Radial Fourier multipliers in high dimensions

TL;DR

This paper characterizes radial Fourier multipliers in high dimensions for certain L^p spaces and establishes new regularity results for wave equation solutions in high-dimensional settings.

Contribution

It provides a simple characterization of radial multipliers in high dimensions and introduces new L^q regularity results for wave equations.

Findings

Characterization of radial multipliers for large dimensions

New L^q regularity results for wave equation solutions

Applicable for fixed p not equal to 2 in high-dimensional spaces

Abstract

Given a fixed $p\neq 2$, we prove a simple and effective characterization of all radial multipliers of $\cF L^p(\Bbb R^d)$, provided that the dimension $d$ is sufficiently large. The method also yields new $L^q$ space-time regularity results for solutions of the wave equation in high dimensions.