Multilinear weighted convolution of L² functions, and applications to non-linear dispersive equations

2000/05/31 by Terence Tao · 5 citations
Mathematics · #math.AP #msc:35G25 #msc:42B35 #msc:35L70 #msc:35Q53 #msc:35Q55

paper · pdf

published as Amer. J. Math. 123 (2001), 839-908 · 50 pages. An incorrect estimate has been fixed

arxiv created 2004/03/04 · arxiv updated 2009/11/30

Abstract

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 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.

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