Multilinear weighted convolution of L2 functions, and applications to nonlinear dispersive equations

<abstract abstract-type="TeX"><p>The <i>X<sup>s,b</sup></i> spaces, as used by Beals, Bourgain, Kenig-Ponce-Vega, Klainerman-Machedon and others, are fundamental tools to study the low-regularity behavior of nonlinear 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 <i>L</i><sup>2</sup> norms of the component functions. In this paper we systematically study weighted convolution estimates on <i>L</i><sup>2</sup>. As a consequence we obtain sharp bilinear estimates for the KdV, wave, and Schrödinger <i>X<sup>s,b</sup></i> spaces.

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