A sharpened Strichartz inequality for the wave equation

We disprove a conjecture of Foschi, regarding extremizers for the Strichartz inequality with data in the Sobolev space ${\dot{H}}^{1\mathrm{/}2}\times {\dot{H}}^{-1\mathrm{/}2}({\mathbb{R}}^d)$, 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.