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arXiv · 1607.02620

The H\"ormander multiplier theorem I: The Linear Case

Loukas Grafakos·Danqing He·Petr Honzík·Hanh Nguyen
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Abstract

We discuss $L^p(\mathbb R^n)$ boundedness for Fourier multiplier operators that satisfy the hypotheses of the H\"ormander multiplier theorem in terms of an optimal condition that relates the distance $|\frac 1p-\frac12|$ to the smoothness $s$ of the associated multiplier measured in some Sobolev norm. We provide new counterexamples to justify the optimality of the condition $|\frac 1p-\frac12|<\frac sn$ and we discuss the endpoint case $|\frac 1p-\frac12|=\frac sn$.

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Loukas Grafakos, Danqing He, Petr Honzík, Hanh Nguyen. 2016-07-09. The H\"ormander multiplier theorem I: The Linear Case. https://arxiv.org/abs/1607.02620

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