On sharp bilinear Strichartz estimates of Ozawa–Tsutsumi type

Abstract

We provide a comprehensive analysis of sharp bilinear estimates of Ozawa–Tsutsumi type for solutions $u$ of the free Schrödinger equation, which give sharp control on $|u|^2$ in classical Sobolev spaces. In particular, we generalise their estimates in such a way that provides a unification with some sharp bilinear estimates proved by Carneiro and Planchon–Vega, via entirely different methods, by seeing them all as special cases of a one-parameter family of sharp estimates. The extremal functions are solutions of the Maxwell–Boltzmann functional equation and hence Gaussian. For $u^2$ we argue that the natural analogous results involve certain dispersive Sobolev norms; in particular, despite the validity of the classical Ozawa–Tsutsumi estimates for both $|u|^2$ and $u^2$ in the classical Sobolev spaces, we show that Gaussians are not extremisers in the latter case for spatial dimensions strictly greater than two.