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

Abstract

We prove that for radial Fourier multipliers $m:{\mathbb{R}}^3\to \mathbb{C}$ supported compactly away from the origin, $T_m$ is restricted strong type (p,p) if $K=\hat{m}$ is in $L^p({\mathbb{R}}^3)$, 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 $m:{\mathbb{R}}^4\to \mathbb{C}$ supported compactly away from the origin, $T_m$ is bounded on $L^p({\mathbb{R}}^4)$ if and only if $K=\hat{m}$ is in $L^p({\mathbb{R}}^4)$, 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 three-dimensional annuli to control numbers of tangencies between pairs of annuli in three and four dimensions.

arXiv comment41 pages

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