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 $|x|^{\alpha}$, it is known that the spherical maximal operator on $\mathbb{R}^d$ is bounded on $L^p(|x|^{\alpha})$ only if $1-d\leq \alpha<(d-1)(p-1)-d$ 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, $\alpha=1-d$.Comment: 14 page