The critical weighted inequalities of the spherical maximal function

Weighted inequality on the Hardy-Littlewood maximal function is completely understood while it is not well understood for the spherical maximal function. For the power weight $\mathrm{\mid }x{\mathrm{\mid }}^{\alpha }$, it is known that the spherical maximal operator on ${\mathbb{R}}^d$ is bounded on $L^p(\mathrm{\mid }x{\mathrm{\mid }}^{\alpha })$ only if $1-d\le \alpha <(d-1)(p-1)-d$ and under this condition, it is known to be bounded except $\alpha =1-d$. In this paper, we prove the case of the critical order, $\alpha =1-d$.