Radial Fourier multipliers in ℝ3 and ℝ4

Cladek, Laura

doi:10.2140/apde.2018.11.467

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Radial Fourier multipliers in $\mathbb{R}^3$ and $\mathbb{R}^4$

Laura Cladek

Vol. 11 (2018), No. 2, 467–498

DOI: 10.2140/apde.2018.11.467

Abstract

We prove that for radial Fourier multipliers $m : ℝ^{3} \to ℂ$ supported compactly away from the origin, $T_{m}$ is restricted strong type $(p, p)$ if $K = \hat{m}$ is in $L^{p} (ℝ^{3})$ , in the range $1 < p < \frac{13}{12}$ . We also prove an $L^{p}$ characterization for radial Fourier multipliers in four dimensions; namely, for radial Fourier multipliers $m : ℝ^{4} \to ℂ$ supported compactly away from the origin, $T_{m}$ is bounded on $L^{p} (ℝ^{4})$ if and only if $K = \hat{m}$ is in $L^{p} (ℝ^{4})$ , in the range $1 < p < \frac{36}{29}$ . Our method of proof relies on a geometric argument that exploits bounds on sizes of multiple intersections of 3-dimensional annuli to control numbers of tangencies between pairs of annuli in three and four dimensions.

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Keywords

Fourier multipliers, radial functions, incidence geometry, local smoothing

Mathematical Subject Classification 2010

Primary: 42B15

Milestones

Received: 12 February 2017

Revised: 9 July 2017

Accepted: 10 August 2017

Published: 17 October 2017

Authors

Laura Cladek
Department of Mathematics University of Wisconsin Madison, WI United States

Vol. 11, No. 2, 2018