Sharp power-weight estimates for the fractional spherical maximal operator

For $0\le \alpha <n$, let $A_{\ast }^{\alpha }$ denote the fractional spherical maximal operator. In this paper we characterise the range of $\beta$ for which the $L^p-L^q$ power-weight estimate $${\parallel A_{\ast }^{\alpha }f\parallel }_{L^q(\mathrm{\mid }x{\mathrm{\mid }}^{q\beta })}\le C{\parallel f\parallel }_{L^p(\mathrm{\mid }x{\mathrm{\mid }}^{p\beta })}$$ 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.