Illinois Journal of Mathematics

The Hörmander multiplier theorem, I: The linear case revisited

Loukas Grafakos, Danqing He, Petr Honzik, and Hanh Van Nguyen

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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 $\vert \frac{1}{p}-\frac{1}{2}\vert =\frac{s}{n}$.

Article information

Source
Illinois J. Math., Volume 61, Number 1-2 (2017), 25-35.

Dates
Received: 2 December 2016
Revised: 17 November 2017
First available in Project Euclid: 3 March 2018

Permanent link to this document
https://projecteuclid.org/euclid.ijm/1520046207

Digital Object Identifier
doi:10.1215/ijm/1520046207

Mathematical Reviews number (MathSciNet)
MR3770834

Zentralblatt MATH identifier
1395.42025

Subjects
Primary: 42B15: Multipliers 42B20: Singular and oscillatory integrals (Calderón-Zygmund, etc.) 42B30: $H^p$-spaces 42B99: None of the above, but in this section

Citation

Grafakos, Loukas; He, Danqing; Honzik, Petr; Van Nguyen, Hanh. The Hörmander multiplier theorem, I: The linear case revisited. Illinois J. Math. 61 (2017), no. 1-2, 25--35. doi:10.1215/ijm/1520046207. https://projecteuclid.org/euclid.ijm/1520046207


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