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 \(1-d\le \alpha <(d-1)(p-1)-1\) . Within this range, boundedness has been proven except for the endpoint case \(\alpha =1-d\) . In this paper, we establish boundedness of the spherical maximal operator in the critical order case \(\alpha =1-d\) when \(d=2\) . Also, we give a partial result when \(d\ge 2\) .
