Vol. 11, No. 2, 2018

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

Vol. 11 (2018), No. 2, 467–498
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 $\left(p,p\right)$ if $K=\stackrel{̂}{m}$ is in ${L}^{p}\left({ℝ}^{3}\right)$, in the range $1. 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 ${L}^{p}\left({ℝ}^{4}\right)$ if and only if $K=\stackrel{̂}{m}$ is in ${L}^{p}\left({ℝ}^{4}\right)$, in the range $1. 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.

Keywords
Fourier multipliers, radial functions, incidence geometry, local smoothing
Primary: 42B15