Optimal dispersive estimates for the wave equation with $C^{(n-3)/2}$ potentials in dimensions $4\le n\le 7$

TL;DR

This paper establishes optimal dispersive estimates for the wave equation with certain regularity and decay conditions on the potential in dimensions 4 through 7, advancing understanding of wave behavior under less smooth potentials.

Contribution

It provides the first proof of optimal dispersive estimates for wave equations with potentials in the class $C^{(n-3)/2}$ for dimensions 4 to 7, under specific decay conditions.

Findings

Proved optimal dispersive estimates for wave groups with $C^{(n-3)/2}$ potentials.

Extended dispersive estimate results to higher dimensions $4 o 7$.

Demonstrated decay conditions on potentials ensure dispersive behavior.

Abstract

We prove optimal dispersive estimates for the wave group $e^{it\sqrt{-\Delta+V}}$ for a class of real-valued potentials $V\in C^{(n-3)/2}({\bf R}^n)$, $4\le n\le 7$, such that $\partial_x^\alpha V(x)=O(|x|^{-\delta})$ for $|x|>1$, where $\delta>(n+1)/2$.