Spherical maximal operators on radial functions

Abstract

Let $A_{t}f(x)=\int f(x+ty)d\sigma (y)$ denote the spherical means in ${\mathbb{R}}^d$ ($d\sigma$ is surface measure on $S^{d-1}$, normalized to $1$). We prove sharp estimates for the maximal function $M_{E}f(x)={\sup }_{t\in E}|A_{t}f(x)|$ where $E$ is a fixed set in ${\mathbb{R}}^{+}$ and $f$ is a {\it radial} function $\in L^p({\mathbb{R}}^d)$. Let $p_d=d/(d-1)$ (the critical exponent of Stein's maximal function). For the cases (i) $p<p_d$, $d\ge 2$ and (ii) $p=p_d$, $d\ge 3$, and for $p\le q\le \infty$ we prove necessary and sufficient conditions for $L^p\to L^{p,q}$ boundedness of the operator $M_E$.

Cited by in corpus (1)

• Spherical maximal functions and fractal dimensions of dilation sets