Power weight inequalities for spherical maximal functions

Abstract

This paper is about spherical maximal functions with general dilation sets acting on functions in weighted $L^p(|x|^\alpha)$ spaces. Aside from endpoint cases, a complete description of the allowable ranges of $p$, $\alpha$ is given in terms of the Legendre--Assouad function of the dilation set. This settles, up to endpoints, an open problem of Duoandikoetxea and Seijo.