The Hörmander multiplier theorem I: The Linear Case

Abstract

We discuss $L^p({\mathbb{R}}^n)$ boundedness for Fourier multiplier operators that satisfy the hypotheses of the Hörmander multiplier theorem in terms of an optimal condition that relates the distance $|\frac{1}{p}-\frac{1}{2}|$ to the smoothness $s$ of the associated multiplier measured in some Sobolev norm. We provide new counterexamples to justify the optimality of the condition $|\frac{1}{p}-\frac{1}{2}|<\frac{s}{n}$ and we discuss the endpoint case $|\frac{1}{p}-\frac{1}{2}|=\frac{s}{n}$.

15 pages