Let φ denote the normalised surface measure on the unit sphere Sn-1. We shall be interested in the weighted Lp estimate for Stein’s maximal function Mφf, namely ∥Mφf∥Lp(w) ≤ Cp,w ∥f∥ Lp(w) , f ∈ Lp(w), where w is an Ap weight, especially for 1 < p ≤ 2. Using the Mellin transformation approach, we prove that the estimate holds for every weight wδ where w ∈ Ap and 0 ≤ δ < (p(n – 1) – n)/(n(p – 1)), for n ≥ 3 and n/(n – 1) < p ≤ 2.

https://doi.org/10.1017/s0004972700015057