Functions, and Applications to Non-linear 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 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.