Multilinear weighted convolution of \(L^2\) functions, and applications to nonlinear dispersive equations. (English) Zbl 0998.42005

Let \(G\) denote either \(\mathbb R^d\) or \(\mathbb T^d\) for \(d\in \mathbb N^{\ast}\). For \(f\in L^1(G)\), denote by \(\widehat{f}\) its Fourier transform on the dual group \(Z\). If \(h\) is a suitable function on \(Z\), let \(H\) be the associated Fourier multiplier, given by \(\widehat{Hf}(\xi)=h(\xi)\widehat{f}(\xi)\). The paper deals with nonlinear Cauchy problems of the form (Cauchy) \(\varphi_t=2i\pi H\varphi+F(\varphi)\) with initial data \(\varphi(0)=\varphi_0\) in some Sobolev space \(H^s\) and \(F\) is a non-linearity. The study of these problems involves the \(X^{s,b}\) spaces, which are spaces of functions \(\varphi\) on \(G\times \mathbb R\) such that \[ \|\varphi\|_{X^{s,b}(G\times \mathbb R)} = \|\langle \xi\rangle^s \langle \tau-h(\xi)\rangle^b \widehat{\varphi}(\xi,\tau)\|_{L^2(G^{\ast}\times \mathbb R)}<+\infty, \] where \(\langle \xi\rangle =(1+|\xi|^2)^{1/2}\) and \(G^{\ast}\) is the dual group of \(G\).
These \(X^{s,b}\) spaces have already been introduced in the study of the wave equation by M. Beals [Ann. Math. (2) 118, 187-214 (1983; Zbl 0522.35064)], and were applied to local well-posedness problems by J. Bourgain [Geom. Funct. Anal. 3, No. 2, 107–156 (1993; Zbl 0787.35097); ibid., No. 3, 209-262 (1993; Zbl 0787.35098)]. Indeed, local well-posedness results for the Cauchy problem (cf. (Cauchy)) may be consequences of estimates in \(X^{s,b}\) spaces for multilinear operators \(F\), see [J. Bourgain (log. cit.)], [C. E. Kenig, G. Ponce and L.Vega, J. Am. Math. Soc. 9, No. 2, 573–603 (1996; Zbl 0848.35114)], and [S. Klainerman, Prog. Nonlinear Differ. Equ. Appl. 29, 29–69 (1997; Zbl 0909.53051)]).
In the case when the operator \(F\) is translation-invariant, this issue reduces to multiplier estimates. More precisely, let \(k\geq 2\) and \(\Gamma_k(Z)=\{(\xi_1,\dots ,\xi_k)\in Z^k; \xi_1+\dots +\xi_k=0\}\), equipped with the measure \[ \int_{\Gamma_k(Z)} f=\int_{Z^{k-1}} f(\xi_1,\dots ,\xi_{k-1}, -\xi_1-\dots -\xi_{k-1}) d\xi_1\dots d\xi_{k-1}. \] If \(m\) is a complex-valued function on \(\Gamma_k(Z)\), let \(\|m\|_{[k;Z]}\) be the best constant such that the inequality \[ \left |\int_{\Gamma_k(Z)} m(\xi)\prod\limits_{j=1}^k f_j(\xi_j)\right|\leq \|m\|_{[k;Z]} \prod\limits_{j=1}^k \|f_j\|_{L^2(Z)} \] is true for all functions \(f_j\) on \(Z\).
The author gives a comprehensive study of \(\|m\|_{[k;Z]}\) when \(k=3\) (the bilinear case). Sharp estimates for the KdV equation, the wave equation and the Schrödinger equation are derived.


42B15 Multipliers for harmonic analysis in several variables
35Q53 KdV equations (Korteweg-de Vries equations)
35L05 Wave equation
35Q55 NLS equations (nonlinear Schrödinger equations)
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