Sharp power-weight estimates for the fractional spherical maximal operator

Abstract

For $0\leqα<n$, let $A_{*}^α$ denote the fractional spherical maximal operator. In this paper we characterise the range of $β$ for which the $L^p-L^q$ power-weight estimate \[\left\|A_{*}^α f\right\|_{L^q(|x|^{qβ})} \leq C \left\|f\right\|_{L^p(|x|^{pβ})}\] holds. In the diagonal case $p=q$, our work gives an affirmative answer to a conjecture of Lee. We prove analogous results for the lacunary fractional spherical maximal operator.