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 α , 0 < α < Q on Carnot group G on generalized weighted Morrey spaces M p,ϕ (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 α ] 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 Carnot group setting.