Multilinear weighted convolution of L 2 functions, and applications to nonlinear dispersive equations

Abstract

Abstract. The X s,b spaces, as used by Beals, Bourgain, Kenig-Ponce-Vega, Klainerman-Machedon and others, are fundamental tools to study the lowregularity behaviour of non-linear dispersive equations. It is of particular interest to obtain bilinear or multilinear estimates involving these spaces. By Plancherel's theorem and duality, these estimates reduce to estimating a weighted convolution integral in terms of the L 2 norms of the component functions. In this paper we systematically study weighted convolution estimates on L 2 . As a consequence we obtain sharp bilinear estimates for the KdV, wave, and Schrödinger X s,b spaces. IntroductionLet Z be any abelian additive group with an invariant measure dξ. For instance, Z could be Euclidean space R d+1 with Lebesgue measure, or the space Z d × R with the product of counting and Lebesgue measure.For any integer k ≥ 2, we let Γ k (Z) denote the "hyperplane"with we endow with the obvious measureNote that this measure is symmetric with respect to permutation of the co-ordinates.We define a [k; Z]-multiplier to be any function m : ] to be the best constant such that the inequalityholds for all test functions f i on Z. It is clear that m [k;Z] determines a norm on m, for test functions at least; we are interested obtaining good bounds on this norm. We will also define m [k;Z] in situations when m is defined on all of Z k by restricting to Γ k (Z). This general problem occurs frequently in the study of non-linear dispersive equations in both the periodic and non-periodic setting. For instance, let G be either R d or T d for some d ≥ 1, and let H be given by a real Fourier multiplier h(ξ) onwhere the Fourier transformf is defined 1 bŷWe consider non-linear Cauchy problems of the formwhere φ = φ(x, t) is a field on G×R which can either be scalar or vector-valued, the initial data φ 0 lives in some Sobolev space H s , and F is a nonlinearity containing second-order and higher order terms. We call the equation τ = h(ξ) the dispersion relation of the Cauchy problem.Examples of such problems include the modified KdV family of Cauchy problems, and non-linear Schrödinger Cauchy problemsin which h(ξ) := |ξ| 2 , and F is some polynomial in the indicated variables. Nonlinear wave (and Klein-Gordon equations) can also be placed in this framework, by writing a second-order wave equation as a first order system and setting h(ξ) = ±|ξ|.Experience has shown that if the regularity H s of the initial data is sub-critical (i.e. if s > s c , whereḢ sc is the scale-invariant regularity), then the Cauchy problem (2) can often be satisfactorily studied using the X s,b τ =h(ξ) (G × R) spaces 2 , which are spaces of functions on G × R defined via the Fourier transform aswhere ξ := (1 + |ξ| 2 ) 1/2 and G * is the dual group of G. For brevity we shall often abbreviate XIndeed, one can usually obtain local well-posedness in (2) by the method of Picard iteration provided that one can prove a multilinear estimate such asfor some b > 1/2, assuming that

A convolution estimate for two-dimensional hypersurfaces

“…We remark that the linear transformation T does not explicitely appear in the estimate (9). Instead, the size…”

“…Other examples are the bounds for the KP-I equation considered in [7]. A large class of bilinear and multilinear estimates have been systematically studied in [9]; however, this does not include the present setup.…”

Well-posedness for the nonlinear fractional Schrödinger equation and inviscid limit behavior of solution for the fractional Ginzburg-Landau equation

“…The trilinear estimates will be obtained by using [k; Z]-multiplier method [10]. We firstly list some useful notations and properties for multi-linear expressions.…”

Global Well-Posedness for Dissipative Korteweg-de Vries Equations

“…; and using Tao's ½k; Z-multiplier norm estimates [14], one could get wellposedness results for the IVP (1.8) in H s ðTÞ, s >s s a withs s 0 ¼ À1=2 and s s 1 ¼ À1. We do not pursue this issue here.…”