The Hörmander multiplier theorem, I: The linear case revisited

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 $\vert \frac{1}{p}-\frac{1}{2}\vert $ to the smoothness $s$ of the associated multiplier measured in some Sobolev norm. We provide new counterexamples to justify the optimality of the condition $\vert \frac{1}{p}-\frac{1}{2}\vert <\frac{s}{n}$ and we discuss the endpoint case $\vert \frac{1}{p}-\frac{1}{2}\vert =\frac{s}{n}$.