  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 Schr\"odinger $X^{s,b}$ spaces.
