Boundedness of iterated spherical average

The iterated spherical average∆(A1)^(N)is an important operator in harmonic analysis,and has very important applications in approximation theory and probability theory,where∆is the Laplacian,A_(1)is the unit spherical average and(A1)^(N)is its iteration.In this paper,we mainly study the sufficient and necessary conditions for the boundedness of this operator in Besov-Lipschitz space,and prove the boundedness of the operator in Triebel-Lizorkin space.Moreover,we use above conclusions to improve the existing results of the boundedness of this operator in L^(p)space.