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Multilinear weighted convolution of L2 functions, and applications to nonlinear dispersive equations
- American Journal of Mathematics
- Johns Hopkins University Press
- Volume 123, Number 5, October 2001
- pp. 839-908
- 10.1353/ajm.2001.0035
- Article
- Additional Information
The Xs,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 Xs,b spaces.