Characterizations for the Fractional Integral Operators in Generalized Morrey Spaces on Carnot Groups

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