Radial Fourier multipliers in ℝ3 and ℝ4

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.