Power weight inequalities for spherical maximal functions

This paper is about spherical maximal functions with general dilation sets acting on functions in weighted $L^p(\mathrm{\mid }x{\mathrm{\mid }}^{\alpha })$ 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.