Endpoint power weight inequalities for spherical maximal functions

We prove sharp $L^p(\mathrm{\mid }x{\mathrm{\mid }}^{\alpha })$ estimate for spherical maximal functions. We resolve the remaining open problem for the full maximal operator for $\alpha =1-d$ and $p<2$ 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 $[1,2]$.