A sharp version of the H\"ormander multiplier theorem

Abstract

We provide an improvement of the H\"ormander multiplier theorem in which the Sobolev space $L^r_s(\mathbb R^n)$ with integrability index $r$ and smoothness index $s>n/r$ is replaced by the Sobolev space with smoothness $s$ built upon the Lorentz space $L^{n/s,1}(\mathbb R^n)$.