Optimal local well-posedness for the periodic derivative nonlinear Schrodinger equation
aa r X i v : . [ m a t h . A P ] M a y OPTIMAL LOCAL WELL-POSEDNESS FOR THE PERIODIC DERIVATIVENONLINEAR SCHR ¨ODINGER EQUATION
YU DENG, ANDREA R. NAHMOD, AND HAITIAN YUE
Abstract.
We prove local well-posedness for the periodic derivative nonlinear Schr¨odinger’s equa-tion, which is L critical, in Fourier-Lebesgue spaces which scale like H s ( T ) for s >
0. In particularwe close the existing gap in the subcritical theory by improving the result of Gr¨unrock and Herr[25], which established local well-posedness in Fourier-Lebesgue spaces which scale like H s ( T ) for s > . We achieve this result by a delicate analysis of the structure of the solution and the con-struction of an adapted nonlinear submanifold of a suitable function space. Together these allowus to construct the unique solution to the given subcritical data. This constructive procedure isinspired by the theory of para-controlled distributions developed by Gubinelli-Imkeller-Perkowski[26] and Cantellier-Chouk [10] in the context of stochastic PDE. Our proof and results however, arepurely deterministic. Introduction
The derivative nonlinear Schr¨odinger’s equation iu t + ∂ x u = i∂ x ( | u | u ) , (1.1)where ( t, x ) ∈ ( − T, T ) × T (periodic) or ( − T, T ) × R (non-periodic), is a Hamiltonian PDE intro-duced as a model for the propagation of nonlinear waves in plasma physics and nonlinear optics [55].It is well-known as a completely integrable system [40, 37, 56, 38], and in particular conserves massand energy. The Cauchy problem for (1.1) is scale invariant for data in L , that is, if u ( t, x ) is asolution then so is u λ ( t, x ) = λ u ( λ t, λx ) with the same L norm. Thus a priori one expects localwell-posedness for (1.1) with initial data data in H s for s ≥
0. However, while local well-posednessin H s for (1.1) is known for s ≥ [56, 38], one has ill-posedness in H s for s < [2, 56, 38].One way to close the gap between the scaling heuristics and actual local well-posedness resultsis by considering data in the Fourier-Lebesgue spaces H σp , where p ≥
2. These spaces are definedas k u k H σp := kh k i σ b u ( k ) k L pk (1.2)(with L pk replaced by ℓ pk in the periodic case). These spaces have naturally arisen in the literatureand we refer the reader to e.g. [41, 60, 23, 13, 24, 22] for some instances. Note in particular, thatin one dimension the Fourier-Lebesgue space H σp has the same scaling as the Sobolev space H s for s = σ + 1 p −
12 ; (1.3)in particular, H ∞ has the scaling of L , and H = H .In the non-periodic case, Gr¨unrock [23] proved optimal local well-posedness for (1.1) in H σp ( R )for σ ≥ and p < ∞ , which allows the corresponding Sobolev regularity s to be arbitrarilyclose to 0, thus covering the full subcritical range. The proof combines the gauge transformation Mathematics Subject Classification.
35, 42.Andrea R. Nahmod is partially supported by NSF-DMS-1463714 and NSF-DMS-1800852. Here and henceforth we mean that the homogenous part of the Fourier-Lebesgue norm scales like the correspondinghomogeneous Sobolev norm introduced in [35] (used also in [36, 37, 56]) and new bilinear and trilinear estimates for the gaugedequation in an appropriate variant of Bourgain’s Fourier restriction norm spaces [7] (see Section 3below for details) which follow from the dispersion and the smoothing properties of the Schr¨odingerpropagator on R .In the periodic case, however, local well-posedness for (1.1) in H σp ( T ) is only known for σ ≥ and 2 ≤ p <
4, which scaling-wise correspond to Sobolev regularity s > . This is the work ofGr¨unrock and Herr [25]. Their proof is based on the adapted periodic gauge transformation in [38]and new multilinear estimates for the gauged equation in adapted variants of the Fourier restrictionnorm spaces. Moreover, it is proved in [25] that the crucial multilinear estimates become falsewhen p ≥
4, so this result as well as the existing gap in the local well-posedness theory between s > / s >
0, cannot be improved within the framework of [25].In this paper we close this existing gap in the periodic case. More precisely, we prove optimallocal well-posedness for (1.1) in H σp ( T ) for σ ≥ and p < ∞ which covers the entire subcriticalregime, hence yielding optimal local well-posedness. Our main theorem is stated as follows: Theorem 1.1.
Fix σ ≥ and p < ∞ . For any A > , there exists T = T ( p , A ) > , suchthat if k u k H σp ≤ A , then there exists a unique solution u ∈ Z ⊂ C t H σp ( J ) to (1.1) with initialdata u (0) = u , where J = [ − T, T ] . Here Z is an explicitly defined sub-manifold of C t H σp ( J ) , seeDefinition 4.3 below. The map u u is continuous with respect to the C t H σp ( J ) metric.Remark . The solution we construct solves (1.1) in the sense that it solves the integral equation u ( t ) = e it∂ x u + Z t e i ( t − s )∆ ∂ x ( | u ( s ) | u ( s )) d s. (1.4)It is also the unique limit of smooth solutions: given A >
0, and any smooth initial data u inthe A -ball of H σp , the classical solution exists for time T = T ( p , A ) >
0, and the data-to-solutionmap extends continuously to all of this ball. Moreover, if p <
4, our solution coincides with thesolution constructed in [25], for as long as the latter exists.
Remark . We will only prove Theorem 1.1 with σ = and p ≥
4. The extension to σ > isstandard (see Proposition 7.1 for a sketch), and when 2 ≤ p < Remark . Global-posedness for the Cauchy problem for (1.1) is known to hold for data in H s , s ≥ both on R [14, 15, 44] and on T [38, 61, 45]. Furthermore, one has almost sure globalwell posedness for data in Fourier Lebesgue spaces H σp ( T ) that have the scaling of H − ε ( T ) , ε > H ( T )which requires quite different techniques such as for example exploiting the integrability of theequation and seeking suitable new conservation laws below H .1.1. The standard approach, and difficulties.
Generally speaking the difficulty one faces insolving (1.1) is a derivative loss arising from the term i | u | u x in the nonlinearity of (1.1), andhence for low regularity data the key is to somehow make up for this loss. The first step towardsthis goal is a gauge transformation [35, 36, 37, 56, 38] which removes this bad resonant term inthe nonlinearity that loses derivatives and makes the estimates uncontrollable. Matters are thenreduce to studying the gauged derivative nonlinear Schr¨odinger equation which we schematicallywrite as ( ∂ t − i∂ x ) v = C ( v, v, v ) + Q ( v, · · · , v ) , (1.5) More precisely the trilinear estimates containing as one of their inputs the derivative term.
PTIMAL LOCAL WELL-POSEDNESS FOR THE DNLS EQUATION 3 where the nonlinearity ∂ x ( | u | u ) has been transformed into the sum of the ‘better’ cubic term C ( v, v, v ) ∼ iv x · v , plus a quintic term which contains no derivatives terms in it and which we momentarily neglect inthis discussion as being ‘lower order’. Once the bad nonlinear term is gauged away from (1.1), thesolution v to (1.5) is constructed by a fixed point argument, which follows from proving multilinearestimates in suitably Fourier restriction norm function spaces adapted to the data space. In the non-periodic case [23] these spaces in conjunction with the dispersion and smoothing effects availableon R suffice, as we mentioned above, to prove optimal local well-posedness for (1.5) and hencefor (1.1) in H σp , where σ ≥ and p < ∞ . In the periodic case [25], however, the authors need tointroduce a fourth parameter q in the Fourier restriction norm function spaces, namely they define k u k X σ,bp,q = kh k i σ h ξ + k i b b u ( k, ξ ) k ℓ pk L qξ and prove that, if σ = and p < (cid:13)(cid:13)(cid:13)(cid:13) Z t e i ( t − s ) ∂ x ( ∂ x v · v v ) d s (cid:13)(cid:13)(cid:13)(cid:13) X ,bp,q . Y j =1 k v j k X ,bp,q (1.6)holds true for ( b, q ) = ( + , σ = and p ≥
4, the trilinear estimate (1.6) fails for any choice of ( b, q ) [25]. In other words,when σ = and p ≥
4, which in the Sobolev scale corresponds to regularity 0 < s ≤ , the localsolution to (1.5) cannot be constructed directly by contraction mapping .To prove our Theorem 1.1 we must and will take a different approach. After performing thegauge transformation, our point of departure is the following observation: let σ = and p ≥ X σ,bp,q spaces. Therefore it isreasonable to imagine that, the solution v to (1.5) still exists in one of these spaces –say b = +and q = 2 for definiteness– but will have some specific structure such that it precisely avoids thecounterexamples constructed in [25]. To that effect we will construct v in a nonlinear submanifold W of the Banach space X σ, + p, containing functions of a specific structure whence the trilinearestimate (1.6) will actually hold true σ = and p ≥
4. The heart of this paper will be to identifythis precise structure.In order to motivate our approach we take a step back and review some of the methods developedin the probabilistic (random data, or stochastically forced) context. We note in passing that animmediate corollary of our main Theorem 1.1 is that for random initial data of form u ω (0) := X k ∈ Z g k ( ω ) h k i + θ e ikx (1.7)where g k are i.i.d Gaussian random variables, h k i := p | k | , and θ > Local well-posedness for the gauged equation (1.5) implies local existence, uniqueness and continuity of the flowmap for (1.1) [38, 25]. When p = q = 2 these spaces coincide with Bourgain’s Fourier restriction norm spaces associated to theSchr¨odinger equation, and are simply denoted by X s,b . In principle, it might be possible that a trilinear estimate holds in some exotic Banach space not of form X s,bp,q but, if not unlikely, this would at least require a rather sophisticated construction. YU DENG, ANDREA R. NAHMOD, AND HAITIAN YUE
Ideas from probabilistic setting.
In the probabilistic PDE context (i.e. random datatheory for dispersive and wave equations or parabolic stochastic PDEs) where one deals withrandomized initial data or a random forcing term, the idea of exploiting the structure of thesolution has been used for a long time, see for example Bourgain [5, 6] in the context of thedefocusing (Wick ordered) cubic nonlinear Schr¨odinger equation , and Da Prato-Debussche [17, 18]in the context of the stochastic Navier-Stokes and the stochastic quantization equations. Morerecently this idea has been exploited in a large body of work by many authors. See for example[5, 6, 58, 16, 9, 20, 52, 39, 21, 3, 4, 62] and references therein for some works on the random datalocal Cauchy theory in the context of nonlinear Schr¨odinger equations. The key point is that, ifone considers the linear evolution of random data (or random forcing), then almost surely, it enjoysmuch better estimates than arbitrary functions of the same regularity. In turn this allows one to re-center the solution around the linear evolution of random data (or around higher order iterates),and conclude that the difference between the two belongs to a Banach space of higher regularitythan the one dictated by the (weaker) regularity of the random initial data.For example, in Bourgain [5], which deals with the cubic nonlinear Schr¨odinger equation on T ,the initial data φ ω belongs to Sobolev H − ε almost surely, for any ε >
0, whence its linear evolutiononly belongs to the Fourier restriction norm spaces X − ε, + almost surely. On the other hand, theequation is L critical, so if one were to try to prove local well posedness via a fixed point argument,the needed trilinear estimates would fail for arbitrary functions in X − ε, . Instead, Bourgain [5]constructed solutions u centered around the random linear evolution Ψ ω := e it ∆ φ ω . That is of theform: u = Ψ ω + R, where ( i∂ t + ∆) R = N (Ψ ω + R ) , (1.8)and where we have denoted by N the Wick ordered cubic nonlinearity. Then, almost surely, theneeded trilinear estimates for N (Ψ ω + R ) hold true and the solution R to the difference equation in(1.8) can be constructed in a smoother space X ε, + by a contraction mapping argument. Heuris-tically, one should view (1.8) as a ‘hybrid equation’ which on the one hand behaves subcriticallyin R , thus locally well-posed in H ε ; while on the other hand the random linear evolutions Ψ ω behave better than an arbitrary function in X − ε, + when they are entries in N (Ψ ω + R ) thanksto large deviation estimates. A similar phenomenon happens in Da Prato-Debussche’s argumentfor-for example-the stochastic Navier-Stokes equation on T with spacetime white noise forcing ζ [17] where the role of Ψ ω is replaced by Z , the linear evolution of white noise, Z t + ∆ Z = ζ. In both cases, the method can be understood as constructing solutions in a (random affine)submanifold W consisting of functions belonging to a ball in a smoother space, centered at therandom linear evolution.In the past few years, Gubinelli, Imkeller and Perkowski [26, 27] (see also [10] and [28]) devel-oped a far-reaching generalization of this re-centering method based on the idea of para-controlleddistributions . This is an analytic counterpart to the theory of regularity structures developed byHairer [31, 32, 33, 34]. Roughly speaking, in addition to the linear evolution and possibly (suitablyrenormalized) higher order expressions of the linear evolution, one moves to the new ‘center’ termsthat are ‘para-controlled’ by such expressions. Here a function f is said to be para-controlled by afunction g if, up to some smoother ‘remainder’ terms, f can be written as the Bony para-productbetween high frequencies of g and low frequencies of some auxiliary function h , namely that f = Π > ( g, h ) + R := X N P N g · P ≪ N h + R, (1.9)where for dyadic frequencies N , P N and P ≪ N are the standard Littlewood-Paley operators pro-jecting onto frequencies ∼ N and ≪ N respectively, and R is smoother than f . An example is See also, more recent work by Burq and Tzvetkov in the context of nonlinear wave equations [8].
PTIMAL LOCAL WELL-POSEDNESS FOR THE DNLS EQUATION 5 the (parabolic) Φ model, which is the cubic heat equation on T with white noise forcing [26, 10],where one constructs solutions of the form u = Z + I ( P ( Z )) + I Π > ( P ( Z ) , u − Z ) + R, (1.10)where Z is the linear evolution of the white noise, P ( Z ) consists of other structured componentswhich themselves are given in term of (suitably renormalized cubic) powers of Z , I is the Duhameloperator ( ∂ t − ∆) − , and R is a remainder, that has higher regularity. Here then the solutionconsists of the linear evolution Z , a higher order expression I ( P ( Z )), a para-controlled part term I Π > ( Z , u − Z ) and a remainder and thus belongs to a random submanifold which is much morenonlinear.These ideas have been extensively used in various stochastic contexts in recent years by manyauthors. We refer the reader for example to work by Mourrat and Weber [48] and to Mourrat,Weber and Xu [49] and to references therein for further work in the context of the Φ model andto Chandra and Weber [11] and references therein for a nice survey of these ideas. See also [1]. Wealso refer to recent work by Gubinelli, Koch and Oh [29, 30] where these ideas were applied to thestochastic nonlinear wave equation with quadratic nonlinearity in T and in T respectively.1.3. The deterministic context of DNLS.
Inspired by the ideas in the probabilistic settingdescribed above, in this paper we develop a new deterministic method to describe the structure ofsolutions v to the Cauchy initial value problem for (1.1) with data at almost critical regularity. Areview of all of the above examples suggests that, if we were in the probabilistic setting (i.e. (1.7)),we should look for solutions essentially of form u = w + (terms para-controlled by w ) + (smooth remainders) , where w is the combination of the random linear evolution, and multilinear expressions dictated bythe random linear evolution. The choice of such w is forced upon us (one can at most choose theorder of expansion) by the fact that one needs to (and indeed can) gain from the exact Gaussianstructure.In the deterministic setting, there is no gain from randomness. One could try to mimic theprobabilistic construction of para-controlled terms in previous works, and arrive at the ansatz v = w + I Π (1) > ( ∂ x w, v, v ) + I Π (2) > ( ∂ x w, w, v ) + (smooth remainders) , (1.11)where I is the Duhamel operator IF ( t ) = Z t e i ( t − s ) ∂ x F ( s ) d s and the cubic para-products are defined byΠ (1) > ( ∂ x w, v, v ) = X N P N ∂ x w · ( P ≪ N v ) , Π (2) > ( ∂ x w, w, v ) = X N P N ∂ x w · ( P N wP ≪ N v ) . (1.12)However unlike the probabilistic setting, we are no longer guided by the Gaussians and need tofind the right w ourselves. The naive choice of linear evolution for w is doomed to fail, and even ifone includes multilinear expressions of the linear evolution, calculations show that in the absenceof randomness, one would need to expand to a very high (if not infinite) order before unearthing‘smooth remainders’ that have enough regularity (namely H − due to [25]) to close the estimates.With a high order of expansion, the terms involved then quickly become too complex to control inour setting. Work in this direction was considered by the second author together with Chanillo,Czubak, Mendelson and Staffilani in the context of the nonlinear wave equation with quadraticderivative nonlinearities, see [12] for details. YU DENG, ANDREA R. NAHMOD, AND HAITIAN YUE
To get out of this maze, in this paper we will give up the idea of fixing w to be some explicitmultilinear expression dictated by the linear evolution. Instead we will construct this w , whichpara-controls the solution v , dynamically . That is, we take all the linear and higher order terms inthe above-mentioned expansion, as well as the presumed smooth error terms, and put them into asingle ‘center’ w . This leads to the new ansatz v = w + I Π (1) > ( ∂ x w, v, v ) + I Π (2) > ( ∂ x w, w, v ) , (1.13)where w is the ‘center’ which itself moves together with v and belongs to some subspace of X σ, + p, , p ≥
4. As it turns out, uncovering the final structure of v is slightly more complex but(1.13) conveys the main philosophy (see Section 4 for details). Since w does not have a specificmultilinear structure, one difficulty is identifying the right space where w will lie. By carefullyanalyzing the terms that are expected to appear in w , we can specify this space to be X , − p, ∞− .A final complication comes from the fact that unlike the parabolic setting where the Duhameloperator I automatically gains two derivatives, such gain is not automatic for the Schr¨odingerequation. Rather, it has to be manually induced by performing a frequency cut-off also in theFourier variable of time, so as to restrict to the region where the parabolic weight in frequency(which is the one that appears in the X s,bp,q norms) is large. In principle this would require thatwe replace in our ansatz (1.13) the Duhamel operator I by a frequency cut-off version of it, whichwould introduce non-locality in time which could be incompatible with local in time solutions.Fortunately the frequency cut-off can be substituted by a suitable time convolution e IF ( t ) := Z t χ ( k ( t − s )) e i ( t − s ) ∂ x F ( s ) ds (1.14)which has the same effect for some carefully chosen χ . See Section 4.1 for details.With the above discussion, we can now fix the submanifold W , in which the solution v to (1.5)is uniquely constructed, to be W = (cid:8) v ∈ X σ, + p, : v = w + e I Π (1) > ( ∂ x w, v, v ) + e I Π (2) > ( ∂ x w, w, v ) , w ∈ X , − p, ∞− (cid:9) . (1.15)We will show that the submanifold W is well-defined, parametrized by w ∈ X , − p, ∞− , and that thetrilinear estimates (1.6), which fail for arbitrary input functions in X σ, + p, , p ≥ W . These together will allow one to construct the solution v ∈ W by acontraction mapping argument. Finally, by inverting the gauge transform, one can construct thesolution u to (1.1) in Z , which is the preimage of W under the gauge transform. See Section 4 fordetails. Remark . We conclude this introductory discussion by noting that there is a large body of workthat has contributed to our current understanding of the Cauchy problem for (1.1) for data in theSobolev spaces H s , s ≥ both in the periodic and non-periodic settings; we refer the reader to[40, 59, 35, 36, 37, 53, 56, 57, 2, 14, 15, 38, 61, 44, 47, 54, 42, 43, 46] and references therein for amore comprehensive treatment. Note that the decomposition of v is nonlinear both in w and in the para-controlled terms. More precisely for δ > q = δ , the right space is X , − δp,q . See Section 3 for precise definitions. PTIMAL LOCAL WELL-POSEDNESS FOR THE DNLS EQUATION 7
Plan of the paper.
The paper is organized as follows. In Section 2 we recall the periodicgauge transformation used in [25] and perform such transformation to (1.1). Then, we lay out theset up and frequency interactions splitting of the nonlinearities in the gauged derivative Schr¨odingerequation which will guide our analysis. In Section 3 we define and set up our function spaces, provethe main linear estimates and prove an improved divisor bound which is used in some our estimates.In Section 4 we discuss the structure of the solution, identifying the para-controlling terms and theprecise solution submanifold W where v will belong. In Section 5 we prove a prior bounds for thepara-controlling terms. Section 6 constitutes the heart of the paper. Here we find w and proveall the underlying multilinear estimates involved in its construction. In the course of the proofwe show in particular that all relevant nonlinearities are well defined as space-time distributionswhence the integral equation (1.4) for u will be equivalent to the integral equation formulation of(1.5) (see Section 2.4 for details). Finally in Section 7 we prove a preservation of regularity result.1.5. Notations and parameters.
We will use the notation P h = 12 π Z T h d x, P =0 h = h − P h. The space, time and spacetime Fourier transforms are respectively defined as b u ( k ) = F x u ( k ) = 12 π Z T e − ikx u ( x ) d x, b u ( ξ ) = F t u ( ξ ) = 12 π Z R e − iξt u ( t ) d t, b u ( k, ξ ) = F u ( k, ξ ) = 1(2 π ) Z R × T e − i ( kx + ξt ) u ( t, x ) d t d x. so F is reserved for the spacetime Fourier transform. As for b u , whether it means space, time orspacetime Fourier transform will be clear from the context. The integral over the set { ( λ , · · · , λ r ) : λ ± · · · ± λ r = µ } for fixed µ will be with respect to the Lebesgue measure d λ · · · d λ r − . We denote by P thecharacteristic function of a set or property P .Recall that p is fixed; we will fix a small parameter 0 < δ ≪ p , and define theother parameters ( b , b , q , q , r , r , r ) as follows: b = 1 − δ, b = 1 − δ, q = 14 δ , q = 1(4 . δ , r = 12 + δ, r = 12 + 2 δ, r = 12 + 3 δ. (1.16)We also use θ to denote a generic positive quantity that is sufficiently small depending on δ (so θ may have different values at different instances.)We will fix A as in the statement of Theorem 1.1, and let A be large depending on A , A belarge depending on A , etc. All implicit constants below will depend on these A j ’s and the aboveparameters. The time length T will also be fixed, and small enough depending on these implicitconstants. 2. The gauge transform and other reductions
The gauge transform.
Notice that P | u | is conserved under the flow of (1.1). Consider thegauge transform, see [25], v ( t, x ) = ( G u )( t, x ) := ( G u ) (cid:0) t, x − P | u | t (cid:1) , ( G u )( t, x ) := e − iG ( t,x ) · u ( t, x ) , (2.1)where G = ∂ − x P =0 ( | u | ) (2.2) YU DENG, ANDREA R. NAHMOD, AND HAITIAN YUE is the unique mean-zero antiderivative of P =0 | u | . This gauge transform is easily inverted, withinverse given by u ( t, x ) = ( G − v )( t, x ) = e iG ( t,x ) · v ( t, x ) , G = ∂ − x P =0 | v | , (2.3)where v ( t, x ) = v ( t, x + 2 P | v | t ) . (2.4) Proposition 2.1.
The maps G and G − are continuous from C t H p ( J ) to itself for any interval J , and map bounded sets to bounded sets.Proof. Notice that G = G G , where G u = exp( − i∂ − x P =0 | u | ) · u, G − u = exp( i∂ − x P =0 | u | ) · u, and G u ( t, x ) = u ( t, x − P | u | t ) , G − u ( t, x ) = u ( t, x + 2 P | u | t ) . In [25], Lemma 6.2 and Lemma 6.3, it is proved that G : H p → H p is locally bi-Lipschitz, andthat G : C t H p ( J ) → C t H p ( J ) is a homeomorphism. Moreover, it is easily checked that kG u k C t H p ( J ) = kG − u k C t H p ( J ) = k u k C t H p ( J ) , so G and G − map bounded sets to bounded sets. (cid:3) The transformed equation.
We calculate that v = G u satisfies the equation( ∂ t − i∂ x ) v = C ( v, v, v ) + Q ( v, · · · , v ) , (2.5)where the cubic and quintic nonlinearities are defined as F x C ( v , v , v )( k ) = X V k M ( k, k , k , k ) · b v ( k ) b v ( k ) b v ( k ) , (2.6)and F x Q ( v , · · · , v )( k ) = X V M ( k, k , · · · , k ) · b v ( k ) b v ( k ) b v ( k ) b v ( k ) b v ( k ) . (2.7)The sets V and V are defined by V = (cid:8) ( k , k , k ) ∈ Z : k + k − k = k, | k | ≥ | k | , k
6∈ { k , k } (cid:9) ∪ { ( k, k, k ) } , V = (cid:8) ( k , · · · , k ) ∈ Z : k − k + k − k + k = k (cid:9) , (2.8)and the coefficients M j are explicitly defined functions, with | M j | . j ∈ { , } . They alsohave the right symmetry so that (2.5) conserves P | v | . See [25] for the precise formulas. Remark . For integers k , k , k and k such that k + k − k = k , we will rely throughout theproofs on the quantity ∆ := k + k − k − k . PTIMAL LOCAL WELL-POSEDNESS FOR THE DNLS EQUATION 9
Splitting the cubic nonlinearity.
We will further split the cubic nonlinearity C into fourparts: a “high-high” part , a “low-low” part, a “semilinear” part and a “non-resonant” part.Decompose V into four subsets: X H = (cid:8) ( k , k , k ) ∈ V : | k | ≥ − | k | (cid:9) , X L = (cid:8) ( k , k , k ) ∈ V : | k | < − | k | (cid:9) , X S = (cid:8) ( k , k , k ) ∈ V : 2 − | k | ≤ | k | < − | k | (cid:9) , X N = V − ( X H ∪ X L ∪ X S ) . (2.9)The following properties of this splitting are elementary and so we omit the proof. Proposition 2.3.
We have the following properties for the sets X ∗ where ∗ ∈ { H, L, S, N } : (1) For ( k , k , k ) ∈ X H we have | k | ≥ | k | ≥ − | k | . (2) For ( k , k , k ) ∈ X L we have | k | / ≤ | k | ≤ | k | and min( | k | , | k | ) ≥ max( | k | , | k | ) . (3) For ( k , k , k ) ∈ X S we have | k | / ≤ | k | ≤ | k | , | k | ≥ | k | and | k | ≥ − | k | . (4) For ( k , k , k ) ∈ X N we have | k | ≥ − max( | k | , | k | ) and min( | k | , | k | ) ≥ | k | . (5) For ( k , k , k ) ∈ X H ∪ H S we have | k | · ( h k ih k ih k i ) − . h k i − . (2.10)(6) For ( k , k , k ) ∈ X L ∪ X N we have | ∆ | ∼ h k ih k i , where ∆ = k + k − k − k = 2( k − k )( k − k ) . (2.11)We also need the following result, which will be used in analyzing the quintic terms in Section6. Once again these properties are elementary. We omit the proof. Proposition 2.4.
Suppose k + k ′ − k = k, ( k , k , k ′ ) ∈ X ∗ ; k + k − k = k ′ , ( k , k , k ) ∈ X , where ∗ , ∈ { H, L, S, N } and where here ∆ = k + k − k ′ − k and ∆ ′ = k ′ + k − k − k . Moreover let us define, α := | k || k |h ∆ i , β := | k || k |h ∆ ih ∆ ′ i , γ := | k || k |h ∆ ′ i . Then we have the followings: (1)
Assume ∗ ∈ {
H, S } and ∈ { L, N } . Then either (1 a ) | γ | . or (1 b ) ∗ = H and | k | ≥ | k ′ | . In case (1 b ) , (i) if L , or if N and | k | ≤ | k ′ | , then we have that, | k | / ≤ | k | ≤ | k | , | k | ≥ max ≤ j ≤ | k j | , | γ | . h k i max( h k i , h k i , h k i , h k i ) , that k = k , and that max( | k | , | k | , | k | ) = | k j | for j = 3 if L and, j ∈ { , } if N ; (ii) if N and | k | ≥ | k ′ | , then we have that | k | / ≤ | k | ≤ | k | , | k | / ≤ | k | ≤ | k | , | k | ≥ max( | k | , | k | ) , | α | . h k i max( h k i , h k i , h± k ± k i ) . Here “high” and “low” are with respect to the frequencies k and k . (2) Assume ∗ , ∈ { L, N } , then either (2 a ) | α | . or (2 b ) N and | k | ≥ | k | . In case (2 b ) , (i) if ∗ = L or if ∗ = N and | k | ≤ | k | , then we have | k | / ≤ | k | ≤ | k | , | k | ≥ max j ∈{ , , } | k j | , | γ | . h k i max( h k i , h k i , h k i , h k i ) , that k = k , and that max( | k | , | k | , | k | ) = | k j | for j = 1 if ∗ = L and, j ∈ { , } if ∗ = N ; (ii) if ∗ = N and | k | ≥ | k | , then we have that | k | / ≤ | k | ≤ | k | , | k | / ≤ | k | ≤ | k | , | k | ≥ max( | k | , | k | ) , | α | . h k i max( h k i , h k i ) , | k | 6 = | k | . (3) Assume ∗ , ∈ { L, N } , then we have that | β | . h k ih± k ± k i , h k i & h k i . For ∗ ∈ {
H, L, S, N } define C ∗ by F x C ∗ ( v , v , v )( k ) = X X ∗ k M ( k, k , k , k ) · b v ( k ) b v ( k ) b v ( k ) , (2.12)then we have C = C H + C L + C S + C N . (2.13)2.4. The full setup.
When all the relevant nonlinearities are well-defined as spacetime distribu-tions, which we will see in the course of the proof, the integral equation (1.4) for u will be equivalentto the integral equation v ( t ) = e it∂ x v + I ( C ( v, v, v ) + Q ( v, · · · , v )) , IF ( t ) = Z t e i ( t − s ) ∂ x F ( s ) d s, (2.14)for v , where the nonlinearities C and Q are as in (2.6) ∼ (2.8), and the initial data v = exp( − i∂ − x P =0 | u | ) · u , (2.15)which satisfies k v k H p ≤ A given that k u k H p ≤ A .In the proof we will be extending the function v , which is defined on J = [ − T, T ], to the wholeline R t ; to this end we fix a smooth function ϕ ( t ) that is 1 for | t | ≤ | t | ≥
2, and definethe truncated versions of the linear solution and Duhamel operator ψ ( t ) = ϕ ( t ) · e it∂ x v , I F ( t ) = ϕ ( t ) · I ( ϕ ( s ) · F ( s )) . (2.16)For later uses we will also define ϕ T ( t ) = ϕ ( T − t ).3. Preparations
In this section we define and set up our function spaces, prove the main linear estimates andprove an improved divisor bound which is used in some our estimates.
PTIMAL LOCAL WELL-POSEDNESS FOR THE DNLS EQUATION 11
Function spaces.
We begin by properly defining the functions spaces that play a role in ourproof. Denote the Fourier-Lebesgue norms H sp ( T ), where p ∈ [2 , ∞ ) an s ∈ R , by k u k H sp = kh k i s b u ( k ) k ℓ pk . (3.1)In one dimension, H sp has the same scaling as the Sobolev space H γ for γ = s + 1 p −
12 (3.2)When p > γ < s , which allows the regularity index γ to decrease while keeping s ≥ .The associated Fourier restriction norm spaces X s,bp,q , where p, q ∈ [2 , ∞ ) an s, b ∈ R , are thendefined by k u k X s,bp,q = kh k i s h ξ + k i b b u ( k, ξ ) k ℓ pk L qξ . (3.3)For 2 ≤ p < ∞ fixed and 0 < δ ≪ p also fixed, let the parameters( b , b , q , q , r , r , r ) be defined as in (1.16). We define the four spaces in which the estimates areproved as follows: Y = X , p ,r , Y = X , p ,r ,Z = X ,b p ,q , Z = X ,b p ,q . (3.4)Note that by H¨older we have X s,bp,q ⊂ X s ′ ,b ′ p ′ ,q ′ , provided p ≤ p ′ , q ≤ q ′ ; s + 1 p < s ′ + 1 p ′ , b + 1 q < b ′ + 1 q ′ , (3.5)in particular Z ⊂ Y ⊂ C t H p . Finally, for any finite interval I and any spacetime norm Y , define k u k Y ( I ) = inf (cid:8) k v k Y : v = u on I (cid:9) . (3.6)3.2. Linear estimates.
We will be using the following notation for a spacetime function F : e F ( k, λ ) = X F ( k, λ ) := b F ( k, λ − k ) , (3.7)where b F is the spacetime Fourier transform. Lemma 3.1.
Define the function K ( λ, σ ) = i (cid:20) Z R b ϕ ( λ − µ ) b ϕ ( µ − σ ) µ d µ − b ϕ ( λ ) Z R b ϕ ( µ − σ ) µ d µ (cid:21) , (3.8) where integrations are defined as principal value limits, then it satisfies | K ( λ, σ ) | . B (cid:18) h λ i B + 1 h λ − σ i B (cid:19) h σ i (3.9) for any B > , and we have f I F ( k, λ ) = Z R K ( λ, σ ) e F ( k, σ ) d σ. (3.10) Proof.
In [19], Lemma 3.3, it is derived that f I F ( k, λ ) = c Z R b ϕ ( λ − µ ) µ d µ Z R b ϕ ( µ − σ ) e F ( k, σ ) d σ + c b ϕ ( λ ) · Z R d µµ Z R b ϕ ( µ − σ ) e F ( k, σ ) d σ, where c and c are numerical constants, and integrations are defined as principal value limits. Byour convention with Fourier transform, we can calculate that c = i and c = − i , which gives theformula (3.8). The bound (3.9) follows easily, using that b ϕ is a Schwartz function. (cid:3) Note that in [25] this same space is denoted by b H sp ′ ( T ) where p + p ′ = 1. Proposition 3.2.
Suppose u is a smooth function such that u (0) = 0 . Then we have the estimates k ϕ T · u k Y . T θ k u k Y , k ϕ T · u k Z . T θ k u k Z . (3.11) Proof.
First notice that, by (3.5), k u k X ,b p ,q . k u k X ,b p ,q = k u k Z . Then, by separating different Fourier modes and conjugating by e ± itk at Fourier mode e ikx , itsuffices to prove that for any function g = g ( t ) satisfying g (0) = 0, kh λ i b ( b g ∗ c ϕ T )( λ ) k L q . T e q − q kh σ i b b g ( σ ) k L e q , (3.12)provided ∞ > q > e q > b + e q > > b . Let g = g + g where b g ( σ ) = | σ |≥ T − ( σ ) b g ( σ ) , b g ( σ ) = | σ |
0. Since by elementary calculation we can prove Z R min(1 , | T σ | ) | b g ( σ ) | d σ . kh σ i b b g ( σ ) k L e q · k min(1 , | T σ | ) h σ i − b k L γ . T b − γ = T b + e q − , γ = 1 − e q , and that k T h T λ i − B h λ i b k L q . T − b − q , we deduce (3.14). (cid:3) PTIMAL LOCAL WELL-POSEDNESS FOR THE DNLS EQUATION 13
Remark . The requirement u (0) = 0 is necessary. Below (3.11) will be applied only for those u of form u ( t ) = Z t (expression) , namely u = I ( · · · ) or u = E ∗∗ ( · · · ), see Section 4.1 for the definition of E , so u (0) = 0 will alwaysbe true.3.3. A divisor bound.
Finally, in this subsection we prove and record an improved divisor boundthat will be handy later on in some parts of the proof . Lemma 3.4. (1) Let R = Z or Z [ ω ] , where ω = exp(2 πi/ , and fix ε > . Let k, q ∈ R and ρ > be such that | q | ≥ | k | ε > . Then the number of divisors r ∈ R of k that satisfies | r − q | ≤ ρ is atmost O ε ( ρ ε ) .(2) Consider the system ( ± a ± b ± c = const . ∓ a ∓ b ∓ c = const . (3.15) where the signs are arbitrary, but the signs of ± a and ∓ a etc. are always the opposite. Assumealso that there is no pairing , where a pairing means that (say) a = b and the signs of a and b in(3.15) are the opposite. Then the number of solutions that satisfy | a | ∼ N , | b | ∼ N and | c | ∼ N is . ε N ε , where N is the second largest of the N j ’s.Proof. (1) It is well-known that R has unique factorization and satisfies the standard divisor bound:the number of divisors of k = 0 is at most O ε ( | k | ε ). Thus the result is trivial if ρ ≥ | k | δ , where δ = ε . Now suppose | ρ | ≤ | k | δ (and | k | is large enough), we claim that the number of divisors r isat most m −
1, where m ∼ ε − is an integer.In fact, suppose d j , where 1 ≤ j ≤ m are distinct divisors, then by unique factorization we knowthat k is divisible by lcm( d , · · · , d m ), and hence divisible by Q mj =1 d j Q ≤ i In this section we discuss the structure of the solution, identifying the para-controlling termsand the precise solution submanifold W where v will belong. From now on we will focus on theequation (2.14). The submanifold Z in Theorem 1.1 will be defined as Z = G − W , where G isthe gauge transform (2.1), and W is a submanifold of Y ( J ) ⊂ C t H p ( J ), in which the solutionsolution v of (2.14) will be constructed. To define W we need some further preparations.4.1. Splitting the Duhamel operator. Let η ( t ) be a Schwartz function that satisfies the can-cellation condition b η (1) = 0 , H b η (1) = 1 , (4.1)where H is the Hilbert transform (principal value convolution by 1 /ξ ). For ∗ ∈ { N, L } , considerthe trilinear operator E ∗ := I C ∗ . Recall that E ∗ satisfies that F x E ∗ ( v , v , v )( k, t ) = X X ∗ k M ( k, k , k , k ) Z t e − i ( t − s ) k b v ( s, k ) b v ( s, k ) b v ( s, k ) d s. (4.2)As before let ∆ = k + k − k − k (we always have | ∆ | ≥ E Y ∗ and E X ∗ by F x E Y ∗ ( v , v , v )( k, t ) = X X ∗ k M ( k, k , k , k ) Z t e − i ( t − s ) k b η (∆( t − s )) b v ( s, k ) b v ( s, k ) b v ( s, k ) d s (4.3)( Y indicated this term is to be estimated in the Y space) and F x E X ∗ ( v , v , v )( k, t ) = X X ∗ k M ( k, k , k , k ) Z t e − i ( t − s ) k [1 − b η (∆( t − s ))] b v ( s, k ) b v ( s, k ) b v ( s, k ) d s (4.4)( X for “extra”). Clearly I C ∗ = E ∗ = E X ∗ + E Y ∗ . As with I , we will also define the time truncatedversions E Y ∗ ( v , v , v ) = ϕ ( t ) · E Y ∗ ( ϕ ( s ) v , v , v ) , E X ∗ ( v , v , v ) = ϕ ( t ) · E X ∗ ( ϕ ( s ) v , v , v ) . (4.5) Proposition 4.1. For ∗ ∈ { N, L } we have the expressions: X E Y ∗ ( v , v , v )( k, λ ) = X X ∗ k M ( k, k , k , k ) Z R K Y ∆ ( λ, σ ) d σ Z λ + λ − λ = σ − ∆ e v ( k , λ ) e v ( k , λ ) e v ( k , λ ) , X E X ∗ ( v , v , v )( k, λ ) = X X ∗ k M ( k, k , k , k ) Z R K X ∆ ( λ, σ ) d σ Z λ + λ − λ = σ − ∆ e v ( k , λ ) e v ( k , λ ) e v ( k , λ ) , (4.6) where recall ∆ = k + k − k − k , and the functions K Y ∆ and K X ∆ satisfy the bounds | K Y ∆ ( λ, σ ) | . B h λ − σ i B min (cid:18) h ∆ i , h σ i (cid:19) + 1 h λ − σ i min (cid:18) h ∆ i , h λ i (cid:19) , (4.7) | K X ∆ ( λ, σ ) | . B h λ i B h σ i + h σ − ∆ ih λ − σ i B h σ i min (cid:18) h ∆ i , h σ i (cid:19) + h λ − ∆ ih λ − σ i min (cid:18) h ∆ i , h λ i (cid:19) . (4.8) Proof. Fix ∗ ∈ { N, L } . Let K Y ∆ be the integral kernel of the linear operator F ( s ) Z t η (∆( t − s )) F ( s ) d s. PTIMAL LOCAL WELL-POSEDNESS FOR THE DNLS EQUATION 15 on the Fourier side, i.e. F t (cid:18) Z t η (∆( t − s )) F ( s ) d s (cid:19) ( λ ) = Z R K Y ∆ ( λ, σ ) b F ( σ ) d σ, and K X ∆ = K − K Y ∆ where K is defined in (3.8). By making Fourier expansion in x twisting by e ± itk on the time-Fourier side at mode k , one can see that (4.6) holds with exactly the same kernels K Y ∆ and K X ∆ .It then suffices to calculate these kernels; by an argument similar to [19], Lemma 3.3, we have K Y ∆ ( λ, σ ) = i Z R b ϕ ( λ − µ ) b ϕ ( µ − σ ) 1∆ ( H b η ) (cid:18) µ ∆ (cid:19) d µ − i Z R b ϕ ( λ − µ ) 1∆ b η (cid:18) µ ∆ (cid:19) ( H b ϕ )( µ − σ ) d µ. (4.9)Since b η and b ϕ are Schwartz functions, their Hilbert transforms will decay like h λ i − , thus (cid:12)(cid:12)(cid:12)(cid:12) 1∆ ( H b η ) (cid:18) µ ∆ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) . min (cid:18) h ∆ i , h µ i (cid:19) , | ( H b ϕ )( µ − σ ) | . h µ − σ i . Then, by elementary estimates of the integral, the first term on the right hand side of (4.9) isbounded by the first term on the right hand side of (4.7), and the second term on the right handside (4.9) is bounded by the second term on the right hand side of (4.7).As with K X ∆ , using (4.9) and (3.8) we can calculate K X ∆ ( λ, σ ) = i Z R b ϕ ( λ − µ ) b ϕ ( µ − σ ) (cid:20) µ − 1∆ ( H b η ) (cid:18) µ ∆ (cid:19)(cid:21) d µ + i Z R b ϕ ( λ − µ ) 1∆ b η (cid:18) µ ∆ (cid:19) ( H b ϕ )( µ − σ ) d µ − i b ϕ ( λ ) Z R b ϕ ( µ − σ ) µ d µ. (4.10)The third term on the right hand side of (4.10) is bounded by the first term on the right hand sideof (4.8). The first term on the right hand side of (4.10) can be bounded by the second term on theright hand side of (4.8), once we can prove (cid:12)(cid:12)(cid:12)(cid:12) µ − 1∆ ( H b η ) (cid:18) µ ∆ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) . h µ − ∆ ih µ i min (cid:18) h ∆ i , h µ i (cid:19) for | µ | ≥ 1, but this follows from rescaling and the assumption H b η (1) = 1. Similarly, the secondterm on the right hand side of (4.10) can be bounded by the third term on the right hand side of(4.8), due to the estimate (cid:12)(cid:12)(cid:12)(cid:12) b η (cid:18) µ ∆ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) . h µ − ∆ i min (cid:18) h ∆ i , h µ i (cid:19) and the fact that b η (1) = 0. (cid:3) Remark . Note that the first term on the right hand side of (4.7) is bounded by the second term,so we have | K Y ∆ | . h λ − σ i min (cid:18) h ∆ i , h λ i (cid:19) . (4.11)Moreover, by (4.8) we can write K X ∆ = K X, + K X, +∆ , where | K X, | . h σ i & h ∆ i h λ i B h ∆ i + h λ − σ i & h σ − ∆ i · min (cid:18) h ∆ i , h λ i (cid:19) , (4.12) | K X, +∆ | . B h σ i≪h ∆ i h λ i B h σ i + h σ − ∆ ih λ − σ i B h σ i min (cid:18) h ∆ i , h σ i (cid:19) + h λ − σ i≪h σ − ∆ i h σ − ∆ ih λ − σ i min (cid:18) h ∆ i , h λ i (cid:19) . (4.13)We will define the terms E X, ∗ and E X, + ∗ accordingly, for ∗ ∈ { N, L } .4.2. The submanifold W . We can now define W as follows. Definition 4.3. Recall that A , A and T are fixed. Let J = [ − T, T ]. We define W = (cid:8) v ∈ Y ( I ) : k v k Y ( I ) ≤ A , and there exists w with k w k Z ( I ) ≤ A , such that v = w + E YN ( w, w, v ) + E YL ( w, v, v ) (cid:9) . (4.14)This is a submanifold of Y ( I ) ⊂ C t H p ( I ). Moreover, we will define the submanifold Z of C t H p ( I )in the statement of Theorem 1.1 by Z = G − W .We will need the following proposition, whose proof is postponed to Section 5. Proposition 4.4. For any w which satisfies k w k Z ( I ) ≤ A there is a unique v satisfying k v k Y ( I ) ≤ A , such that v = w + E YN ( w, w, v ) + E YL ( w, v, v ) . (4.15) This mapping w v = v [ w ] is Lipschitz from the A -ball of Z ( I ) to the A -ball of Y ( I ) . Thesubmanifold W of Y ( I ) is the image of this mapping. Reducing to an equation for w . The next step is to reduce (2.14) to an equation for w .We will construct a function w satisfying k w k Z ( I ) ≤ A , such that the function v = v [ w ] definedby Proposition 4.4 satisfies (2.14). By direct calculation, we see that (2.14) reduces to w = e it∂ x v + I Q ( v, · · · , v ) + I C H ( v, v, v ) + I C S ( v, v, v )+ I ( C N ( v, v, v ) − C N ( w, w, v )) + I ( C L ( v, v, v ) − C L ( w, v, v ))+ E XN ( w, w, v ) + E XL ( w, v, v ) . (4.16)where v = v [ w ] (we will always assume this below) and satisfies v = w + E YN ( w, w, v ) + E YL ( w, v, v ) . (4.17)It is now clear that Theorem 1.1 will be a consequence of the following Proposition 4.5. The mapping that maps w to the right hand side of (4.16) is a contractionmapping from the A -ball of Z ( I ) to itself. This proposition will be proved in Section 6.5. Proof of Proposition 4.4 In this section we prove a prior bounds for the para-controlling terms which will crucially enterin the next section We start by noting that Z ( I ) ⊂ Y ( I ). In order to prove Proposition 4.4, itsuffices to prove the trilinear estimates k E YN ( v , v , v ) k Y ( I ) . T θ k v k Z ( I ) k v k Z ( I ) k v k Y ( I ) , (5.1) k E YL ( v , v , v ) k Y ( I ) . T θ k v k Z ( I ) k v k Y ( I ) k v k Y ( I ) . (5.2)In fact, these would imply that given w which satisfies k w k X , p,q ( I ) ≤ A , the mapping v w + E YN ( w, w, v ) + E YL ( w, v, v ) PTIMAL LOCAL WELL-POSEDNESS FOR THE DNLS EQUATION 17 is a contraction mapping from the A -ball of Y ( I ) to itself. It then has a unique fixed point v = v [ w ], and the Lipschitz property of the mapping w v is also easily checked.In order to prove (5.1) and (5.2), we will assume that w + and v + are extensions of w and v respectively, such that k w + k Z ≤ A and k v k Y ≤ A . Recall that ϕ T ( t ) = ϕ ( T − t ), clearly ϕ T ·E YN ( w + , w + , v + ) and ϕ T ·E YL ( w + , v + , v + ) are extensions of E YN ( w, w, v ) and E YL ( w, v, v ) respectively.Using also Proposition 3.2, we can reduce Proposition 4.4 to the following Proposition 5.1. We have the following bounds kE YN ( v , v , v ) k Y . k v k Z k v k Z k v k Y , (5.3) kE YL ( v , v , v ) k Y . k v k Z k v k Y k v k Y . (5.4) Proof. Let ∗ ∈ { N, L } , using the embedding Z ⊂ Y , we only need to prove the stronger result kE Y ∗ ( v , v , v ) k Y . k v k Z k v k Y k v k Y . (5.5)Let E = E Y ∗ ( v , v , v ), we may assume the norms on the right hand side are all equal to 1. Recallfrom (4.6) and (4.11) that | e E ( k, λ ) | . X X ∗ | k | min (cid:18) h ∆ i , h λ i (cid:19) Z λ + λ + λ − λ = λ − ∆ h λ i Y j =1 | e v j ( k j , λ j ) | , (5.6)where ∆ = 2( k − k )( k − k ) as before. we may restrict to the dyadic region h k i ∼ N and h k i ∼ N (so N & N ), where N and N are powers of two.Recall that r = ( ) + 3 δ (so r < r ). Notice that kh k i e v k L λ ℓ p k . kh k i h λ i b e v k L q λ ℓ p k . kh k i h λ i b e v k ℓ p k L q λ . kh k j i (1 −√ δ ) /p e v j k L λ ℓ r k . kh k j i (1 −√ δ ) /p h λ j i e v j k L r λ ℓ r k . kh k j i (1 −√ δ ) /p h λ j i e v j k ℓ r k L r λ . kh k j i h λ j i e v j k ℓ p k L r λ . j ∈ { , } , we may then fix ( λ , λ , λ ) which we eventually intergate over, and denote | e v ( k , λ ) | = h k i − f ( k ) , | e v j ( k j , λ j ) | = N − (1 −√ δ ) /p j f j ( k j ) , ≤ j ≤ , where (after a further normalization) k f k ℓ p k . , k f j k ℓ r k . k = 2 , , (5.7)and it will suffice to prove that for any fixed µ (= λ + λ − λ ) ∈ R , (cid:13)(cid:13)(cid:13)(cid:13) h k i h λ i X X ∗ h k i min (cid:18) h ∆ i , h λ i (cid:19) h λ − ∆ − µ i Y j =1 f j ( k j ) (cid:13)(cid:13)(cid:13)(cid:13) ℓ p k L r λ . ( N N ) (1 −√ δ ) /p − θ . (5.8)In the above summation over ( k , k , k ) ∈ X ∗ , we may first fix ∆ and sum over ( k , k , k ) ∈ X ∗ that corresponds to this fixed ∆.We first assume ∗ = L , which is the slightly harder case. Note that h k i ∼ h k i , by Lemma 3.4,we can bound the left hand side of (5.8) by (cid:13)(cid:13)(cid:13)(cid:13) h k ih λ i X ∆ F ( k, ∆) min (cid:18) h ∆ i , h λ i (cid:19) h λ − ∆ − µ i (cid:13)(cid:13)(cid:13)(cid:13) ℓ p k L r λ , (5.9) where F ( k, ∆) = X ( k ,k ,k ) ∈ X L k + k − k − k =∆ 3 Y j =1 f j ( k j ) . N θ (cid:18) X ( k ,k ,k ) ∈ X L k + k − k − k =∆ 3 Y j =1 f j ( k j ) r (cid:19) r . Using the facts that min (cid:18) h ∆ i , h λ i (cid:19) . h ∆ i − h λ i − , h ∆ i ∼ h k i and by Schur’s estimate, we can bound (cid:13)(cid:13)(cid:13)(cid:13) h k ih λ i X ∆ F ( k, ∆) min (cid:18) h ∆ i , h λ i (cid:19) h λ − ∆ − µ i (cid:13)(cid:13)(cid:13)(cid:13) L r λ . k F ( k, ∆) k ℓ r for each fixed k . By the definition of F ( k, ∆), it then suffices to prove that N θ (cid:13)(cid:13)(cid:13)(cid:13)(cid:18) X X L Y j =1 f j ( k j ) r (cid:19) r (cid:13)(cid:13)(cid:13)(cid:13) ℓ p k . ( N N ) (1 −√ δ ) /p N − θ . (5.10)Let f j ( k j ) r = g j ( k j ) and β = ( p /r ), it suffices to prove (for a possibly different θ ) that (cid:13)(cid:13)(cid:13)(cid:13) X | k |∼ N | k |∼ N g ( k + k − k ) g ( k ) g ( k ) (cid:13)(cid:13)(cid:13)(cid:13) ℓ βk . ( N N ) r (1 −√ δ ) /p N − θ . As k g k ℓ βk = k f k r ℓ p k . 1, by Minkowski we can bound the above by k g k ℓ k k g k ℓ k = k f k r ℓ r k k f k r ℓ r k . ∗ = L .When ∗ = N , we will further assume h k i ∼ N and h k i ∼ N , then all the proof will be thesame as above, using the fact that h k i h k i ∼ N N ∼ h ∆ i . The sum over N and N is then taken care of using the positive power of N on the right handside of (5.10), and the fact that N & max( N , N ) when ( k , k , k ) ∈ X N . (cid:3) Proof of Proposition 4.5 This section constitutes the heart of the paper. Here we find w and prove all the underlyingmultilinear estimates involved in its construction. In the course of the proof we show in particularthat all relevant nonlinearities are well defined as space-time distributions whence the integralequation (1.4) for u will be equivalent to the integral equation formulation of (1.5) from Section2.4.Given w satisfying k w k Z ( I ) ≤ A , let w + be an extension of w such that k w + k Z ≤ A . Bythe proof of Proposition 4.4 in Section 5, we know that there is a unique v + = v + [ w + ] such that k v + k Y ≤ A , and v + = w + + ϕ T · E YN ( w + , w + , v + ) + ϕ T · E YL ( w + , v + , v + ) . (6.1)Moreover this v + is an extension of v = v [ w ]. Therefore, recall that ψ := ϕ ( t ) e it∂ x v , the function z := ψ + ϕ T · IQ ( v + , · · · , v + ) + ϕ T · IC H ( v + , v + , v + ) + ϕ T · IC S ( v + , v + , v + )+ ϕ T · I ( C N ( v + , v + , v + ) − C N ( w + , w + , v + )) + ϕ T · I ( C L ( v + , v + , v + ) − C L ( w + , v + , v + ))+ ϕ T · E XN ( w + , w + , v + ) + ϕ T · E XL ( w + , v + , v + ) (6.2) PTIMAL LOCAL WELL-POSEDNESS FOR THE DNLS EQUATION 19 will be an extension of the right hand side of (4.16).6.1. Splitting the formula of z . Now that w + , v + and z are defined for all time, we can furthermanipulate the expression of z , as this manipulation sometimes requires inserting time-frequencycutoffs. We will analyze each term in (6.2) separately. The initial data term ψ is trivial. Forthe other terms, we will remove the ϕ T factor in front, and bound the corresponding terms in thestronger space Z ; Proposition 3.2 then allows us to gain a factor T θ which provides the requiredsmallness.(1) The term IQ ( v + , · · · , v + ). This is a single term, we will name it z = IQ ( v + , · · · , v + ) . (6.3)(2) The term ϕ T · I ( C H + C S )( v + , v + , v + ). Here decomposing v + by (6.1), we can obtain thefollowing terms z = I ( C H + C S )( w + , w + , w + ) ,z = I ( C H + C S )( ϕ T · E YN ( w + , w + , v + ) , v + , v + ) ,z = I ( C H + C S )( ϕ T · E YL ( w + , v + , v + ) , v + , v + ) ,z = I ( C H + C S )( w + , ϕ T · E YN ( w + , w + , v + ) , v + ) ,z = I ( C H + C S )( w + , ϕ T · E YL ( w + , v + , v + ) , v + ) ,z = I ( C H + C S )( w + , w + , ϕ T · E YN ( w + , w + , v + )) ,z = I ( C H + C S )( w + , w + , ϕ T · E YL ( w + , v + , v + )) . (6.4)Here z is a cubic expression, and the others are quintic expressions.(3) The term I ( C N ( v + , v + , v + ) − C N ( w + , w + , v + )). Similar to (2), we can obtain the terms z = I ( C N ( ϕ T · E YN ( w + , w + , v + ) , v + , v + ) ,z = I ( C N ( ϕ T · E YL ( w + , v + , v + ) , v + , v + ) ,z = I ( C N ( w + , ϕ T · E YN ( w + , w + , v + ) , v + ) ,z = I ( C N ( w + , ϕ T · E YL ( w + , v + , v + ) , v + ) . (6.5)They are all quintic expressions.(4) The term I ( C L ( v + , v + , v + ) − C L ( w + , v + , v + )). In the same way we get two terms z = ϕ T · I ( C L ( E YN ( w + , w + , v + ) , v + , v + ) ,z = I ( C L ( ϕ T · E YL ( w + , v + , v + ) , v + , v + ) . (6.6)They are both quintic expressions.(5) The term E XN ( w + , w + , v + ) + E XL ( w + , v + , v + ). This term requires a little more care. Let ∗ ∈{ N, L } , recall that from Proposition 4.1 and Remark 4.2, we have X E X, + ∗ ( v , v , v )( k, λ ) = X X ∗ k M ( k, k , k , k ) Z R K X, +∆ ( λ, σ ) d σ Z λ + λ − λ = σ − ∆ e v ( k , λ ) e v ( k , λ ) e v ( k , λ ) . (6.7)We may further decompose this expression into E X, + ∗ = E X, ∗ + E X, ∗ + E X, ∗ , where in E X,j ∗ we makethe restriction | λ j | = max ≤ ℓ ≤ | λ ℓ | , | λ j | & | σ − ∆ | . Now if ∗ = N and j = 3, or ∗ = L and j ∈ { , } , we will make decompose the v + correspondingto frequency λ j using (6.1). This gives the following terms z = ( E X, N + E X, N + E X, N )( w + , w + , v + ) ,z = E X, N ( w + , w + , w + ) ,z = E X, N ( w + , w + , ϕ T · E YN ( w + , w + , v + )) ,z = E X, N ( w + , w + , ϕ T · E YL ( w + , v + , v + )) ,z = ( E X, L + E X, L )( w + , v + , v + ) ,z = E X, L ( w + , w + , v + ) ,z = E X, L ( w + , ϕ T · E YN ( w + , w + , v + ) , v + ) ,z = E X, L ( w + , ϕ T · E YL ( w + , v + , v + ) , v + ) ,z = E X, L ( w + , v + , w + ) ,z A = E X, L ( w + , v + , ϕ T · E YN ( w + , w + , v + )) ,z B = E X, L ( w + , v + , ϕ T · E YL ( w + , v + , v + )) . (6.8)Some of these are cubic expressions, and some of them are quintic.(6) An operation on quintic terms. Each of the above z jℓ ’s is a multilinear expression, eithercubic or quintic; we will always list its input functions from left to right. Consider now a generalquintic term. Let k and k j , where 1 ≤ j ≤ 5, are the (space) frequencies of the output and inputfunctions, then it will involve a summation X ± k ···± k = k (expression) . As with Lemma 3.4, we say a pairing ( i, j ) happens, if k i = k j and the signs of k i and k j in theexpression ± k · · · ± k are the opposite.For each tuple ( k j ), we will choose an index i ∈ { , · · · , } as follows: if there is no pairing, thenlet i ∈ { , · · · , } be such that | k i | is the maximum; if there is a pairing, say (1 , { , , } , then let i ∈ { , , } such that | k i | is the maximum; if there is a pairing in { , , } , say (3 , i = 5. It is clear that we always have | k i | & | k | .This procedure then decomposes this quintic term into five parts; once an i is fixed, and if theinput function corresponding to this i in this quintic term happens to be v + (instead of w + ), wewill decompose this v + using (6.1), so that this quintic term is decomposed into a quintic and twoseptic terms.(7) Summary. Now we have decomposed z into a superposition of multilinear expressions z jℓ (including those coming from step (6) above), either cubic or quintic or septic, with input functionsbeing either w + or v + . Moreover, if we consider two different w and w ′ , then we may chooseextensions w + and ( w ′ ) + such that k ( w ′ ) + − w + k X , p,q ≤ k w ′ − w k X , p,q ( I ) . Let v + and ( v ′ ) + be defined from w + and ( w ′ ) + by (6.1), then we also have k ( v ′ ) + − v + k Y p . k ( w ′ ) + − w + k X , p,q . k w ′ − w k X , p,q ( I ) . Then z and z ′ , which are defined by (6.2) using w and w ′ , satisfy that z − z ′ is an extension of thedifference of the right hand sides of (4.16) corresponding to w and w ′ . Therefore, in order to proveProposition 4.5, it will suffice to prove the following PTIMAL LOCAL WELL-POSEDNESS FOR THE DNLS EQUATION 21 Proposition 6.1. All these terms z jℓ , including those coming from step (6) above, satisfy themultilinear estimates k z jℓ ( v , · · · , v r ) k Z . k v k · · · k v r k , where r ∈ { , , } , and for each i , v i is measured in the Z norm if the corresponding input functionin z jℓ is w + , and in the Y norm if the input is v + . For example the estimate for z will be k z ( v , · · · , v ) k Z . k v k Z Y j =2 k v j k Y . Remark . We make a further remark about the operation in step (6) above. For some quinticterms z jℓ this operation is necessary; for others it is not. However, even in the latter case, performingthis operation will not affect the proof: if z jℓ itself satisfies a multilinear estimate where this inputfunction v + is measured in the Y norm, then by Propositions 3.2 and 5.1, after decomposing this v + using (6.1), the resulting quintic and septic terms will also satisfy the right multilinear estimate.For example, we will see below that k z ( v , · · · , v ) k Z . Y j =1 k v j k Z · k v k Y . Then, even after performing this operation (with the chosen index i = 5) we still have k z ( v , · · · , v ) k Z . Y j =1 k v j k Z · k v k Y , k z ( v , · · · , v , ϕ T · E YN ( v , v , v )) k Z . Y j =1 k v j k Z · k v k Y , k z ( v , · · · , v , ϕ T · E YN ( v , v , v )) k Z . Y j =1 k v j k Z · k v k Y k v k Y . The following subsections are devoted to the proof of Proposition 6.1.6.2. Cubic terms. In this subsection we treat the cubic terms, which are z and the cubic z ∗ terms. First we deal with z term in the following Proposition 6.3. Proposition 6.3. z is defined in (6.4). We have the following bound k z ( v , v , v ) k Z . Y j =1 k v j k Z . (6.9) Proof. Let ∗ ∈ { H, S } , we need to show the following bound kIC ∗ ( v , v , v ) k Z . Y j =1 k v j k Z . (6.10)We may assume the norms on the right hand side are all equal to 1. Recall from (3.9) and (3.10)that for any B > | f IC ∗ ( v , v , v )( k, λ ) | . X X ∗ | k | Z λ + λ − λ = σ − ∆ h λ i B + 1 h λ − σ i B ! h σ i Y j =1 | e v j ( k j , λ j ) | , (6.11)where ∆ = 2( k − k )( k − k ) as before. It will suffice to prove that (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) h k i h λ i b X X ∗ | k | Z λ + λ − λ = σ − ∆ h λ i h λ − σ i Y j =1 | e v j ( k j , λ j ) | (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ℓ p k L q λ . Y norm (3.4) and the following inequality h λ i B + 1 h λ − σ i B ! h σ i . h λ ih λ − σ i (6.13)for B large enough.Recall that b = 1 − δ and 1 /q = 4 δ and hence similar to the proof of Proposition 5.1 we havefor j ∈ { , , }kh k j i h λ j i δ e v j k L λ ℓ p k . kh k j i h λ j i b e v j k L q λ ℓ p k . kh k j i h λ j i b e v j k ℓ p k L q λ . λ , λ , λ ) which we eventually integrate over, anddenote | e v j ( k j , λ j ) | = h k j i − h λ j i − δ f j ( k j ) (1 ≤ j ≤ µ (= λ + λ − λ ) ∈ R , (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) h k i h λ i δ X X ∗ | k |h k i h k i h k i h λ − ∆ − µ ih λ i δ h λ i δ h λ i δ Y j =1 | f j | (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ℓ p k L q λ . , (6.14)and then applying the inequality (2.10), it will suffice to prove (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) h λ i δ X X ∗ h λ − ∆ − µ ih λ i δ h λ i δ h λ i δ Y j =1 | f j | (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ℓ p k L q λ . , (6.15)In the above summation over ( k , k , k ) ∈ X ∗ , we again first fix ∆ and sum over ( k , k , k ) ∈ X ∗ corresponds to this fixed ∆. Using the fact that( h λ ih λ ih λ ih λ ih λ − ∆ − µ i ) δ & h ∆ i δ (6.16)and by the standard divisor bound , we can bound the left side of (6.15) by (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)X ∆ h λ − ∆ − µ i − δ h ∆ i δ F ( k, ∆) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ℓ p k L q λ (6.17)where F ( k, ∆) = X ( k ,k ,k ) ∈ X ∗ k − k )( k − k )=∆ 3 Y j =1 | f j | . h ∆ i θ X ( k ,k ,k ) ∈ X ∗ k − k )( k − k )=∆ 3 Y j =1 | f j | p p . (6.18)By our choice we have δ < p and θ < δ , so by Schur’s estimate, we can bound (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)X ∆ h λ − ∆ − µ i − δ h ∆ i δ F ( k, ∆) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L q λ . (cid:13)(cid:13)(cid:13)(cid:13) F ( k, ∆) h ∆ i δ (cid:13)(cid:13)(cid:13)(cid:13) ℓ p . (6.19) The divisor bound applies when ∆ = 0; however when ∆ = 0 we must have k = k = k = k by the definitionof V , so the bound is still true. PTIMAL LOCAL WELL-POSEDNESS FOR THE DNLS EQUATION 23 Then we may sum over k and we obtain that (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)X ∆ h λ − ∆ − µ i − δ h ∆ i δ F ( k, ∆) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ℓ p k L q λ . Y j =1 k f j k ℓ p k (6.20)by (6.18) and (6.19). Finally we integrate (6.20) over ( λ , λ , λ ) and it finishes this proof. (cid:3) Next let’s consider the cubic z ∗ terms (i.e. z , z , z , z and z ). The following Proposition6.4 gives the suitable bounds for z , z , z , z and z in Proposition 6.1. Proposition 6.4. For ∗ ∈ { N, L } , j ∈ { , , , } and E X,j ∗ defined in (4.6) and the descriptionabove (6.8), we have the following bounds.(1) If j = 0 , we obtain that (cid:13)(cid:13) E X, ∗ ( v , v , v ) (cid:13)(cid:13) Z . k v k Z k v k Y k v k Y . (6.21) (2) If j = 1 , we obtain that (cid:13)(cid:13) E X, ∗ ( v , v , v ) (cid:13)(cid:13) Z . k v k Z k v k Y k v k Y . (6.22) (3) If j ∈ { , } and i ∈ { , } − { j } , we obtain that (cid:13)(cid:13) E X,j ∗ ( v , v , v ) (cid:13)(cid:13) Z . k v k Z k v j k Z k v i k Y (6.23) Proof. Recall from (4.6) that | g E X, ∗ ( v , v , v )( k, λ ) | . X X ∗ | k | Z λ + λ − λ = σ − ∆ | K X, ( λ, σ ) | Y j =1 | e v j ( k j , λ j ) | , (6.24)and for j ∈ { , , }| g E X,j ∗ ( v , v , v )( k, λ ) | . X X ∗ | k | Z λ + λ − λ = σ − ∆ | λ j | =max ≤ ℓ ≤ | λ ℓ | | K X, +∆ ( λ, σ ) | Y j =1 | e v j ( k j , λ j ) | , (6.25)where ∆ = 2( k − k )( k − k ) as before.(1) Let’s consider the case when j = 0 and ∗ ∈ { N, L } , and then left side of the bound (6.21)can be bounded by (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) h k i h λ i b X X ∗ | k | Z λ + λ − λ = σ − ∆ | K X, ( λ, σ ) | Y j =1 | e v j ( k j , λ j ) | (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ℓ p k L q λ . (6.26)Recall (4.12), it will suffice to prove that (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) h k i h λ i b X X ∗ | k | Z λ ,λ ,λ h λ i δ h ∆ i − δ Y j =1 | e v j ( k j , λ j ) | (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ℓ p k L q λ . k v k Z k v k Y k v k Y (6.27)and then by Minkowski’s inequality and integrating over λ the left side of (6.27) can bounded by (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) h k i X X ∗ | k | Z λ ,λ ,λ h ∆ i − δ Y j =1 | e v j ( k j , λ j ) | (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ℓ p k . (6.28)We may then fix ( λ , λ , λ ) which we eventually integrate over. In the above summation over( k , k , k ) ∈ X ∗ , we may first fix ∆ and sum over ( k , k , k ) ∈ X ∗ that corresponds to this fixed ∆. Moreover, as before we may restrict to the dyadic region h k i ∼ N and h k i ∼ N (so N & N ),where N and N are dyadic numbers. It will suffice to bound (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) h k i X ∆ | k | h ∆ i − δ X ( k ,k ,k ) ∈ X ∗ ( k − k )( k − k )=∆ | k |∼ N , | k |∼ N Y j =1 | e v j ( k j , λ j ) | (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ℓ p k (6.29)where h ∆ i ∼ h k ih k i and N ∼ | k | & max( | k | , | k | ) (by Proposition 2.3). By the standard divisorbound and H¨older’s inequality as the proof of Proposition 5.1 we obtain that(6 . . kh k i e v k ℓ p k kh k i p e v k ℓ r k kh k i p e v k ℓ r k N p − δ − θ N p . (6.30)Then we may integrate over λ λ and λ and sum over ( N , N ). By using the negative power of N (suppose δ < / (4 p )) and the following facts (similar as before): kh k i e v k L λ ℓ p k . k v k Z , kh k i p e v k L λ ℓ r k . k v k Y , kh k i p e v k L λ ℓ r k . k v k Y , (6.31)this finishes the proof of (6.21).Before we start to prove the parts (2) and (3), we may first hold a easier bound for | K X, +∆ | .Suppose | λ j | = max ≤ ℓ ≤ and | λ j | & | σ − ∆ | . Recall that b = 1 − δ and b = 1 − δ , and then weobtain that h λ i b h λ j i b − δ | K X, +∆ | . h ∆ i − δ h σ − λ i . (6.32)We may then fix the other two λ ℓ ( ℓ = j ) and ∆, and then we integrate over λ j and λ . We canobtained the following bound: (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Z λ j h λ + λ − λ − λ − ∆ i (cid:16) h λ j i b e v j ( k j , λ j ) (cid:17)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L q λ . (cid:13)(cid:13)(cid:13) h λ j i b − δ e v j ( k j , λ j ) (cid:13)(cid:13)(cid:13) L q λj (6.33)by Schur’s inequalities. For | λ j | = max ℓ ∈{ , , } | λ ℓ | , to prove the parts (2) and (3), it will suffice toconsider the norm (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) h k i X X ∗ | k | h ∆ i − δ (cid:13)(cid:13)(cid:13) h λ j i b − δ e v j ( k j , λ j ) (cid:13)(cid:13)(cid:13) L q λj Y ℓ ∈{ , , }−{ j } k e v ℓ ( k ℓ , λ ℓ ) k L λℓ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ℓ p k (6.34)By (6.32) and (6.33).(2) Let’s consider the case when j = 1. By (6.34), it will suffice to bound (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) h k i X X ∗ h k i h k i p h k i p h ∆ i − δ Y ℓ =1 | f ℓ ( k ℓ ) | (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ℓ p k , (6.35)where f ( k ) = h k i kh λ i b − δ e v k L q λ and f ℓ ( k ℓ ) = h k ℓ i p k e v ℓ k L λ for ℓ = 2 , 3. Similar to the proofof Proposition 5.1, we also have similar bounds: k f k ℓ p k . k v k Z , k f ℓ k ℓ r k . k v ℓ k Y for ℓ = 2 , 3. We may use dyadic decomposition on ( k , k ) and sum over ( k , k , k ) that correspondsto ∆ and then over ∆ and k . Following the same proof as in the part (1), the negative power of N help us bound (6.35) by k v k Z k v k Y k v k Y , when δ < / (6 p ). This finish the proof of (6.22). PTIMAL LOCAL WELL-POSEDNESS FOR THE DNLS EQUATION 25 (3) Let’s consider the case when j ∈ { , } and denote i is the other number in { , } . Similarlyby (6.34), it will suffice to bound (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) h k i X X ∗ h k i h k i p h k i p h ∆ i − δ Y ℓ =1 | f ℓ ( k ℓ ) | (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ℓ p k , (6.36)where f ( k ) = h k i k e v k L λ , f j ( k j ) = h k j i p kh λ j i b − δ e v j k L q λ and f i ( k i ) = h k i i p k e v i k L λ . Similarto the proof of Proposition 5.1, we also have similar bounds: k f ℓ k ℓ p k . k v ℓ k Z , k f i k ℓ r k . k v i k Y for ℓ = 1 , j . Following the same proof of the part (2), (6.36) can be bounded by k v k Z k v j k Z k v i k Y when δ < / (6 p ). This finishes the proof of (6.23). (cid:3) The canonical quintic term. The majority of z jℓ are quintic terms; in fact the majority ofthem can be treated in the same way, using the following estimate. Proposition 6.5. Consider a quintic expression R that satisfies | X R ( v , · · · , v )( k, λ ) | . X ± k ±···± k = k Z R d σ h λ i − θ h λ − σ i − θ Z ± λ ± λ ±···± λ = σ − Ξ 5 Y j =1 | e v j ( k j , λ j ) | , (6.37) where Ξ := k ∓ k ∓ · · · ∓ k (the signs are arbitrary, but the signs of ± k j and ∓ k j are alwaysthe opposite). Then after the operation in Section 6.1, step (6), the resulting terms satisfy thecorresponding multilinear estimates. In particular, suppose the chosen index during this operationis i = 1 , then we have kR ( v , · · · , v ) k Z . k v k Z Y j =2 k v j k Y , (6.38) kR ( ϕ T · E YN ( v , v , v ) , v , · · · , v ) k Z . k v k Z k v k Z Y j =3 k v j k Y , (6.39) kR ( ϕ T · E YL ( v , v , v ) , v , · · · , v ) k Z . k v k Z Y j =2 k v j k Y . (6.40) Proof. (1) We first prove (6.38). Assume all the norms on the right hand side are 1. By a dyadicdecomposition, we may restrict to the region where h k j i ∼ N j for 2 ≤ j ≤ 5; by symmetry we mayassume N ≥ · · · ≥ N . As in the proof of Proposition 5.1 we have kh k i e v k L λ ℓ p k . kh k j i (1 −√ δ ) /p e v j k L λ ℓ k . kh k j i (1 −√ δ ) /p e v j k L λ ℓ r k . ≤ j ≤ 5. We may then again fix λ j for 1 ≤ j ≤ 5, which we eventually integrate over, andassume f ( k ) = h k i | e v ( k , λ ) | , f j ( k j ) = N (1 −√ δ ) /p j | e v j ( k j , λ j ) | (2 ≤ j ≤ , such that (after a further normalization) k f k ℓ p k ≤ , k f j k ℓ k . ≤ j ≤ . (6.41) Using also that h k i . h k i , it then suffices to prove that (cid:13)(cid:13)(cid:13)(cid:13) h λ i b X ± k ±···± k = k Z R h λ i − θ h λ − Ξ − µ i − θ Y j =1 f j ( k j ) (cid:13)(cid:13)(cid:13)(cid:13) ℓ p k L q λ . N − θ ( N N N N ) (1 −√ δ ) /p (6.42)for any fixed µ (which is a linear combination of λ j for 1 ≤ j ≤ b < − θ and Schur’s estimate, we can bound for fixed k that (cid:13)(cid:13)(cid:13)(cid:13) h λ i b X ± k ±···± k = k Z R h λ i − θ h λ − Ξ − µ i − θ Y j =1 f j ( k j ) (cid:13)(cid:13)(cid:13)(cid:13) L q λ . k F ( k, Ξ) k ℓ q , where F ( k, Ξ) = X ± k ···± k = k ± k ···± k =Ξ − k Y j =1 f j ( k j ) . As Ξ is determined by ( k , k , k , k ) and hence the number of different Ξ’s does not exceed O ( N ),we can bound the ℓ q Ξ norm of F ( k, Ξ) by N q times its ℓ ∞ Ξ norm.(a) Assume N ≥ N √ δ . For fixed k and Ξ, by assumption we know that either there is no pairing,or there is a pairing, say (2 , { , , } , or there are two pairings, say(2 , 3) and (4 , k , k ) (or ( k , k )), the number of choices for ( k , k , k )(or ( k , k , k )) is at most O ( N θ ) by Lemma 3.4, so we can bound using (6.41) that F ( k, Ξ) . X k ,k X k ,k ,k f ( k ) f ( k ) f ( k ) + X k ,k X k ,k ,k f ( k ) f ( k ) f ( k ) . N θ sup | k − k | . N f ( k ) . In the other two cases this estimate can be similarly established. This gives k F ( k, Ξ) k ℓ ∞ Ξ . N θ (cid:18) X | k − k | . N | f ( k ) | p (cid:19) p , and hence the left hand side of (6.42) is bounded by N θ + q (cid:13)(cid:13)(cid:13)(cid:13)(cid:18) X | k − k | . N | f ( k ) | p (cid:19) p (cid:13)(cid:13)(cid:13)(cid:13) ℓ p k . N θ + q + p using (6.41). As N ≥ N √ δ , 4 /q = O ( δ ), δ is small enough depending on p and θ is small enoughdepending on δ , this implies (6.42).(b) Assume N ≤ N √ δ . Then in estimating F ( k, Ξ), we may fix the choices of ( k , k , k ) andeventually sum over them. In this process we lost at most N O ( √ δ )2 . Then, with ( k , k , k ) fixed,we know that ( k , k ) is uniquely determined by k and Ξ, since by assumption (1 , 2) cannot be apairing. Thus, with ( k , k , k ) fixed, we have | F ( k, Ξ) | . sup k f ( k − ℓ − k ) f ( k ) , where ℓ is a linear combination of ( k , k , k ). As (cid:13)(cid:13) sup k f ( k − ℓ − k ) f ( k ) (cid:13)(cid:13) ℓ p k . (cid:13)(cid:13)(cid:13)(cid:13)(cid:18) X k f ( k − ℓ − k ) p f ( k ) p (cid:19) p (cid:13)(cid:13)(cid:13)(cid:13) ℓ p k = k f k ℓ p k k f k ℓ p k and k f k ℓ p k . k f k ℓ k . 1, this bounds the left hand side of (6.42) by N q + O ( √ δ )2 , which also sufficesas δ is small enough depending on p . PTIMAL LOCAL WELL-POSEDNESS FOR THE DNLS EQUATION 27 (2) Next we will prove (6.39) and (6.40). By (6.37) and (4.6) ∼ (4.7), we can write (where ∗ ∈ { N, L } ) | e R ( k, λ ) | . X ± k ′ ± k ±···± k = k Z R d σ h λ i − θ h λ − σ i − θ Z ± λ ′ ± λ ±···± λ = σ − Ξ 7 Y j =4 | e v j ( k j , λ j ) | X k + k − k = k ′ ( k ,k ,k ) ∈ X ∗ | k | Z R T b ϕ ( T ( λ ′ − µ ′ )) d µ ′ Z R min (cid:18) h ∆ ′ i , h µ ′ i (cid:19) d σ ′ h µ ′ − σ ′ i Z λ + λ − λ = σ ′ − ∆ ′ Y j =1 | e v j ( k j , λ j ) | . (6.43)Here Ξ = k ∓ ( k ′ ) ∓ k ∓ · · · ∓ k , and ∆ ′ = ( k ′ ) + k − k − k . This can be reduced to X ± k ±···± k = k | k || ∆ ′ | Z R R ( λ, τ ) d τ Z ± λ ±···± λ = τ − Ξ ′ Y j =1 | e v j ( k j , λ j ) | , (6.44)where Ξ ′ = k ∓ k ∓ · · · ∓ k , and the kernel | R ( λ, τ ) | . Z R d σ h λ i − θ h λ − σ i − θ Z R T b ϕ ( T ( ξ − σ )) d ξ h τ − ξ i . Here we can verify that λ ′ − σ ′ = τ − σ , and ξ is the variable such that ξ − σ = λ ′ − µ ′ and τ − ξ = µ ′ − σ ′ . Using the fact that | T b ϕ ( T ξ ) | . h ξ i − , we can easily bound the above by | R ( λ, τ ) | . h λ i − θ h λ − τ i − θ . We may then restrict to the dyadic region h k i ∼ N , h k ′ i ∼ N ′ and h k j i ∼ N j for 1 ≤ j ≤ 7. Let N + be the maximum of all the N j ’s. Then we have N ′ & N , and | ∆ ′ | ∼ N ′ N .(a) Assume N . N ′ , we will then measure v in the Y norm. By repeating the above proof andfixing λ j for 1 ≤ j ≤ 7, we may reduce to proving k F ( k, Ξ ′ ) k ℓ p k ℓ ∞ Ξ ′ . N − N ′ N · ( N · · · N ) (1 −√ δ ) /p − (1 /q ) ( N + ) − θ , (6.45)where F ( k, Ξ ′ ) = X ± k ±···± k = k ± k ±···± k = k − Ξ ′ Y j =1 f j ( k j ) , (6.46)and k f k ℓ p k ≤ , k f j k ℓ k ≤ ≤ j ≤ . (6.47)First assume max( N , · · · , N ) ≥ ( N + ) √ δ , then with fixed k , k and Ξ, by using Lemma 3.4 andsimilar arguments as in the above proof, we can easily show that (whether or not there is anypairing in { , , · · · , } ) X ± k ±···± k =const . ± k ±···± k =const . Y j =2 f j ( k j ) . max( N , · · · , N ) θ Y j =2 k f j k ℓ k , (6.48)therefore | F ( k, Ξ ′ ) | . max( N , · · · , N ) θ X k f ( k ) . max( N , · · · , N ) θ N − ( p )18 YU DENG, ANDREA R. NAHMOD, AND HAITIAN YUE pointwise in ( k, Ξ ′ ), thus k F ( k, Ξ ′ ) k ℓ p k ℓ ∞ Ξ ′ . N p N − ( p )1 . Using the fact that N ′ & max( N , N ) and max( N , · · · , N ) ≥ ( N + ) √ δ , this easily implies (6.45).Next assume max( N , · · · , N ) ≤ ( N + ) √ δ , then N + ∼ N ′ , and by fixing ( k , · · · , k ) we easilydeduce that | F ( k, Ξ ′ ) | . ( N ′ ) O ( √ δ ) , from which (6.45) follows trivially.(b) Assume N ≫ N ′ , then we must have ∗ = N and N ∼ N ≫ N ′ & N . In particular v willbe measured in the Z norm, so we may reduce to proving k F ( k, Ξ ′ ) k ℓ p k ℓ ∞ Ξ ′ . N − N ′ N · ( N · · · N ) (1 −√ δ ) /p ( N + ) − θ − (6 /q ) , (6.49)where F ( k, Ξ ′ ) is as (6.46), and k f k ℓ p k ≤ , k f k ℓ p k ≤ , k f j k ℓ k ≤ ≤ j ≤ . (6.50)Here we argue in the same way as case (1), using (6.48), but make the additional observation thatfor fixed k we must have | k − k | ∼ N ′ . Therefore X ± k ±···± k =const . ± k ±···± k =const . Y j =2 f j ( k j ) . ( N + ) θ Y j =3 k f j k ℓ k · k f · | k − k |∼ N ′ k ℓ k . ( N + ) θ ( N ′ ) ( ) − ( p ) , and hence k F ( k, Ξ ′ ) k ℓ p k ℓ ∞ Ξ ′ . N p N − ( p )1 ( N ′ ) ( ) − ( p ) ( N + ) θ . Using the fact that N . N ′ , this implies (6.49). (cid:3) Remark . From the proof above we actually deduce something slightly stronger: the bounds(6.38) ∼ (6.40) remain true if the right hand side of (6.37) gets multiplied by h k + i θ where | k + | isthe maximum of all relevant frequencies, unless h k i ∼ h k i & N . This fact will be used in theanalysis of the z ∗ terms of Section 6.4.To apply Proposition 6.5, we will verify that z , z ℓ (2 ≤ ℓ ≤ z ℓ (1 ≤ ℓ ≤ 4) and z ℓ (1 ≤ ℓ ≤ 2) all have the form (6.37). The claim for z follows from (3.9). For the other terms, let us lookat z as an example. By (3.9) and (4.6) we have, for z = z ( v , · · · , v ), that | f z ( k, λ ) | . B X k + k − k ′ = k ( k ′ ,k ,k ) ∈ X H ∪ X S | k ′ | Z R (cid:18) h λ i B + 1 h λ − τ i B (cid:19) d τ h τ i Z λ + λ − λ ′ = τ − ∆ | e v ( k , λ ) || e v ( k , λ ) | X k + k − k = k ′ ( k ,k ,k ) ∈ X N | k | Z R T b ϕ ( T ( λ ′ − µ ′ )) d µ ′ Z R min (cid:18) h ∆ ′ i , h µ i (cid:19) h µ ′ − σ ′ i d σ ′ Z λ + λ − λ = σ ′ − ∆ ′ Y j =1 | e v j ( k j , λ j ) | , (6.51)where ∆ = k + ( k ′ ) − k − k and ∆ ′ = ( k ′ ) + k − k − k . Note that | ∆ ′ | ∼ h k ih k i , the abovecan be written as X k − k − k + k + k = k Z R R ( λ, σ ) d σ Z λ − λ − λ + λ + λ = σ − Ξ 5 Y j =1 | e v j ( k j , λ j ) | , where Ξ = ∆ − ∆ ′ = k − k + k + k − k − k , and | R ( λ, σ ) | . Z R (cid:18) h λ i B + 1 h λ − τ i B (cid:19) d τ h τ i Z R T b ϕ ( T ( σ − ξ )) . d ξ h ξ − τ i PTIMAL LOCAL WELL-POSEDNESS FOR THE DNLS EQUATION 29 Here we can verify that λ ′ − σ ′ = σ − τ , and ξ is the variable such that σ − ξ = λ ′ − µ ′ and ξ − τ = µ ′ − σ ′ . The above integral can easily be bounded by h λ i − θ h λ − σ i − θ , so Proposition6.5 can be applied.The other z ℓ , z ℓ and z ℓ terms can be treated in the same way; in fact the kernel R ( λ, σ ) willhave exactly the same form, the only difference is that the weight | k | · | k || ∆ ′ | will be replaced by different weights depending on which input function gets substituted by E Y ,and which X ∗ subset we are in. For the terms z , z , z , z , z and z one can directly checkthat this weight is . 1; for the terms z , z , z and z , this weight is . | k | & | k | when ( k , k , k ) ∈ X H ∪ X S ∪ X N . Thus Proposition 6.1 has beenproved for these terms.6.4. Remaining quintic terms. The remaining quintic terms, namely z , z and quintic z ∗ terms, may not have the canonical form (6.37). In fact these terms will be estimated directlywithout preforming the operation in step (6) of Section 6.1, see Remark 6.2. For them we need twoextra estimates, stated in the following two propositions. Proposition 6.7. Suppose a quintic term R satisfies | e R ( k, λ ) | . X ± k ±···± k = k α ( k, k , · · · , k ) Z R h λ i − θ h λ − σ i − θ Z ± λ ±···± λ = σ − Ξ 5 Y j =1 | e v j ( k j , λ j ) | , (6.52) where as usual Ξ = k ∓ k ∓ · · · ∓ k . Then we have the following (below | k + | will denote themaximum of all relevant frequencies):(1) Assume | k | / ≤ | k | ≤ | k | , | k | ≥ | k | , | k | ∼ max( | k | , | k | , | k | ) and | α | . h k + i θ h k i max( h k i , h k i , h k i , h k i ) , (6.53) moreover assume (1 , is not a pairing. Then we have that kRk Z . k v k Z k v k Z k v k Z · k v k Y k v k Y ; (6.54) (2) Assume | k | / ≤ | k | ≤ | k | , | k | / ≤ | k | ≤ | k | , | k | ≥ max( | k | , | k | ) and | α | . h k + i θ h k i max( h k i , h k i ) . (6.55) Moreover assume (1 , is not a pairing, and that, either | k | 6 = | k | , or the stronger bound | α | . h k + i θ h k i max( h k i , h k i , h± k ± k i ) (6.56) holds. Then we have kRk Z . k v k Z k v k Z k v k Z k v k Z · k v k Y . (6.57) Proof. We may restrict to the region where h k j i ∼ N j , where 1 ≤ j ≤ 5, and h k i ∼ N and h k + i ∼ N + .(1) By the same arguments as in the proof of Propositions 5.1 and 6.5, we may fix λ j (1 ≤ j ≤ (cid:13)(cid:13)(cid:13)(cid:13) h λ i b X ± k ±···± k = k Z R h λ i − θ h λ − Ξ − µ i − θ Y j =1 f j ( k j ) (cid:13)(cid:13)(cid:13)(cid:13) ℓ p k L q λ . ( N + ) − θ N ′ N ( N N ) (1 −√ δ ) /p N , (6.58) where N ′ = max( N , N ), and f j satisfies that k f j k ℓ p k ≤ , ≤ j ≤ k f j k ℓ k ≤ , ≤ j ≤ . (6.59)By the same argument as in the proof of Proposition 6.5, we may apply Schur’s estimate and reduceto proving k F ( k, Ξ) k ℓ p k ℓ q . ( N + ) − θ N ′ N ( N N ) (1 −√ δ ) /p N , F ( k, Ξ) := X ± k ±···± k = k ± k ±···± k = k − Ξ 5 Y j =1 f j ( k j ) . (6.60)By fixing ( k , k ) we get that k F ( k, Ξ) k ℓ q . k f k ℓ k k f k ℓ k sup ℓ,ρ k F ℓ,ρ ( k, Ξ) k ℓ p k ℓ q , F ℓ,ρ ( k, Ξ) := X ± k ± k ± k = k + ℓ ± k ± k ± k = k − Ξ+ ρ Y j =1 f j ( k j ) , while since there is no pairing in { , , } , by the standard divisor estimate we have | F ℓ,ρ ( k, Ξ) | . ( N + ) θ (cid:18) X ± k ± k ± k = k + ℓ ± k ± k ± k = k − Ξ+ ρ Y j =1 f j ( k j ) p (cid:19) p , and hence k F ℓ,ρ ( k, Ξ) k ℓ p k ℓ q . k F ℓ,ρ ( k, Ξ) k ℓ p k ℓ p . ( N + ) θ . Using also H¨older we obtain k F ( k, Ξ) k ℓ p k ℓ q . ( N N ) ( N + ) θ . Comparing with (6.60) and using that N ′ ∼ max( N , N ) and max( N , N ) . N , we see that(6.60) is proved, except for the loss ( N + ) θ . Clearly this loss can be covered if N ′ & N / ; nowsuppose max( N , N , N , N ) ≪ N / , then since (1 , 2) is not a pairing, we must have | Ξ | & N + ,which gives max( | λ | , · · · , | λ | , | λ | , | λ − Ξ − µ | ) & N + , where µ is a linear combination of λ , · · · λ . Now, in estimating (6.58) we can gain a power h λ i (1 − θ ) − b ≥ h λ i δ/ ; in the process of fixing λ j we can also gain a power h λ j i δ/ , as kh λ j i δ/ h k j i (1 −√ δ ) /p e v j k L λ ℓ k . kh λ j i δ/ h k j i (1 −√ δ ) /p e v j k L λ ℓ r k . kh k j i (1 −√ δ ) /p h λ j i e v j k L r λ ℓ r k . kh k j i (1 −√ δ ) /p h λ j i e v j k ℓ r k L r λ . kh k j i h λ j i e v j k ℓ p k L r λ . Finally, in the process of using Schur’s estimate to reduce (6.58) to (6.60), we can also replace thepower h λ − Ξ − µ i − θ by a slightly larger power gain a power h λ − Ξ − µ i δ/ . In this way we cangain a power of at least ( N + ) δ/ which suffices to cover the ( N + ) θ loss.(2) If there is no pairing in { , , } , then similar to (1), we may fix λ j and reduce to proving k F ( k, Ξ) k ℓ p k ℓ q . ( N + ) − θ N ′ N · N (1 −√ δ ) /p N , F ( k, Ξ) := X ± k ±···± k = k ± k ±···± k = k − Ξ 5 Y j =1 f j ( k j ) , (6.61)where N + = max( N , N ), N ′ = max( N , N ) and k f j k ℓ p k ≤ , ≤ j ≤ k f k ℓ k ≤ . (6.62)Then we may fix k and k and argue as in part (1) to get k F ( k, Ξ) k ℓ p k ℓ q . ( N + ) θ k f k ℓ k k f k ℓ k . ( N + ) θ N − ( p )3 N , PTIMAL LOCAL WELL-POSEDNESS FOR THE DNLS EQUATION 31 which implies (6.61) except for the loss ( N + ) θ , which can be covered in the same way as part (1)by considering Ξ.If there is a pairing in { , , } , say (1 , ≤ N /N ≤ 2. If (2 , 4) is not a pairing, thenwe can fix ( k , k ) and repeat the above argument to get the same (in fact better) estimate; sowe may assume (2 , 4) is also a pairing. This forces Ξ = 0 and k = k , in particular the strongerbound (6.56) holds. Let h± k ± k i ∼ N and N ′′ = max( N ′ , N ). In this case we will still fix λ j (1 ≤ j ≤ 4) but will not fix λ . Instead, let µ be a linear combination of λ j (1 ≤ j ≤ 4) and isthus fixed, and notice that Ξ = 0, we have | e R ( k, λ ) | . N − ( N ′′ ) − X | k |∼| k |∼ N |± k ± k |∼ N Z R d σ h λ i − θ h λ − σ i − θ f ( k ) f ( k ) f ( k ) f ( k ) | e v ( k, ± σ ± µ ) | , where k f j k ℓ p k ≤ ≤ j ≤ 4. This implies that kh λ i b e R ( k, λ ) k L q λ . N − ( N ′′ ) − X | k |∼| k |∼ N |± k ± k |∼ N f ( k ) f ( k ) f ( k ) f ( k ) kh λ i e v ( k, λ ) k L r λ , and hence kh k i h λ i b e R ( k, λ ) k ℓ p k L q λ . N − ( N ′′ ) − kh k i h λ i e v ( k, λ ) k ℓ p k L r λ · X | k |∼| k |∼ N |± k ± k |∼ N f ( k ) f ( k ) f ( k ) f ( k ) , while the latter sum is bounded by X k f ( k ) f ( k ) · N − (2 /p )6 k f k ℓ pk k f k ℓ pk . ( N N ) − (2 /p ) , which gives the desired estimate as N ′′ & N . (cid:3) Proposition 6.8. Suppose a quintic term R satisfies | e R ( k, λ ) | . h λ i δ X ± k ±···± k = k β ( k, k , · · · , k ) Z R Y j =1 | e v j ( k j , λ j ) | d λ · · · d λ , (6.63) where (as usual | k + | is the maximum of all relevant frequencies) | β | . h k + i √ δ h k ih± k ± k i , h k i & h k i . (6.64) then we have kRk Z . Y j =1 k v j k Y . (6.65) Note that all the norms on the right hand side are Y (in particular the bound is symmetric in v and v , v and v ).Proof. As before we will restrict to the region where h k j i ∼ N j for 1 ≤ j ≤ h k i ∼ N , and h± k ± k i ∼ N . Let N + ∼ h k + i . This time we will not fix λ j ; instead we first integrate in them.We may assume all the norms on the right hand side are 1. Let N j k e v j ( k j , λ j ) k L λ = f j ( k j ) , then k f j k ℓ p k . v j ∈ Y . By (6.63), it suffices to prove that (cid:13)(cid:13)(cid:13)(cid:13) h λ i b h λ i − − δ X ± k ±···± k = k Y j =1 f j ( k j ) (cid:13)(cid:13)(cid:13)(cid:13) ℓ p k L q λ . ( N + ) − √ δ · N ( N N N N N N ) , (6.66)where k f j k ℓ p k ≤ , ≤ j ≤ . (6.67)By symmetry we may assume N ≤ N and N ≤ N . By the choice of power of λ , the L q λ part iseasily estimated, so we only need to bound the ℓ p k norm (cid:13)(cid:13)(cid:13)(cid:13) X ± k ±···± k = k Y j =1 f j ( k j ) (cid:13)(cid:13)(cid:13)(cid:13) ℓ p k . By Young’s inequality, this is bounded by the ℓ p k of f (which is ∼ X k ,k ,k ,k f ( k ) f ( k ) f ( k ) f ( k ) . The sum over k and k gives (by H¨older) ( N N ) − ( p ) ; when k is fixed the sum over k gives N − ( p )6 as | ± k ± k | . N , and finally the sum over k gives N − ( p )3 . This gives the bound( N N N N ) − ( p ) . ( N + ) − / (2 p ) N ( N N N N N N ) , as N ≤ N , N ≤ N and N & N . (cid:3) Using Propositions 6.7, we can easily deal with the terms z and z . For these two terms, byrepeating the arguments for z detailed above, we are led to considering the tuple ( k , k , k ′ ) and( k , k , k ), such that k + k ′ − k = k, ( k , k , k ′ ) ∈ X H ∪ X S ; k + k − k = k ′ , ( k , k , k ) ∈ X N ∪ X L , and a weight α ( k, k , · · · , k ) ∼ | k || k |h ∆ ′ i , noticing that | ∆ ′ | ≥ 1. By Proposition 2.4, this term can be bounded using either Proposition 6.5,or Proposition 6.7, (1) or (2).6.4.1. The z ∗ terms. Finally let us consider quintic z ∗ terms. By (4.6) ∼ (4.7) and (4.13) we write,where z ∗ = z ∗ ( v , · · · , v ) and ∗ , • ∈ { N, L } , that (strictly speaking z and z have a differentformula, but taking into account that the set X L is symmetric with respect to k and k - apartfrom the artificial restriction | k | ≥ | k | - they can be treated in exactly the same way): | f z ∗ ( k, λ ) | . B X k + k ′ − k = k ( k ,k ,k ′ ) ∈ X ∗ | k | Z R (cid:20) h λ i B h τ i + h τ − ∆ ih λ − τ i B h τ i min (cid:18) h ∆ i , h τ i (cid:19) + h τ − ∆ ih λ − τ i min (cid:18) h ∆ i , h λ i (cid:19) (cid:21) d τ Z λ + λ ′ − λ = τ − ∆ | λ ′ | & | τ − ∆ | | e v ( k , λ ) || e v ( k , λ ) | X k + k − k = k ′ ( k ,k ,k ) ∈ X • | k | Z R T b ϕ ( T ( λ ′ − µ ′ )) d µ ′ Z R min (cid:18) h ∆ ′ i , h µ ′ i (cid:19) h µ ′ − σ ′ i d σ ′ Z λ + λ − λ = σ ′ − ∆ ′ | e v ( k , λ ) || e v ( k , λ ) || e v ( k , λ ) | , (6.68) PTIMAL LOCAL WELL-POSEDNESS FOR THE DNLS EQUATION 33 where ∆ = k + k − k − ( k ′ ) and ∆ ′ = ( k ′ ) + k − k − k . The above can be reduced to X − k + k − k + k + k = k Z R R ( λ, σ ) d σ Z − λ + λ − λ + λ + λ = σ − Ξ 5 Y j =1 | e v j ( k j , λ j ) | , (6.69)where Ξ = ∆ + ∆ ′ = k + k − k + k − k − k , and the kernel R ( λ, σ ) = Z R (cid:20) h λ i B h τ i + h τ − ∆ ih λ − τ i B h τ i min (cid:18) h ∆ i , h τ i (cid:19) + h τ − ∆ ih λ − τ i min (cid:18) h ∆ i , h λ i (cid:19) (cid:21) d τ | k k | | λ ′ | & | τ − ∆ | Z R T b ϕ ( T ( λ ′ − µ ′ )) min (cid:18) h ∆ ′ i , h µ ′ i (cid:19) h µ ′ − σ ′ i d µ ′ . (6.70)Here λ ′ = τ − ∆ + λ − λ and σ ′ = λ + λ − λ + ∆ ′ are defined in terms of τ and ( k j , λ j ). Firstfix τ and integrate in µ ′ ; this integral is bounded by Z R h λ ′ − µ ′ i min (cid:18) h ∆ ′ i , h µ ′ i (cid:19) h µ ′ − σ ′ i d µ ′ , and we separate two cases.(1) Assume | σ ′ | ≪ | λ ′ | , then we can calculate that1 h ∆ ′ i Z R h λ ′ − µ ′ ih µ ′ − σ ′ i d µ ′ . h ∆ ′ i h λ ′ − σ ′ i − θ ∼ h ∆ ′ i h λ ′ i − θ . Note that | λ ′ | & | τ − ∆ | , we can then bound the resulting integral in τ by Z | τ − ∆ | . | λ − λ | (cid:20) h λ i B h τ i + h τ − ∆ ih λ − τ i B h τ i min (cid:18) h ∆ i , h τ i (cid:19) + h λ − τ i≪h τ − ∆ i h τ − ∆ ih λ − τ i min (cid:18) h ∆ i , h λ i (cid:19) (cid:21) h ∆ ′ i h τ − ∆ i − θ d τ, which is then bounded by 1 h ∆ ih ∆ ′ i (max j h k j i ) √ δ h λ i − − δ by actually performing the integration in τ . By bounding the weight β = | k || k |h ∆ ih ∆ ′ i using Proposition 2.4, we can apply Proposition 6.8 and conclude the estimate for this term.(2) Assume | σ ′ | & | λ ′ | , then we can calculate that Z R h λ ′ − µ ′ ih µ ′ ih µ ′ − σ ′ i d µ ′ . h λ ′ i − θ h λ ′ − σ ′ i − θ . Note that λ ′ − σ ′ = τ − σ , and using the fact that | λ ′ | & | τ − ∆ | , we can bound the resulting integralin τ by Z R (cid:20) h σ i≪h ∆ i h λ i B h τ i + h τ − ∆ ih λ − τ i B h τ i min (cid:18) h ∆ i , h τ i (cid:19) + h τ − ∆ ih λ − τ i min (cid:18) h ∆ i , h λ i (cid:19) (cid:21) h τ − ∆ i − θ h τ − σ i − θ d τ, which can be bounded by (max j h k j i ) θ h λ i − θ h λ − σ i − θ h ∆ i . This kernel depends on k j and λ j , but we will write it as R ( λ, σ ) for simplicity. By bounding the weight α = | k || k |h ∆ i using Proposition 2.4, we can apply either Proposition 6.5, or Proposition 6.7, (1) or (2).In the case we apply Proposition 6.5, we will also use Remark 6.6 to cover the loss (max j h k j i ) θ ,which can be done unless for some j we have | k j | ∼ | k | & max ℓ = j | k l | ; in this final case we cancheck that the stronger bound | α | . | k | − holds, so the loss can still be covered. This completesthe proof of Proposition 6.1. 7. Preservation of regularity Finally in this section we prove a preservation of regularity result. More precisely, we prove theproperties of our solution stated in Remark 1.2. The following proposition is standard: Proposition 7.1. Given s > and ≤ p < ∞ , all the arguments in the previous sections carryover to H sp (and correspondingly X s,b j p ,q and X s, p ,r j for j ∈ { , } ). Moreover, in these arguments T still depends only on the H p (instead of H sp ) size of the initial data.Proof. This follows from the elementary inequality that h k i s − ( ) . max ≤ j ≤ r h k j i s − ( ) , if k = ± k ± · · · ± k r , r ∈ { , , } . Thus, any previously proved multilinear estimate will continue to be true if the exponent in theoutput function space is replaced by s , provided that the exponent in one appropriate inputfunction space is replaced by s .Suppose the initial data has H p norm A and H sp norm L , then for T = T ( A ), all the X ,b j p ,q ( I )- and similarly for X , p ,r j ( I ) - contraction mappings proved before will still be contraction mappingunder the norm k · k X ,bjp ,q ( I ) + L − k · k X s,bjp ,q ( I ) , similarly k · k X , p ,rj ( I ) + L − k · k X s, p ,rj ( I ) . (cid:3) Now consider a smooth initial data u . Proposition 7.1 implies that, if k u k H p ≤ A , then for T = T ( A ), we can construct a solution to (1.1) on J = [ − T, T ] that belongs to C t H sp ( J ) for s sufficiently large. This is clearly the classical solution to (1.1). If a sequence of smooth initial data u ( n )0 satisfies k u ( n )0 k H p ≤ A and u ( n )0 → u in H p , then by continuity of the data-to-solution mapin H p (which follows from the previous proofs), the corresponding solutions u ( n ) will converge to u in C t H p , where u is the solution we construct in Theorem 1.1 with initial data u . This showsthat our solution is the unique limit of smooth solutions.Finally, suppose p < 4, then the gauged solution v we construct belongs to the space X , p ,r ( J )where r < 2. It can be shown that X , p ,r ( J ) ⊂ X , p , ( J ) ∩ X , p , ( J )for any interval J of length not exceeding 1, so our solution belongs to the function space definedin [25] in which the authors have proved uniqueness. Therefore when p < 4, our solution mustcoincide with the one constructed in [25], as long as the latter exists. PTIMAL LOCAL WELL-POSEDNESS FOR THE DNLS EQUATION 35 References [1] Bailleul, I. and Bernicot, F., Heat semigroup and singular PDEs , J. Funct. 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