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Abstract: 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 R-d is bounded on L-p(|x|(alpha)) only if 1-d <= 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 >= 2.

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