A sharpened Strichartz inequality for the wave equation

Abstract

We disprove a conjecture of Foschi, regarding extremizers for the Strichartz inequality with data in the Sobolev space ${\dot{H}}^{1/2}\times {\dot{H}}^{-1/2}({\mathbb{R}}^d)$, for even $d\ge 2$. On the other hand, we provide evidence to support the conjecture in odd dimensions, and refine his sharp inequality in ${\mathbb{R}}^{1+3}$, adding a term proportional to the distance of the initial data from the set of extremizers. The proofs use the conformal compactification of the Minkowski space-time given by the Penrose transform.

23 pages, 2 tables, 1 figure. By editorial request, Section 6 from the previous versions has been removed, to improve readability. To appear in Annales Scientifiques de l'École Normale Supérieure

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