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2018 Radial Fourier multipliers in $\mathbb{R}^3$ and $\mathbb{R}^4$
Laura Cladek
Anal. PDE 11(2): 467-498 (2018). DOI: 10.2140/apde.2018.11.467

Abstract

We prove that for radial Fourier multipliers m:3 supported compactly away from the origin, Tm is restricted strong type (p,p) if K=m̂ is in Lp(3), in the range 1<p<1312. We also prove an Lp characterization for radial Fourier multipliers in four dimensions; namely, for radial Fourier multipliers m:4 supported compactly away from the origin, Tm is bounded on Lp(4) if and only if K=m̂ is in Lp(4), in the range 1<p<3629. 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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Laura Cladek. "Radial Fourier multipliers in $\mathbb{R}^3$ and $\mathbb{R}^4$." Anal. PDE 11 (2) 467 - 498, 2018. https://doi.org/10.2140/apde.2018.11.467

Information

Received: 12 February 2017; Revised: 9 July 2017; Accepted: 10 August 2017; Published: 2018
First available in Project Euclid: 20 December 2017

zbMATH: 1378.42009
MathSciNet: MR3724494
Digital Object Identifier: 10.2140/apde.2018.11.467

Subjects:
Primary: 42B15

Keywords: Fourier multipliers , incidence geometry , local smoothing , radial functions

Rights: Copyright © 2018 Mathematical Sciences Publishers

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Vol.11 • No. 2 • 2018
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