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 provide a generalization of 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. We show that the extremal functions are solutions of the Maxwell-Boltzmann functional equation and provide a new proof that this equation admits only Gaussian solutions. We also make a connection to certain sharp estimates on $u^2$ involving certain dispersive Sobolev norms.

17 pages, references updated

References in corpus (3)

• A sharp bilinear estimate for the Klein-Gordon equation in arbitrary space-time dimensions
• Some sharp bilinear space-time estimates for the wave equation
• On characterization of the sharp Strichartz inequality for the Schrödinger Equation

Cited by in corpus (5)

• A sharp trilinear inequality related to Fourier restriction on the circle
• Extremizers for Fourier restriction on hyperboloids
• A sharp $k$-plane Strichartz inequality for the Schrödinger equation
• Some recent progress on sharp Fourier restriction theory
• Bilinear identities involving the $k$-plane transform and Fourier extension operators