An inverse theorem for the bilinear L² Strichartz estimate for the wave equation

2009/04/19 by Terence Tao, Tao, Terence · 2 citations
Mathematics · #35L05 #Advanced Harmonic Analysis Research #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Mathematical Analysis and Transform Methods

paper · pdf · doi:10.48550/arxiv.0904.2880

openalex publication_date 2009/04/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

A standard bilinear L² 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 L² with a quantitative bound. In this paper we consider the inverse problem for this estimate: if the product of two waves has large L² 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 \citetao:heatwave4 of the author on the global regularity problem for wave maps.

Citations

• Space‐time estimates for null forms and the local existence theorem

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