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 $\vert \frac{1}{p}-\frac{1}{2}\vert =\frac{s}{n}$.
Citation
Loukas Grafakos. Danqing He. Petr Honzik. Hanh Van Nguyen. "The Hörmander multiplier theorem, I: The linear case revisited." Illinois J. Math. 61 (1-2) 25 - 35, Spring and Summer 2017. https://doi.org/10.1215/ijm/1520046207
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