Radial Fourier multipliers in high dimensions

Given a fixed $p\mathrm{\ne }2$, we prove a simple and effective characterization of all radial multipliers of $\text{cF}{L}^p({\mathbb{R}}^d)$, 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.