ON RADIAL WEIGHTS FOR THE SPHERICAL SUMMATION OPERATOR

In this paper we give a sufficient condition for radial weights ω such that the spherical summation operator is bounded on L2(Rn, ω(x)dx), n ≥ 2. When applied to power weights, ω(x) = |x|x, these conditions are also necessary. Moreover, in the general case we shall show that our condition is almost necessary. Furthermore, we give an example which shows that the conjecture in Anderson (Proc. Amer. Math. Soc. 83, No. 2 (1981), 269-275) fails to hold. © 1990.