Strichartz Estimates and Maximal Operators for the Wave Equation in R^3

TL;DR

This paper establishes sharp Strichartz estimates and maximal operator bounds for the wave equation in three dimensions, leading to improved well-posedness results for certain semilinear wave equations.

Contribution

It provides new sharp Strichartz estimates, including reverse norms, and refines maximal operator bounds, advancing the analysis of wave equations with potentials.

Findings

Proved sharp Strichartz estimates in R^3 for wave equations.

Established a sharper version of maximal operator estimates.

Demonstrated local and global well-posedness for small data in semilinear wave equations.

Abstract

We prove sharp Strichartz-type estimates in three dimensions, including some which hold in reverse spacetime norms, for the wave equation with potential. These results are also tied to maximal operator estimates studied by Rogers--Villaroya, of which we prove a sharper version. As a sample application, we use these results to prove the local well-posedness and the global well-posedness for small initial data of semilinear wave equations in R^3 with quintic or higher monomial nonlinearities.