Vol. 11, No. 2, 2018

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

Laura Cladek

Vol. 11 (2018), No. 2, 467–498

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 < 13 12. 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 < 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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Fourier multipliers, radial functions, incidence geometry, local smoothing
Mathematical Subject Classification 2010
Primary: 42B15
Received: 12 February 2017
Revised: 9 July 2017
Accepted: 10 August 2017
Published: 17 October 2017
Laura Cladek
Department of Mathematics
University of Wisconsin
Madison, WI
United States