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_λ(H^p) . We prove that the degenerate expression of average on sphere of an L^p function is convergent almost everywhere, and the speed of the approximation depends on λ. And when λ ≥ 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.