Characterizations for the fractional integral operators in generalized Morrey spaces on Carnot groups

Abstract

In this paper, we study the boundedness of the fractional integral operator I_α on Carnot group G in the generalized Morrey spaces M_{p, φ}(G). We shall give a characterization for the strong and weak type boundedness of I_α on the generalized Morrey spaces, respectively. As applications of the properties of the fundamental solution of sub-Laplacian L on G, we prove two Sobolev–Stein embedding theorems on generalized Morrey spaces in the Carnot group setting.