Key points are not available for this paper at this time.
Let unk(f)(x) denote the supremum of the averages of f taken over all (surfaces of) spheres centered at x. Then f --> unk(f) is bounded on L(p)(R(n)), whenever p > n/(n - 1), and n >/= 3.
Elias M. Stein (Thu,) studied this question.