$L^p\to L^q$ bounds for spherical maximal operators

Let $f\in L^p({\mathbb{R}}^d)$, $d\ge 3$, and let $A_{t}f(x)$ the average of $f$ over the sphere with radius $t$ centered at $x$. For a subset $E$ of $[1,2]$ we prove close to sharp $L^p\to L^q$ estimates for the maximal function ${\sup }_{t\in E}\mathrm{\mid }{A}_{t}f\mathrm{\mid }$. A new feature is the dependence of the results on both the upper Minkowski dimension of $E$ and the Assouad dimension of $E$. The result can be applied to prove sparse domination bounds for a related global spherical maximal function.