A Sharp Sobolev--Strichartz estimate for the wave equation

TL;DR

This paper determines the exact best constant and the extremal initial data for a specific Sobolev--Strichartz estimate related to the wave equation in four dimensions, advancing understanding of optimal bounds in this setting.

Contribution

It computes the sharp constant and characterizes extremal initial data for the $L^4$ Sobolev--Strichartz estimate in four-dimensional wave equations, a previously unresolved problem.

Findings

Calculated the sharp constant for the estimate

Characterized extremal initial data in the specified Sobolev space

Enhanced understanding of optimal bounds in wave equation estimates

Abstract

We calculate the the sharp constant and characterise the extremal initial data in $\dot{H}^{\frac{3}{4}}\times\dot{H}^{-\frac{1}{4}}$ for the $L^4$ Sobolev--Strichartz estimate for the wave equation in four space dimensions.