One-dimensional average on spheres and approximation

In this paper, we consider the approximation of the degeneration of classic average on sphere on Hardy-Sobolev spaces I-lambda(H-p) (0<lambda<infinity,1 lambda p function is convergent almost everywhere, and the speed of the approximation depends on lambda. And when lambda >= 2, the approximation of the average operator will be saturated. On the other hand, we also study the generalization of the average on the product of Hardy-Sobolev spaces.