Multilinear weighted convolution of $L^2$ functions, and applications to non-linear dispersive equations

Terence Tao

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Multilinear weighted convolution of $L^2$ functions, and applications to non-linear dispersive equations

Terence Tao

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Abstract

The $X^{s,b}$ spaces, as used by Beals, Bourgain, Kenig-Ponce-Vega, Klainerman-Machedon and others, are fundamental tools to study the low-regularity 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.

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Terence Tao. 2004-03-04. Multilinear weighted convolution of $L^2$ functions, and applications to non-linear dispersive equations. https://arxiv.org/abs/math/0005001

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