Boundedness Of Iterated Spherical Average On Alpha-Modulation Spaces

For the iteration of spherical average (A(1))N and the Laplace operator Delta, we consider the boundedness of the operator Delta(A(1))N on the alpha-modulation spaces M-p,q(s,alpha) . The authors obtain some sufficient and necessary conditions to ensure the boundedness on the alpha-modulation spaces. The main theorems significantly improve some known results.