Endpoint power weight inequalities for spherical maximal functions

Abstract: We prove sharp $L<sup>p(|x|<sup>α)$ estimate for spherical maximal functions. We resolve the remaining open problem for the full maximal operator for $α=1-d$ and $p&lt;2$ and prove a more general result involving a local version of Assouad dimension of the underlying dilation set. We also make a case study with endpoint bounds for a model case involving convex sequences in $[1,2]$.