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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.

##### MSC:
 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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