CHARACTERIZATIONS FOR THE FRACTIONAL INTEGRAL OPERATOR AND ITS COMMUTATORS IN GENERALIZED WEIGHTED MORREY SPACES ON CARNOT GROUPS

In this paper, we shall give a characterization for the strong and weak type Spanne type boundedness of the fractional integral operator I-alpha, 0 < alpha < Q on Carrot group G on generalized weighted Morrey spaces M-p,M-phi (G,w), respectively, where Q is the homogeneous dimension of G. Also we give a characterization for the Spanne type boundedness of the commutator operator [b,I-alpha] on generalized weighted Morrey spaces. As applications of the properties of the fundamental solution of sub-Laplacian L on G, we prove two Sobolev-Stein embedding theorems on generalized weighted Morrey spaces in the Carrot group setting.