Low regularity a priori estimate for KDNLS via the short-time Fourier restriction method

In this article, we consider the kinetic derivative nonlinear Schrödinger equation (KDNLS), which is a one-dimensional nonlinear Schrödinger equation with a cubic derivative nonlinear term containing the Hilbert transformation. For the Cauchy problem, both on the real line and on the circle, we apply the short-time Fourier restriction method to establish a priori estimate for small and smooth solutions in Sobolev spaces Hs\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$H^{s}$\end{document} with s>1/4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$s>1/4$\end{document}.


Introduction
In the present article, we continue our study in [11,12] and consider the kinetic derivative nonlinear Schrödinger equation (KDNLS) on R and on T := R/2πZ: where α, β are real constants and H is the Hilbert transformation.We assume β < 0 throughout this article.
In the periodic case, we proved in [12] that the Cauchy problem has a (forwardin-time) global solution for any initial data in H s (T) if s > 1/4, with the solution map u(0) = u 0 → u(•) being (locally-in-time) continuous in the H s topology away from the origin u 0 = 0.More precisely, we proved the following claims: (i) For any s > 1/4 and any R ≥ r > 0, there exist T > 0 and a solution map u 0 → u on the set {u 0 ∈ H s (T) : u 0 H s ≤ R, u 0 L 2 ≥ r} which gives a solution u ∈ C([0, T ]; H s (T)) to (1.1) on [0, T ] with u(0) = u 0 and is continuous in the H s topology.
(ii) The above (non-trivial) solution u(t) is smooth (especially in H 1 (T)) for t > 0, and then it extends to a global solution by means of the H 1 -upper and L 2 -lower a priori bounds which are obtained for H 1 solutions of arbitrary size.
Note that the trivial solution u ≡ 0 is a global solution for u 0 = 0.The continuity of the solution map at the origin can be verified if s > 1/2 ( [11]), but it is open for 1/2 ≥ s > 1/4.This is because a priori estimates and the local existence time given by the contraction argument depend on the reciprocal of the L 2 norm of solution for 1/2 ≥ s > 1/4.In the non-periodic case, local well-posedness of the Cauchy problem in H s (R) can be proved for s > 3/2 by the energy method, but no result seems to be currently available below H 3/2 .To summarize, on T we have a global solution for s > 1/4, while on R we only have a local solution for s > 3/2.We also note that these solutions to the Cauchy problem are unique in C t H s x if s > 3/2.The goal of this article is to prove an a priori H s estimate for small and smooth solutions to (1.1) in the regularity range 1/2 ≥ s > 1/4.In the periodic case, this and an approximation argument would imply the same estimate for the small (rough) H s solutions constructed in [12], thus verifying the continuity of the solution map at the origin.Although our argument in the present paper is applicable to both periodic and non-periodic problems, we will mainly consider the periodic case, which seems technically more complicated.(See Remark 2.11 below for a comment on the non-periodic case.)Theorem 1.1.Let M = R or T and s > 1/4.Then, there exist δ > 0 and C > 0 such that if 0 < T ≤ 1 and u ∈ C([0, T ]; H ∞ (M)) is a smooth solution to (1.1) on M satisfying u(0) H s ≤ δ, then it holds that To establish the H s a priori bound (1.2), we shall employ the short-time Fourier restriction method.The short-time X s,b norms were introduced by Ionescu, Kenig, and Tataru [9]; the idea is to combine the X s,b analysis implemented in frequencydependent small time intervals with an energy-type argument recovering the estimate on the whole interval.The method has been applied to the modified Benjamin-Ono and the derivative NLS equations by Guo [5] in the non-periodic case, and in the periodic case by Schippa [19], who used the U p -V p type spaces instead of X s,b .The X s,b type spaces are suitable for detailed analysis on the resonance structure, while the U p -V p type spaces work well with sharp cut-off functions in time.For our purpose, the U p -V p type spaces seem to be more convenient.In our argument with the shorttime Fourier restriction method, the modified energy plays a crucial role.Our way of constructing the modified energy is slightly different from that in [14], [5] and [19] because of the presence of the Hilbert transformation in the cubic nonlinearity.To be specific, (1.1) has less symmetry than the DNLS, the cubic NLS and the modified Benjamin-Ono equations.Moreover, it is known that the kinetic term β∂ x H(|u| 2 )u in (1.1) exhibits a kind of dissipation when β < 0 (e.g., we have ∂ t u(t) 2  L 2 ≤ 0 for (smooth) solutions of (1.1), while the L 2 norm is conserved for the DNLS equation).This dissipative nature has to be taken into account in the construction and the estimate of the modified energy, since otherwise there would remain some uncanceled terms with higher order derivatives compared to the corresponding estimate for the nonlinearity α∂ x (|u| 2 u).Here, we do not have to estimate the difference of two solutions, since we only consider the continuity of the solution map at the origin.So, we do not have to consider the modified energy for the difference of two solutions, either.
Remark 1.2.(i) In the case of DNLS, a similar a priori H s estimate was obtained in [5,19] for solutions of arbitrary size by using a rescaling argument.Although the same idea may work for our problem (1.1) to remove the smallness condition in Theorem 1.1, we will focus on small solutions in order to keep the argument not too complicated, and also because of our particular interest in the continuity of the solution map at the origin.
(ii) An adaptation of the theory of low-regularity conservation laws for integrable PDEs by Killip, Vişan, and Zhang [10] might be another possible approach.For the derivative NLS on R and on T, the H s a priori estimate for s > 0 was established in [13] by this method.Of course, KDNLS is not known to be completely integrable, but the method seems also useful to some dissipative perturbations of the integrable dispersive equations (e.g., the KdV-Burgers equation).Unfortunately, this approach has not been successful for KDNLS up to now.
The plan of the present paper is as follows.In Section 2, we describe the definition of function spaces we work with, the short-time Strichartz estimates and the shorttime bilinear Strichartz estimates.Assuming the trilinear estimates and the modified energy estimate which are proved in later sections, we give the proof of our main Theorem 1.1.In Section 3, we give the trilinear estimate on the cubic nonlinearity in terms of short-time norms.In Section 4, we define the modified energy and prove its estimates which are helpful for the short-time argument.
We would like to conclude this section with a couple of comments on Jean Ginibre's work about nonlinear wave and dispersive equations.Ginibre started to study the scattering theory in the finite energy class for nonlinear Klein-Gordon and Schrödinger equations in late 1970's with Giorgio Velo.Since then, he has made the great contribution to nonlinear partial differential equations, specifically, nonlinear wave and dispersive equations.In early 1990's, Bourgain presented the so-called Fourier restriction method to study the well-posedness of the Cauchy problem for nonlinear dispersive equations such as nonlinear Schrödinger equations and the KdV equation.The Fourier restriction method is very powerful, but it is rather complicated.In fact, Bourgain's papers were not very easy to read.Many people hoped the readable exposition on Bourgain's work about the Fourier restriction method.In 1996, Ginibre wrote the nice exposition [4] on the Fourier restriction method, which contained several new and important observations, for example, the relation between the Fourier restriction norm and the interaction representation in quantum physics.This helped the Fourier restriction method to prevail among the community of nonlinear wave and dispersive equations.
• Let P(I) denote the set of all partitions of the interval I: where χ Ω denotes the characteristic function of a set Ω and we set t K+1 := b.The space U p (I) and its norm are defined as follows: • The normed space V p (I) is defined by Recall that U p (I), V p (I) are Banach spaces and their elements are bounded functions from I to L 2 x which have one-sided limits at every point in [a, b].Moreover, u ∈ U p (I) is right continuous and satisfies lim t→a u(t) = 0.As usual, we write V p rc (I) := {v ∈ V p (I) : v is right continuous} and V p −,rc (I) := {v ∈ V p rc (I) : lim t→a v(t) = 0}, which are closed subspaces of V p (I).We have U p (I) ⊂ U q (I) for p < q with continuous inclusion, and similarly for V p (I), V p rc (I), V p −,rc (I).Following [2] (see also [14]), we consider the space DU p (I) := {u : u ∈ U p (I)}, where the derivative is taken in the sense of L 2 x -valued distributions on I.For each f ∈ DU p (I) there is a unique u ∈ U p (I) satisfying f = u , and hence DU p (I) is a Banach space equipped with the norm f DU p (I) = u U p (I) .Finally, we write , and similarly for V p ∆ (I), V p rc,∆ (I), V p −,rc,∆ (I), and We collect some basic properties of these spaces.(i) [Continuous embeddings] For any 1 ≤ p < q < ∞, we have x ) n → L 1 loc,x be an operator such that it is either linear or conjugate linear in each variable.Let 1 ≤ p, q ≤ ∞, and assume that the map x for some A > 0.Then, T can be regarded as a map from (U p ∆ (R)) n to L p t (I; L q x ) by (u 1 , . . ., u n ) → [t → T (u 1 (t), . . ., u n (t))], and it is bounded: Here, (iv) [Interpolation] Let 1 ≤ p < q < ∞, E be a Banach space, and T : U q ∆ (I) → E be a bounded, linear or conjugate linear operator such that T U q ∆ (I)→E ≤ C q , T U p ∆ (I)→E ≤ C p for some 0 < C p ≤ C q < ∞.Then, we have Proof.(i) See, e.g., [ The following property of the U p space was stated in [2, in the paragraph preceding Lemma 2.1].We give a precise statement and a proof for the reader's convenience.Proof.Let ε > 0, and write u as u = j λ j u j with {u j } being U p (I)-atoms, {λ j } ⊂ C such that j |λ j | < u U p (I) + ε.Since u vanishes outside I , it holds that u = j λ j χ [a ,b ) u j .A simple argument shows that µ j χ [a ,b ) u j is a U p (I)-atom for some µ j ≥ 1 unless χ [a ,b ) u j ≡ 0. Hence, replacing λ j by λ j /µ j and excluding the terms with χ [a ,b ) u j ≡ 0, we may assume that each U p (I)-atom u j in the representation u = j λ j u j is zero outside [a , b ).
To show u| I ∈ U p (I ), we define a sequence {u [J] } J≥1 by where {t J } J≥1 ⊂ (a , (a + b )/2) is a decreasing sequence such that lim J→∞ t J = a and u j is constant on [a , a + 2(t J − a )) for all J ≥ j ≥ 1.We observe that This shows that {u [J] | I } ⊂ U p (I ) is a Cauchy sequence.Since U p (I ) is a Banach space, the limit u| I belongs to U p (I ).Furthermore, we have The reverse inequality is immediate, since every U p (I )-atom gives a U p (I)-atom by the zero extension.Now, we define the short-time norms.In this article, we use capital letters N, K, . . .for dyadic integers 1, 2, 4, 8, . . . .Definition 2.4.First of all, we fix a bump function so that 1 = N ≥1 ψ N (ξ) and supp (ψ N ) ⊂ I N , where We define the corresponding Littlewood-Paley projections P N := F −1 ξ ψ N F x .Next, we define frequency-localized short-time norms F N (T ), G N (T ) for functions u : [0, T ] → L 2 with supp (û(t, ξ)) ⊂ [0, T ] × I N by In the definition of the F N (T ) norm, we regard χ I u as a function on R by extending it by zero outside I. Here, we consider half-open intervals I = [a, b) so that χ I u can be right continuous, and we avoid writing the norm To measure the nonlinearity, the following short-time norm is used: We also need the following energy norm:

Proof of the main theorem
It is known (e.g., [2,Lemma 3.1], where the fact shown in Lemma 2.3 is needed) that the norms defined above satisfy the basic linear estimate for any s ∈ R.Then, what we need to show are the trilinear estimate and the energy estimate We will prove the trilinear estimate for general functions u ∈ F s (T ) in Section 3, and the energy estimate for smooth solutions of (1.1) with small initial data in Section 4.
Both of these estimates require s > 1/4, and also have the constants uniform for T ∈ (0, 1] but growing for T > 1.
Let us admit these estimates and prove Theorem 1.1.For 0 < T ≤ T and a smooth solution u with initial data small in H s , define The above three estimates show that On the other hand, it is easy to show (e.g., for Hence, by a bootstrap argument, we have , this concludes the proof of Theorem 1.1.

Short-time L 6 and bilinear Strichartz estimates
Most of Strichartz-type estimates for the non-periodic Schrödinger equation are known to hold for the periodic problem in the short-time setting, and these estimates will be used as basic tools to prove the trilinear and energy estimates.We begin with the following L 6 Strichartz estimate.
Lemma 2.5.For N ≥ 1 and 0 < δ N −1 , we have As a consequence, we have for any interval Proof.The first estimate was shown in [1, Proposition 2.9].To obtain the second claim, we use Lemma 2.2 (iii) with the operator T : φ → P ≤N φ and apply the resulting estimate to Actually, the estimate of this type holds on L p ([0, δ]; L q (T)) for any admissible pair (p, q); 2/p + 1/q = 1/2, 2 ≤ q ≤ ∞.This was proved in [1] on more general compact manifolds, by applying Keel-Tao's abstract theory to the short-time dispersive estimate: The proof of (2.1) for general compact manifolds in [1] was based on semiclassical calculus.For the case of T, there is a more direct proof, which we would like to present below.(We learned this proof from a recent paper [16] studying the Zakharov-Kuznetsov equation on T 2 , but the argument would have been well known since a long time ago.) Proof of (2.1).The claim is reduced to the following pointwise bound on the kernel of P ≤N e it∂ 2 x : where We set Φ(ξ) := yξ − tξ 2 .On one hand, |Φ | ≡ 2|t| and the van der Corput lemma show that |F N,t (y)| |t| −1/2 uniformly in y ∈ R. On the other hand, for |y| ≥ 5, the restrictions |ξ| ≤ 2N and |t| ≤ N −1 imply that Φ(ξ) has no stationary point and Hence, after doing integration by parts twice: which verifies (2.2) and thus (2.1).
As a counterpart of the bilinear Strichartz estimate of Ozawa and Tsutsumi [18, Theorem 2 (1)], we have the following short-time bilinear Strichartz estimate on T. A Fourier analytic proof was given in [17], which we will recall below for completeness.Lemma 2.6.For K ≥ 1 and δ > 0, we have In particular, if N 1 N 2 ≥ 1 and φ 1 , φ 2 satisfy supp ( φj ) ⊂ I N j , then for 0 < δ N −1 we have (2.4) Remark 2.7.The latter estimate (2.4) clearly holds regardless of the complex conjugation, while this is not the case for the former estimate (2.3).For the product without conjugation of two functions of comparable frequencies, we can deduce, for instance, the following result from (2.3): if In particular, if we have Remark 2.8.As for the L 6 Strichartz estimate, from (2.3) and Lemma 2.2 (iii) we immediately obtain the corresponding bilinear estimates in U 2 ∆ : for I = [a, b) with |I| K −1 we have A similar extension is valid also for Lemmas 2.9 and 2.10 below.On the other hand, by the Bernstein and Hölder inequalities and the assumption |I| K −1 , together with the embedding U p ∆ → L ∞ L 2 , we have By applying Lemma 2.2 (iv) to the operator v → P K (uv) with these estimates, we have Proof of Lemma 2.6.If K = 1, the claim follows from the Hölder inequality in t and the Bernstein inequality in x.
Assume K > 1.We observe that By the Plancherel theorem and the change of variable t = −2nt, we have Since the last term is equal to L 2 by the Plancherel theorem, the claimed estimate follows.
In order to deal with the nonlinearity of (1.1) including the Hilbert transformation, we prepare the next two lemmas.These estimates can be shown in the same manner as Lemma 2.6.Lemma 2.9.Let φ 1 , φ 2 , φ 3 ∈ L 2 (T) satisfy supp ( φj ) ⊂ I N j , and assume that N 1 N 2 , N 3 .Then, for 0 < δ N −1 we have The same estimate holds if e it∂ 2 x φ 1 is replaced by e it∂ 2 x φ 1 , and also if H is replaced by any Fourier multiplier with bounded symbol (such as P ≤N ).
Proof.Since H(uv) = H(ūv), we may assume N 2 ≤ N 3 .We observe that and hence , where at the last step we have used the Minkowski inequality to replace the L Hence, we have as desired.
Lemma 2.10.Let φ 1 , φ 2 , φ 3 ∈ L 2 (T) satisfy supp ( φj ) ⊂ I N j , and assume that we have The same estimate holds if e it∂ 2 x φ 3 is replaced by e it∂ 2 x φ 3 , and also if H is replaced by any Fourier multiplier with bounded symbol.
Proof.By an almost orthogonality argument, we can restrict the frequencies of φ 1 and φ 2 onto intervals of length K.Then, the same argument as for the preceding lemma can be used.
Remark 2.11.We note that all the above short-time L 6 and bilinear Strichartz estimates (Lemmas 2.5, 2.6, 2.9, and 2.10) are true in the non-periodic case as well.In fact, these estimates hold on R without restricting to a frequency-dependent short time interval (i.e., with the L 6 t,x or L 2 t,x norm over R × R on the left-hand side).Concerning Lemmas 2.9 and 2.10, this can be shown by a slight modification of the proofs for the periodic estimates given above.

Trilinear estimate in the short-time norms
In this section, we shall prove the following trilinear estimate in the G s (T ) norm.Proposition 3.1.For s > 1/4 and 0 < T ≤ 1, we have Proof.We only consider the second term on the left-hand side with the Hilbert transformation.The first term (for DNLS) was treated in [5,19]; in fact, it can be dealt with in a similar manner but more easily.
We apply dyadic decompositions as where we write N * 1 , . . ., N * 4 to denote the numbers N 1 , . . ., N 4 rearranged in decreasing order.It then suffices to show for each N = (N 1 , . . ., N 4 ) the localized estimate with some C(N ) satisfying , the factor (N * 3 ) 0− allows us to restore the claimed estimate by summing up (3.1) in N .)We will actually obtain (3.1) with smaller C(N ) which satisfies From the definition of the F N (T ), G N (T ) norms, we need to prove sup by Lemma 2.2 (ii) it suffices to prove either or sup for any and any u 4 ∈ V 2 rc,∆ (I 4 ).
When N 4 N * 1 , the time scale on the right-hand side is finer than that on the left-hand side, and therefore we need to first divide I 4 into sub-intervals of size ≤ (N * 1 ) −1 , the number of which is O(N * 1 /N 4 ).Then, to verify (3.3) we need to show for any interval I with In fact, (3.5) implies (3.3) by the Schwarz inequality in t and the Bernstein inequality in x.From now on, we write simply u j for P N j u j .
We simply use the L 6 Strichartz estimate (Lemma 2.5) for each function: This shows (3.5) with In this case, we apply the standard bilinear Strichartz estimate (Lemma 2.6) to the product u 1 ū2 , on which we may put , we apply Lemma 2.6 to the product H(ū 2 )u 3 and follow the argument in the preceding case to obtain (3.5) with In this case, we need to consider the dual estimate (3.4), because we cannot use Lemma 2.6 to the product H(u 1 )u 3 (in the form of Remark 2.7) when the Fourier supports of u 1 and u 3 are overlapping.We first replace u 4 ∈ V 2 rc,∆ (I 4 ) with its extension ũ4 ∈ V 2 −,rc,∆ (R) defined by ũ4 (a) := lim t→a+0 u 4 (t) and ũ4 (t) := 0 for t ∈ [a, b) (recall that I 4 = (a, b)).Next, we decompose I 4 into sub-intervals of length ≤ (N * 1 ) −1 , the number of which is O(1).Then, for each integral on a subinterval I = [a , b ) we apply Lemma 2.6 (in the form obtained in Remark 2.8) to the product u 3 ∂ x ũ4 (on which we may put P ∼N * 1 ), bound the remaining functions H(u 1 ), ū2 in the L ∞ (I; L 2 ) and the L 2 (I; L ∞ ) norms respectively, and finally derive the from the last one by the Hölder inequality in t and the Bernstein inequality in x.The resulting bound is .
(Since we have to bound ũ4 in V 2 ∆ , the bilinear Strichartz estimate is accompanied by a factor (N * 1 ) 0+ .)Now, it is verified directly from the definition of the V 2 ∆ norm that , we can put H on a single function.Then, similarly to the case (Ib-i), we apply Lemma 2.6 to the product of functions corresponding to N * 1 and N * 3 and use the L ∞ embedding for the other one corresponding to N * 4 , to obtain the desired bound.In the remaining case, i.e., if N 3 ∼ N 4 N 1 , N 2 , we apply the first modified bilinear Strichartz estimate (Lemma 2.9) to the left-hand side of (3.5), which gives the same bound.

Case (II) N
We follow the argument in the case (Ia) to obtain (3.5) with C(N ) = (N * 1 ) 1/2 , which satisfies (3.2). ( 2 by considering the following two cases separately. , we first bound the left-hand side of (3.5) by (N * 1 ) and then apply Lemma 2.6 to u 1 ū2 (on which we may put P ∼N * 1 ).This implies , we may put P N 3 on u 1 ū2 .Using the second modified bilinear Strichartz estimate (Lemma 2.10), we obtain (3.5) with min{N 1 , N 2 , N 3 }, we consider the dual estimate (3.4).Note that we can always put H on a single function, since N 3 ∼ N 4 and Then, the argument is parallel to case (Ib-iii).This time we decompose I 4 into sub-intervals of length ≤ (N * 1 ) −1 , the number of which is O(N * 1 /N 4 ), and apply Lemma 2.6 to the product of functions corresponding to N * 1 and N * 3 (= N 4 ).Further, we bound the remaining functions corresponding to N * 2 and N * 4 in the L ∞ (I; L 2 ) and the L 2 (I; L ∞ ) norms, respectively.We then obtain (3.4) with which satisfies (3.2).
We have seen all the possible cases, and this completes the proof of the localized estimate (3.1) with (3.2).

Energy estimate
In this section, we shall prove the following a priori estimate.
In fact, this is the main part of the proof of Theorem 1.1.Recall that the E s (T ) norm takes L ∞ t before the 2 summation over dyadic frequency blocks, and so it is fairly stronger than the L ∞ t H s x norm.

A reduction
First of all, we reduce Proposition 4.1 to the following estimate on a "modified energy".
a(2ξ) a(ξ) for any ξ > 0, a(N 1 ) a(N 2 ) |∂ j ξ a(ξ)| ξ −j a(ξ) for any ξ ∈ R and 1 ≤ j ≤ 5. there exist δ > 0 and C > 0 depending on s and the implicit constants in (4.1) (but not on u) such that if u(0) L 2 ≤ δ then we have (ii) To obtain an E s (T ) bound, one may consider estimating localized H s norms N 2s ψ N (D)u(t) 2 L 2 for dyadic numbers N ≥ 1 and summing them up.This is indeed the approach taken in [9].On the other hand, we will improve the bound by adding a correction term to the energy functional.For this purpose, it will be convenient to introduce a modified energy a(D)u(t) 2 L 2 and estimate it instead of the localized H s norms, where a symbol a(ξ) is chosen so that it is positive everywhere and its derivatives are controlled by itself as |∂ j ξ a(ξ)| ξ −j a(ξ).Such a modified energy has been used for the cubic NLS in [14] and for the modified Benjamin-Ono (and also the DNLS) equation in [5,19].
(iii) Our choice of a(ξ) is slightly simpler than that in [14,5,19] (see the proof of Proposition 4.1 below).Indeed, the modified energies in these papers are defined from a sequence of positive numbers {β N } depending on the initial data, but we do not use such a sequence.
Proof of Proposition 4.1 from Proposition 4.2.Let s > 1/4, ε := s − 1/4 > 0. For each dyadic integer N , we define the positive sequence {a N N } N ≥1 by It is clear that {a N N } is increasing in N .In fact, the growth of {a N N } is controlled as It is easy to see that a N satisfies all the properties in (4.1) with implicit constants independent of N .Applying Proposition 4.2 and restricting the left-hand side of the resulting estimate to the target frequency ξ ∼ N }, we have sup for any smooth solution u ∈ C([0, T ]; H ∞ ) to (1.1) with u(0) L 2 sufficiently small.Summing up in N , we obtain the claimed estimate.

Construction of the modified energies
Now, we start proving Proposition 4.2.The argument is very similar to the estimate of the modified energy with correction terms in the I-method, where an important role is played by various cancellations after symmetrization of the energy functionals (see, e.g., Colliander, Keel, Staffilani, Takaoka and Tao [3]).However, there are less symmetries compared to the DNLS case, and more delicate analysis is required.In particular, some of the highest order terms cannot be canceled out, and we need to recognize these terms to be non-positive by making use of the dissipative nature of the equation.Let a ∈ C ∞ (R) be a symbol satisfying (4.1).Hereafter, the notation ξ ij... = ξ i + ξ j + • • • will be frequently used.Our proof will be designed for the periodic problem; however, in view of Remark 2.11, the same argument can be applied in the non-periodic setting.For a smooth solution u of (1.1), we have The first term is the same as that appears in the DNLS case, and it is symmetrized as follows: We observe that the multiplier part ξ 1 a(ξ 1 ) + • • • + ξ 4 a(ξ 4 ) vanishes for the resonant frequencies: this quadrilinear term can be canceled with the leading term of the derivative of the quadrilinear functional which can be used as an appropriate correction term to E a 0 (u).In the following, we assume α = 0 for simplicity and consider the term This term has less symmetry due to the sign functions.In fact, this is symmetrized as and the multiplier part does not vanish when ξ 23 = 0 = ξ 12 (in this case sgn(ξ 34 ) = −sgn(ξ 12 ) = 0).Now, we observe that the function ξ → ξa(ξ) is odd and strictly increasing on R, and hence for any ξ 1 , ξ 2 ∈ R with ξ 12 = 0.Then, we decompose this term as On one hand, for β < 0 we have On the other hand, it will turn out that the multiplier part of Q 2 vanishes when ξ 12 = 0 and also when ξ 23 = 0. Q 2 is then canceled out by adding the correction term to the modified energy E a 0 (u), where We can show that b a is extended to a smooth function on Γ 4 , where (we put sgn(ξ 12 ) outside in order to make b a 4 smooth).Moreover, ξ 12 ξ 23 = 0 implies b a 4 (ξ 1 , ξ 2 , ξ 3 , ξ 4 ) = 0 (and hence the restriction ξ 12 ξ 23 = 0 for the sum in E a 1 (u) can be disregarded).In fact, noticing that while q(ξ 1 , ξ 2 ) is actually positive and smooth on R 2 , as we see in the following lemma.
the derivatives of q, we may focus on the case ξ max > 1.In the case |ξ 12 | ∼ ξ max , we first observe that On the other hand, using the property |∂ j ξ a(ξ)| ξ −j a(ξ) for j ≤ 5, we have which implies that The claimed estimate follows from these estimates and the expression (4.2).When |ξ 12 | ξ max , we deduce from the expression (4.3) that Recalling that | − ξ 2 + ξ 12 t| ∼ ξ max for t ∈ [0, 1], we have This proves (i).For (ii), we compute Using these expressions instead of (4.2)-(4.3), the desired estimate is verified by a similar argument to (i).
We have the following estimate on E a 1 (u(t)) for each t.
Proof of Proposition 4.2.By Lemma 4.5, Proposition 4.6, and the fact that any smooth solution of (1.1) reduces its L 2 norm, we have sup Assuming u(0) L 2 1, this yields the claimed estimate.

Estimate on the remainder term
It remains to prove Proposition 4.6.A difficulty here is that we cannot directly apply pointwise bounds on the multipliers (as we did in the proof of Lemma 4.5 above), because u ∈ F s (T ) does not in general imply F −1 ξ [|û(t, ξ)|] ∈ F s (T ).Indeed, linear solutions u = e it∂ 2 x u 0 can be considered as counterexamples.We prepare the following lemma, which allows us to separate variables in the multiplier b a 4 .This idea has also been used in [14,8,19 Then, there is a sequence ĉ ∈ 1 (Z 4 ) such that Proof.Following the argument in [8], first construct a smooth function ba 4 (ξ 1 , . . ., ξ 4 ) on I N 1 × • • • × I N 4 which extends b a 4 (ξ 1 , . . ., ξ 4 ) (defined on Γ 4 ) and satisfies We use the following extensions of b a 4 : From Lemma 4.4, we can show that ; Using these bounds, we see that the desired estimates (4.4) hold we define which is a smooth function supported in [−2, 2] 4 and thus can be extended to a 2πperiodic smooth function on R 4 .Let ĉ(k 1 , k 2 , k 3 , k 4 ) be the Fourier coefficients of c, then the claimed identity follows from the Fourier series expansion and the restriction onto Γ 4 .Moreover, we deduce from (4.4) that This completes the proof.
We are now in a position to prove Proposition 4.6.
Proof of Proposition 4.6.As usual, we decompose the sum into dyadic pieces: × sgn(ξ 12 )sgn(ξ 56 )(ψ where for j = 4, 5, 6, ψN j (ξ j ) := ψj (ξ j /N j ) and ψj ∈ C ∞ 0 (R) is chosen so that ψj (•/N j ) ≡ 1 on supp (ψ N j ) and supp ( ψj (•/N j )) ⊂ I N j , with ψ N j , I N j defined as in Definition 2.4.In the following, we write N * 1 , . . ., N * 6 to denote the numbers N 1 , . . ., N 6 rearranged in decreasing order.Note that the range of N 1 , . . ., N 6 , N can be restricted to Applying Lemma 4.7 for each (N, N 4 , N 5 , N 6 ), we have where we write u j = N j u for brevity, j = 1, . . ., 6, and Since multiplication by e iθξ on the Fourier side does not change the F N j (T ) norm of u j , the proof is reduced to estimating In order to obtain a bound with the short-time norms, we have to divide the time interval into small sub-intervals of length ≤ (N * 1 ) −1 (denoted by I), which gives a factor of O(N * 1 ).The strategy in the previous results on DNLS [5,19] is to use two bilinear Strichartz and two For KDNLS, there are some cases where the same argument does not work due to the presence of the Hilbert transformations.For instance, we cannot use the standard biliear Strichartz estimate (Lemma 2.6) with only one of u 1 and u 2 involved.We can indeed use the modified bilinear Strichartz estimates (Lemmas 2.9, 2.10) instead, but the argument will be even more complicated.
For the first term, we can eliminate the Hilbert transformations by the fact that the frequency for H(u 5 ū6 ) must be much bigger than that of u 3 ū4 : There is no contribution from the third term, while the estimate for the fourth term is exactly the same as the first term in Case (IIIc-i), since in the integral we can replace P N 3 (u 5 ū6 ) by u 5 ū6 .For the first two terms, we can separate two functions of high frequency from the Hilbert transformation; for instance, Then, we can obtain by using Lemmas 2.6 and 2.9 when N 3 = N * 3 , or Lemma 2.6 twice when N 5 = N * 3 .Note that we do not actually need the improved bound (4.8) in this subcase.
(Vd-ii) N * 3 ∼ N * 6 .In this case, we use Lemma 2.10 for H(u 1 ū2 )u 3 (noticing that u 1 ū2 may be replaced by P N * 3 (u 1 ū2 )), Lemma 2.5 for the others, to obtain , by the assumption we have N 3 ∼ N 4 ∼ N 5 N 6 .In this case, we make the decomposition u 3 ū4 = P N 3 (u 3 ū4 ) + P ∼N 3 (u 3 ū4 ).put P K on H(u 1 ū2 ).Hence, by applying Lemma 2.10 to HP K (u 1 ū2 )u 3 and using Lemma 2.6 for HP K (u 5 ū6 ), L ∞ embedding for ū4 , we obtain We have thus completed the case-by-case analysis for the proof of (4.7).

Lemma 2 . 3 .
Let I = (a, b), I = (a , b ) with −∞ ≤ a < b ≤ ∞ and a < a < b ≤ b.If u ∈ U p (I) and u vanishes outside I , then u| I ∈ U p (I ) and u| I U p (I ) = u U p (I) .
instance, where η is defined as in Definition 2.4.It is clear that the above defined ba 4 coincides with b a 4 on Γ 4 and satisfies (4.4).