An inverse theorem for the bilinear $L^2$ Strichartz estimate for the wave equation

Details An inverse theorem for the bilinear $L^2$ Strichartz estimate for the wave equation An inverse theorem for the bilinear $L^2$ Strichartz estimate for the wave equation

2009-04-19

arxiv.org/abs/0904.2880v1

Mathematics

Analysis of PDEs

20 pages, no figures. To be submitted in conjunction with other "heatwave" papers

Scientific paper

A standard bilinear $L^2$ Strichartz estimate for the wave equation, which underlies the theory of $X^{s,b}$ spaces of Bourgain and Klainerman-Machedon, asserts (roughly speaking) that if two finite-energy solutions to the wave equation are supported in transverse regions of the light cone in frequency space, then their product lies in spacetime $L^2$ with a quantitative bound. In this paper we consider the \emph{inverse problem} for this estimate: if the product of two waves has large $L^2$ norm, what does this tell us about the waves themselves? The main result, roughly speaking, is that the lower-frequency wave is dispersed away from a bounded number of light rays. This result will be used in a forthcoming paper \cite{tao:heatwave4} of the author on the global regularity problem for wave maps.

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