdt (4.4) To prove this we succesively consider all the terms in Nl(φ,A) in Sections Linear estimates. We now proceed to state and prove a collection of linear lemmas which, together, imply the results stated before in this section. Given an angle A in R with opening less than π, we say that a direction e is admissible with respect to A if ±e A. For 0 we define the following subset of Fourier space A = {(ξ,τ) A R; ξ, τ ξ } We denote by P A, a smooth space-time multiplier with support in A. Then we have Lemma Let A be an angle in R, > 0, I a time interval and e 1, e admissible directions with respect to A. Then for functions f that are frequency localized in A, we have P A, χ I f V e f U e1 (4.43) with an implicit constant that is uniform with respect to pairs e 1,e for which the distances dist(±e 1,,A) lie in a compact set away from zero. This lemma serves to prove the properties (4.16) and (4.18) in Proposition 4.1. We note that two nontrivial properties are coupled in the statement, namely the embedding Ue 1 Ve and the time truncation. We further note that the same estimate holds true if either of the two lateral spaces is replaced by the corresponding vertical space. In that case the time truncation can be absorbed into the vertical space and one is left with just the embedding, for which the proof below still applies. Proof. By scaling we can assume that = 0. We consider the multiplier P which selects a small neighborhood of A 0. Then P is bounded on both V e and U e 1, and so it suffices to show that Pχ I Pf V e f U e1 (4.44) 14

16 Set I = [t 0,t 1 ]. We can harmlessly replace χ I by its mollified version Q 0 χ I as its high modulation part provides no output. We first observe that the simpler bound Pf L, e + Pf L L f U e 1 (4.45) follows easily by reducing to a U e 1 atom where the free waves associated to each step are supported in a small neighborhood of the intersection of A 0 with the paraboloid. Then we apply either the energy estimate or the lateral energy estimate in the e direction for each step of that atom. Then the bound (4.44) reduces to (i t )Pχ I Pf DV e f U e1 Using the duality between DV and U, this is equivalent to the symmetric bound Q R (f,g) f U e1 g U e (4.46) where Q R (f,g) = (i t )χ I Pf,Pg = Pf,χ I (i t )Pg This is not entirely symmetric, and so we also introduce its twin Q L (f,g) = χ I (i t )Pf,Pg Their difference is easy to control. Indeed, we have Q R (f,g) Q L (f,g) = [(i t ),χ I ]Pf,Pg = i t χ I Pf,Pg The time derivative of χ I is a sum of two unit bump functions on a unit time interval around t 0, respectively t 1. Hence using the energy part of (4.45) we obtain Q R (f,g) Q L (f,g) f U e1 g U e (4.47) Given the support of P, we can rewrite Q L and Q R in terms of the sideways evolutions for f and g: Q L (f,g) = χ I P(D e1 L e1 )f,pg, Q R (f,g) = Pf,χ I P(D e1 L e1 )g (4.48) Here the elliptic factor in the factorization of i t is included in P. Thus by a slight abuse of notation we use the same P for different multipliers with similar size and support. It suffices to prove (4.46) for atoms. Thus consider f and g of the form f = χ [ai,b i ](x e 1 )f i, g = η [ci,d i ](x e )g i where f i and g i are homogeneous waves, frequency localized in a small neighborhood of A 1, and with f i (0) L 1, g i (0) L 1 i As f i and g i are free waves frequency localized near the A section on the parabola at frequency one, we can measure their energy in an equivalent way at time t = 0. Instead of the data at time t = 0, it is better to describe f i in terms of its values at x e 1 = a i and at x e 1 = b i. By a slight abuse of notation we denote these two functions by f i (a i ) and f i (b i ). We remar that f i (a i ) and f i (b i ) are related via the sideways evolution and in particular we have f i (a i ) L = f i (b i ) L However, it will be convenient to wor with both of them together rather than separately. 15 i

17 We observe that it suffices to consider the case when b i a i 1 and c i d i 1. Indeed, if for instance b i a i 1 for all i then f L 1 This is easily combined with the following easy consequence of (4.10), to conclude the argument. (i t )Pg L 1, We can also assume without any restriction in generality that a i+1 b i 1 and c i+1 d i 1. To the intervals [a i,b i ] we associate bump functions χ i which equal 1 inside the interval and decay rapidly on the unit scale. By η i we denote similar bump functions associated to [c i,d i ]. Set B ij L := Q L(χ [ai,b i ](x e 1 )f i,η [cj,d j ](x e )g j ) B ij R := Q R(χ [ai,b i ](x e 1 )f i,η [cj,d j ](x e )g j ) We want to be able to use Q L and Q R interchangeably. For that we estimate the difference B ij L Bij R = i tχ I Pχ [ai,b i ](x e 1 )f i,pη [cj,d j ](x e )g j Using the time localization given by t χ I and the finite speed of propagation in time for waves supported in A 0, we obtain a localized analogue of (4.47), namely B ij L Bij R χ iη j f i (t 0 ) L χ i η j g j (t 0 ) L + χ i η j f i (t 1 ) L χ i η j g j (t 1 ) L By Cauchy-Schwarz this implies that which indeed allows us to estimate B ij L (4.48) we have i,j B ij L Bij R 1 (4.49) and Bij R interchangeably. Using the representation of Q L in B ij L := Q L(χ [ai,b i ](x e 1 )f i,η [cj,d j ](x e )g j ) = f i (b i )δ x e1 =b i f i (a i )δ x e1 =a i,p(η [cj,d j ](x e )g j ) A symmetric formula holds for B ij R. For g j we have lateral energy estimates in the e 1 directions, and P has a rapidly decreasing ernel. Hence the above expression is bounded by B L ij η j f i (b i ) L η j g j (b i ) L + η j f i (a i ) L η j g j (a i ) L (4.50) In order to complete the proof of (4.46) for atoms we need to distinguish between different interval balances: A. Unbalanced intervals: Either b i a i d j c j or b i a i d j c j. In this case we will prove that min{ B ij L, Bij R } ( η j(x e )f i (b i ) L + η j (x e )f i (a i ) L ) ( χ i (x e 1 )g j (d j ) L + χ i (x e 1 )g j (c j ) L ) 16 (4.51)

18 for some more relaxed bump functions η j and χ i which share the properties of χ i and η j. Assuming (4.51) is true, the estimate for the corresponding part of (4.46) easily follows from Cauchy-Schwarz: min{ B ij L, Bij R } η j (x e )f i (b i ) L + η j (x e )f i (a i ) L i,j i,j + i,j χ i (x e 1 )g j (d j ) L + χ i (x e 1 )g j (c j ) L i 1 f i (b i ) L + f i (a i ) L + j g j (d j ) L + g j (c j ) L By symmetry supposethat b i a i d j c j. Then (4.51) follows from (4.50) dueto the propagation estimate η j g j (b i ) L + η j g j (a i ) L χ i g j (d j ) L + χ i g j (c j ) L To see this it suffices to consider the Schrödinger propagator from the surfaces x e 1 = a i,b i to the surfaces x e = c j,d j. corresponding to waves which are localized in A 0. On the one hand, with respect to suitable elliptic multiplier weights, this is an L isometry. On the other hand, its ernel decays rapidly outside a conic neighborhood of the propagation cone associated to A 0. Hence all that remains to be seen is that the propagation cone of the interval {x e 1 = a i, x e [c j,d j ]} either intersects the line x e = c j within the interval x e 1 [a i,b i ] or intersects the line x e = d j within the interval x e 1 [a i,b i ]. But this is a geometric consequence of the unbalanced intervals. B. Balanced intervals. Here we consider the case when b i a i d j c j. The first observation is that it suffices to consider a fixed dyadic scale X and assume that b i a i d j c j X The dyadic summation with respect to X will be straightforward since we have l summbability both on the f and on the g side. The simplification that occurs when we fix the interval size is that we are allowed to relax the localization scale in the choice of the functions χ j and η j in (4.50). Precisely, instead of the rapid decay on the unit scale (dictated by the smallest distance to the next interval) we allow them to decay rapidly on the X scale, and denote them by χ X i and ηj X. This maes the following norms equivalent: ηj X f j (b i ) L ηj X f j (a i ) L χ X i f j (c j ) L χ X i f j (c j ) L by standard propagation arguments. By (4.50) this implies the version of (4.51) with the weights χ X i and ηj X. The punch line is then in the i and j summation argument under (4.51). The bumps χ X i and ηj X are wider now, but they are still almost orthogonal since the intervals are now also uniformly spaced at distance X (or above). Our next lemma serves to prove the estimate (4.7), which is needed for Proposition In order to do that we need two more definitions, namely the local energy space (centered at 0) LE and its dual LE. The LE space-time norm adapted to frequency-one functions is defined as Then we have φ LE = φ L ( x 1) +sup j φ L ( x j ) j>0 17

19 Lemma Let s > 0. Then for frequency-one functions u solving (i t )u = f 1 +f, where f 1 has no radial modes, we have the following estimate u LE + r 1 s / s u L u(0) L + r 1 +s / s f 1 L + f LE (4.5) Proof. Our starting point is the standard local energy decay estimate for frequency-one functions, namely u LE u(0) L + f LE (4.53) We expand the function u in (4.5) in an angular Fourier series. This preserves the frequency localization, and it suffices to prove (4.5) for each such mode separately (with uniform constants). Our contention is that for a fixed angular mode the bound (4.5) is a direct consequence of (4.53). To see that let Z and u be of the form u (t,x) = u (r,t)e iθ Then we have r 1 s / s u = r 1 s s u This is easily controlled by the LE norm of u for r. To deal with smaller r we need to use the angular localization. Precisely, we claim that which easily leads to r 1 s P 0 u L (r +) 1 s u L (4.54) and, by duality, s r 1 s P 0 u L u LE P 0 f LE s r 1 +s f L The last two bounds prove that (4.5) follows from (4.53). It remains to establish (4.54). The ernel of P 0 is given by a Schwartz function φ. Then for x we write P 0 u (x) = φ(x y)u(y)dy = ( ) N / N y φ(x y)u(y)dy. We have and therefore Hence (4.54) easily follows. / N y φ(x y) x N x y N P 0 u (x) ( ) r N x y N u(y) dy As a consequence of the above lemma we get the following result, which proves the embedding (4.36) needed in Theorem 1. Lemma The following inequality holds for frequency functions u: (1 s) r 1 s / s u L u U, (4.55) In addition, if u is localized in a unit time interval then (1 s) r 1 s / s u L u l U, (4.56) 18

20 Proof. It suffices to consider 0 < s < 1. For the vertical U space the bound (4.55) is a direct consequence of the previous lemma via the atomic decomposition. It remains to consider a lateral Ue space and a corresponding atom u = χ j u j For each of these atoms we have (1 s) r 1 s / s u j L u j (0) L Thus it would suffice to show that r 1 s / s ( χ j u j ) L r 1 s / s u j L The bound (4.56) also reduces to the same estimate, with the only difference being that the atomic decomposition is now done separately in each sized spatial cube. The last bound reduces to an estimate on the unit circle, χ j u j H s (S 1 ) u j H s (S 1 ), 0 s < 1 It maes no difference whether this is done on the circle or on the real line. The following argument is for the case of the real line. We begin with a simple observation, namely that χ j u j H s u j H s Hence we can drop the cutoffs χ j and instead assume that the u j have disjoint supports in consecutive intervals I j. We use a Littlewood-Paley decomposition, but instead of having sharp Fourier localization it is convenient to choose multipliers P whose ernels have sharp localization in the physical space on the scale. To estimate P ( u j ), we split the intervals I j into long ( I j > 1) and short ( I j < 1). The outputs of long intervals are almost orthogonal, P ( u j ) L P u j L I j long I j long It remains to consider the outputs of short intervals. We still have orthogonality at interval separations of j, and so we can write s P ( I j I u j ) L s P u j L I j short I = I j short But for short intervals we can use the fact that the ernel of P is a bump function with sized support and amplitude to write Hence we obtain s P ( P u j L I j 1 / u j L I j 1 +s / u j H s I j short u j ) L I = I = I j 1 ( I j ) 1 +s u j H s I j I I j I I j ( I j ) s u j H s ( I j ) s u j H s 19 I j I

21 and the summation is straightforward. Finally, the following lemma allows us to move centers and proves the estimate (4.8), which is needed in Proposition 4.14.: Lemma For u localized at frequency one solving (i t )u = f and for any x 0 R, we have x x 0 1 s / x0 s u L u(0) L + r 1 +s / s f L (4.57) Proof. We split u into several spatial regions depending on the ratio of x x 0 and R = x 0. (i) The intermediate region, A med = { x x 0 R}. In this region it suffices to use the local energy decay x x 0 1 s / x0 s χ med u L R 1 χmed u L u LE and then the previous lemma. (ii) The inner region, A in = { x x 0 R}. Here we compute (i t )(χ in u) = f in := χ in u χ in u+χ in f and use the local energy bound for u to estimate f in LE u LE + χ in f LE u LE + r 1 +s / s f L, where at the last step we have used the fact thet χ in is supported in a single dyadic region x R. Then we apply Lemma (iii) The outer region, A out = { x x 0 R}. Here we use the following estimate which applies for frequency one functions x x 0 1 s / x0 s χ out u L x 1 s / s u L +R s x+r 1 s u L, and estimate the second term on the right by the local energy norm. This in turn is proved by complex interpolation between s = 0 and s = 1 since / / x0 = x 0 5. The linear part of N(φ,A) In this section we prove the main estimate (4.41) for the component L of the nonlinearity, see (3.7). For convenience we restate the full result here: Proposition 5.1. Let C = H 1 1 φ 0. Then for s 0 we have Q 1 (C,φ) X σ,s, φ X σ,s φ 0 H s (5.1) Proof. For u 0 = φ 0 we use the multiplicative Sobolev estimate u 0 L 1 + u 0 B 1, s φ 0 H s This is somewhat wasteful if s > 0 but it is tight for s = 0. Using duality we rewrite the bound (5.1) in the more symmetric form Q 1 (C,φ)ψdxdt φ X σ,s ψ X σ, s φ 0 H s 0

22 Integrating by parts it is easy to see that the null form Q 1 can be placed on any two of the factors C, φ, ψ. We use the standard Littlewood-Paley trichotomy. A. High-low interactions. Here we only need Bernstein s inequality to write for > j Q 1 (C,φ j )ψ dxdt +j C L 1 φ j L ψ L (j ) s u 0 B 1, s φ j L L ψ L L (j ) sj φ 0 H s φ j X σ,s ψ X σ, s Thefactor sj is not needed. Thesummation with respect to and j is ensuredby theoff-diagonal decay. B. High-high interactions. This case is equivalent to the one above if s = 0 and better if s > 0. C. Low-high interactions. In this case it suffices to prove the estimate Q 1 (C <,φ )ψ dxdt φ X σ ψ X σ u 0 L 1 (5.) for some choice of σ. This choice is not important due to the nesting property of the X σ spaces. It is easiest to wor with σ = 1. By scaling we can tae = 0. By translation invariance we can tae u 0 = δ 0. Then C is radial, and C <0 (x) = xa( x,t), a(r,t) (1+r +t) 1 Hence which shows that Thus (5.) follows. Q 1 (C <0,φ 0 ) = a( x,t)/ φ 0 Q 1 (C <0,φ 0 ) X 1, 0 φ 0 X Bilinear estimates We definethe temporal frequencylocalization operator Q 0 N to bethe Fourier multiplier withsymbol ψ N (τ) and the modulation localization operator Q N to be the Fourier multiplier with symbol ψ N (τ ξ ). Here ψ N is the same bump function we used in (4.8) to define Littlewood-Paley projections. In the rest of the paper, Q N will be applied to single functions whereas Q 0 N will be applied to bilinear expressions. In the following, we will sometimes use the notation U, which stands for both U and D 1 U e. The same convention holds for V. 1

23 6.1. Pointwise bilinear estimates. These are needed for the case of balanced frequency interactions. Let I λ denote the frequency annulus {ξ R : λ/ ξ λ}. Our first result is Lemma 6.1. Let µ,ν,λ be dyadic frequencies satisfying µ λ and ν µλ. Let φ λ,ψ λ be functions with frequency support contained in I λ. Then P µ Q 0 ν ( φ λ ψ λ ) L µν λ φ λ V, ψ λ V, (6.1) Proof. We first dispense with the high modulations in the inputs. If both are high ( µλ) then by Bernstein we have P µ Q 0 ν (Q µλφ λ Q µλ ψ λ ) L νµ (Q µλ φ λ Q µλ ψ λ ) L 1 µν λ φ λ V ψ λ V If one is high and one is low then we decompose the low modulation factor with respect to small angles and use the lateral energy: P µ Q 0 ν(p e φ λ Q µλ ψ λ ) L νµ 3 (Pe φ λ Q µλ ψ λ ) L,1 µν e λ φ λ D V 1 ψ e λ V Finally if both factors are low modulations then we decompose with respect to small angles and compute P µ Q 0 ν(p e φ λ P e ψ λ ) L νµ (P e φ λ P e ψ λ ) L,1 e A slight sharpening of the above result is as follows: µν λ P eφ λ D 1 e ψ λ Ve P D V 1 e Lemma 6.. Let µ,λ be dyadic frequencies satisfying µ λ. Let φ λ,ψ λ be functions with frequency support contained in I λ. Then P µ H 1 ( φ λ x ψ λ ) L µ φ λ V, ψ λ V, (6.) Proof. While using Bernstein s inequality as in the previous proof leads to a logarithmic divergence and is no longer immediately useful, we can instead use ernel bounds for P µ H 1 with the same effect. The ernel K µ of P µ H 1 satisfies K µ (t,x) µ (1+µ x ) N (1+µ t ) N Then one can repeat the three cases in the previous proof, but using the ernel bounds instead of Bernstein s inequality. 6.. L bilinear estimates for free solutions. We introduce an improved bilinear Strichartz estimate that is a slight generalization of that first shown in [3, Lemma 111]. Lemma 6.3 (Improved bilinear Strichartz). Let u(x,t) = e it u 0 (x),v(x,t) = e it v 0 (x), where u 0,v 0 L (R ). Let Ω 1 denote the support of û 0 (ξ 1 ), Ω the support of ˆv 0 (ξ ), and set Ω = Ω 1 Ω. Assume that Ω 1 and Ω are open and separated by some positive distance. Then u v L t,x sup ξ,τ ξ=ξ 1 ξ χ Ω (ξ 1,ξ )dh 1 (ξ 1,ξ ) τ= ξ 1 ξ dist(ω 1,Ω ) 1/ u 0 L v 0 L (6.3) where dh 1 denotes 1-dimensional Hausdorff measure (on R 4 ) and χ Ω (ξ 1,ξ ) the characteristic function of Ω.

24 Proof. To control u v L t,x, we are led by duality to estimating g(ξ 1 ξ, ξ 1 ξ )û 0 (ξ 1 ) ˆv 0 (ξ )dξ 1 dξ ξ 1,ξ We apply Cauchy-Schwarz and reduce the problem to bounding G := g(ξ 1 ξ, ξ 1 ξ ) dξ 1 dξ (ξ 1,ξ ) Ω Let f : R 4 R 3 be given by R R (ξ 1,ξ ) (ξ 1 ξ, ξ 1 ξ ) =: (ξ,τ) R R. The differential corresponding to this change of coordinates is df = ξ (1) 1 ξ () 1 ξ (1) ξ () The size J 3 f of the 3-dimensional Jacobian of f is defined to be the square root of the sum of the squares of the determinants of the 3 3 minors of the differential df: J 3 f := ( (ξ () ξ () 1 ) +(ξ (1) ξ (1) 1 ) +(ξ () ξ () 1 ) +(ξ (1) 1 ξ (1) 1 )) 1/ Hence J 3 f = C ξ ξ 1 Cdist(Ω 1,Ω ) (6.4) By the coarea formula (see [7, 3]), G = g(ξ 1 ξ, ξ 1 ξ ) dξ 1 dξ (ξ 1,ξ ) Ω = (ξ 1,ξ ) Ω: g(ξ 1 ξ, ξ 1 ξ ) J 3 f 1 (ξ 1,ξ )dh 1 (ξ 1,ξ )dξdτ ξ,τ ξ=ξ 1 ξ τ= ξ 1 ξ g(ξ,τ) (ξ 1,ξ ) Ω: J 3 f 1 (ξ 1,ξ )dh 1 (ξ 1,ξ )dξdτ ξ,τ ξ=ξ 1 ξ τ= ξ 1 ξ g(ξ,τ) dξdτ sup (ξ 1,ξ ) Ω: J 3 f 1 (ξ 1,ξ )dh 1 (ξ 1,ξ ) (6.5) ξ,τ ξ,τ ξ=ξ 1 ξ τ= ξ 1 ξ In view of (6.4), the right hand side of (6.5) is bounded (up to a constant) by g L dist(ω 1,Ω ) 1 sup χ ξ=ξ ξ,τ 1 ξ Ω (ξ 1,ξ )dh 1 (ξ 1,ξ ) τ= ξ 1 ξ A straightforward application of Lemma 6.3 yields Corollary 6.4 (Bourgain s improved bilinear Strichartz estimate [3]). a) Let µ, λ be dyadic frequencies, µ λ. Let φ µ,ψ λ denote free waves respectively localized in frequency to I µ and I λ. Then φ µ ψ λ L µ1/ λ 1/ φ µ(0) L x ψ λ (0) L x (6.6) 3

25 b) If either φ µ or ψ λ is further frequency localized to a box of size α α, then we have the better estimate φ µ ψ λ L α1/ λ 1/ φ µ(0) L x ψ λ (0) L x (6.7) As a corollary of the proof of Lemma 6.3, we obtain the following. Corollary 6.5. Let u(x,t) = e it u 0 (x),v(x,s) = e is v 0 (x), where u 0,v 0 L (R ). Let Ω 1 denote the support of û 0 (ξ 1 ), Ω the support of ˆv 0 (ξ ). Assume that for all ξ 1 Ω 1 and ξ Ω we have Then ξ 1 ξ β u v L s,t,x β 1/ u 0 L v 0 L (6.8) Proof. As in the proof of Lemma 6.3, we use a duality argument. The ey is to bound g(ξ 1 ξ, ξ 1, ξ ) dξ 1 dξ (ξ 1,ξ ) Ω in L. In this setting, the proof is simpler because the change of variables f is given by R R (ξ 1,ξ ) (ξ 1 ξ, ξ 1, ξ ) R R R so that f : R 4 R 4 and df ξ 1 ξ 1. In order to achieve a gain at matched frequencies, we localize the output in both frequency and modulation, seeing to bound P µ Q ν ( φ λ ψ λ ) in L. That Lemma 6.3 may be used efficiently, we introduce an adapted frequency-space decomposition of annuli I λ R that depends upon both the output frequency and modulation cutoff scales µ and ν. Definition 6.6 (Frequency decomposition). Suppose µ, ν, λ are dyadic frequencies satisfying µ λ and ν µλ. We define a partition of I λ into curved boxes as follows. First, partition I λ into λ /ν annuli of equal thicness. Next, uniformly partition the annuli into λ/µ sectors of equal angle. The resulting set of curved boxes we call Q = Q(µ,ν,λ). The curved sides of the boxes in Q have length µ, whereas the straight sides of the boxes have length ν/λ. By adapting a suitable partition of unity to the decomposition, we have f = P R f µ λ R Q(µ,ν,λ) ν µλ Note that we may extend Q(µ,ν,λ) to all smaller dyadic scales λ < λ in the following way: tae the partition Q(µ,ν,λ ) and cut the annuli into λ/λ smaller annuli of equal thicness. In this way we can impose a finer scale on lower frequencies. Corollary 6.7. Let µ,ν,λ be dyadic frequencies satisfying µ λ and ν µλ. Let φ λ,ψ λ be free waves with frequency support contained in I λ. Then P µ Q 0 ν ( φ λ ψ λ ) L ν1/ (µλ) 1/ φ λ L x ψ λ L x (6.9) Proof. The frequency restriction P µ applied to P R φλ P R ψ λ restricts us to looing at the subcollection of boxes R,R Q separated by a distance µ. This subcollection is further restricted by the temporal frequency multiplier Q 0 ν. Let ξ 1 R,ξ R. The modulation τ of the product P R φλ P R ψ λ is given by ξ 1 ξ = (ξ 1 ξ ) (ξ 1 +ξ ) 4

26 Because we apply P µ, ξ 1 ξ µ, and therefore necessarily τ lies in the range τ µλ. We write τ µλcosθ, where θ is the angle between ξ 1 ξ and ξ 1 + ξ. Applying Q ν restricts τ so that τ ν and in particular cosθ ν/(µλ). These restrictions motivate defining the set of interacting pairs of boxes P = P(µ,ν,λ) as the collection of all pairs (R,R ) Q Q (Q = Q(µ,ν,λ)) for which all (ξ 1,ξ ) R R satisfy ξ 1 ξ µ and ξ 1 +ξ ν. Note that, for R Q(µ,ν,λ) fixed, the number p of interacting pairs of boxes P P(µ,ν,λ) containing R is O(1) uniformly in µ,ν,λ. This is a consequence of the restrictions cosθ ν/(µλ) and ξ µ: they jointly enforce at most O(1) translations of a distance ν/λ, which is precisely the scale of the short sides of the boxes. It remains only to show that for (R,R ) P we have sup ξ,τ χ ξ=ξ 1 ξ R (ξ 1 )χ R (ξ )dh 1 (ξ 1,ξ ) ν λ τ= ξ 1 ξ (6.10) Fix ξ R,τ R, ξ 0, and consider the constraint equations { ξ = ξ 1 ξ τ = ξ 1 ξ (6.11) These determine a line in R : τ = (ξ 1 ξ ) (ξ 1 +ξ ) = ξ (ξ ξ 1 ) Suppose this line intersects R. The angle ρ that it forms with the long side length of R satisfies cosρ ν/(µλ) due to the modulation constraint (note that at the scale of these boxes, the effects of curvature can be neglected). Since the long side of R has length µ and the short side length ν/λ, it follows that the total intersection length is O(ν/λ) Extensions. Now we extend the bilinear estimates to U and U D 1 e functions; since these extensions are valid for both spaces, we simplify the notation to U. Our first application of this proposition is in observing that (6.6) of Corollary 6.4 extends to U functions. This follows easily from the atomic decomposition. Corollary 6.8. Let φ µ,ψ λ U be respectively localized in frequency to I µ and I λ, µ λ. Then We may similarly conclude the following. φ µ ψ λ L µ1/ λ 1/ φ µ U ψ λ U (6.1) Corollary 6.9. Let φ 1,φ U be respectively localized in frequency to Ω 1 and Ω, where Ω 1,Ω I λ. Assume that for all ξ 1 Ω 1 and ξ 1 Ω we have Then ξ 1 ξ β. φ 1 φ L s,t,x β 1/ φ 1 U φ U (6.13) Lemma Let Q 1,Q {Q ν1,q ν,q ν3,1 : ν 1,ν,ν 3 dyadic}. Let φ µ,φ λ U have respective frequency supports contained in α boxes lying in I µ and I λ, where µ λ. Then Q 1 φ µ Q φ λ L α1/ λ 1/ φ µ U φ λ U (6.14) 5