Spherical maximal operators on radial functions

Abstract

Let $A_tf(x)=\int f(x+ty)dσ(y)$ denote the spherical means in $\Bbb R^d$ ($dσ$ is surface measure on $S^{d-1}$, normalized to $1$). We prove sharp estimates for the maximal function $M_E f(x)=\sup_{t\in E}|A_tf(x)|$ where $E$ is a fixed set in $\Bbb R^+$ and $f$ is a {\it radial} function $\in L^p(\Bbb R^d)$. Let $p_d=d/(d-1)$ (the critical exponent of Stein&#39;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 $M_E$.