Inhomogeneous Strichartz estimates in some critical cases

TL;DR

This paper investigates the limitations of strong-type inhomogeneous Strichartz estimates for the wave equation outside acceptable regions and introduces weak-type estimates on a critical line, using an abstract framework and Besov space techniques.

Contribution

It establishes weak-type inhomogeneous Strichartz estimates at critical cases where standard estimates fail, extending the theoretical understanding of wave equation dispersive properties.

Findings

Strong-type estimates fail outside acceptable regions.

Weak-type estimates hold on a critical line.

Application to wave equation using Besov space dispersive estimates.

Abstract

Strong-type inhomogeneous Strichartz estimates are shown to be false for the wave equation outside the so-called acceptable region. On a critical line where the acceptability condition marginally fails, we prove substitute estimates with a weak-type norm in the temporal variable. We achieve this by establishing such weak-type inhomogeneous Strichartz estimates in an abstract setting. The application to the wave equation rests on a slightly stronger form of the standard dispersive estimate in terms of certain Besov spaces.