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

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 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 L² norms of the component functions. In this paper we systematically study weighted convolution estimates on L². As a consequence we obtain sharp bilinear estimates for the KdV, wave, and Schrödinger X^s,b spaces.