Morrey spaces for Schrödinger operators with nonnegative potentials, fractional integral operators and the Adams inequality on the Heisenberg groups

Abstract

Let $\mathcal{L}=-{\Delta }_{{\mathbb{H}}^n}+V$ be a Schrödinger operator on the Heisenberg group ${\mathbb{H}}^n$, where ${\Delta }_{{\mathbb{H}}^n}$ is the sublaplacian on ${\mathbb{H}}^n$ and the nonnegative potential $V$ belongs to the reverse Hölder class $R{H}_s$ with $s\in [Q/2,\infty )$. Here $Q=2n+2$ is the homogeneous dimension of ${\mathbb{H}}^n$. For given $\alpha \in (0,Q)$, the fractional integral operator associated with the Schrödinger operator $\mathcal{L}$ is defined by ${\mathcal{I}}_{\alpha }={\mathcal{L}}^{-\alpha /2}$. In this article, the author introduces the Morrey space $L_{\rho ,\infty }^{p,\kappa }({\mathbb{H}}^n)$ and weak Morrey space $W{L}_{\rho ,\infty }^{p,\kappa }({\mathbb{H}}^n)$ associated with $\mathcal{L}$, where $(p,\kappa )\in [1,\infty )\times [0,1)$ and $\rho (\cdot )$ is an auxiliary function related to the nonnegative potential $V$. The relation between the fractional integral operator and the maximal operator on the Heisenberg group is established. From this, the author further obtains the Adams (Morrey-Sobolev) inequality on these new spaces. It is shown that the fractional integral operator ${\mathcal{I}}_{\alpha }={\mathcal{L}}^{-\alpha /2}$ is bounded from $L_{\rho ,\infty }^{p,\kappa }({\mathbb{H}}^n)$ to $L_{\rho ,\infty }^{q,\kappa }({\mathbb{H}}^n)$ with $0<\alpha <Q$, $1<p<Q/\alpha$, $0<\kappa <1-(\alpha p)/Q$ and $1/q=1/p-\alpha /Q(1-\kappa )$, and bounded from $L_{\rho ,\infty }^{1,\kappa }({\mathbb{H}}^n)$ to $W{L}_{\rho ,\infty }^{q,\kappa }({\mathbb{H}}^n)$ with $0<\alpha <Q$, $0<\kappa <1-\alpha /Q$ and $1/q=1-\alpha /Q(1-\kappa )$. Moreover, in order to deal with the extreme cases $\kappa \ge 1-(\alpha p)/Q$, the author also introduces the spaces ${\mathrm{BMO}}_{\rho ,\infty }({\mathbb{H}}^n)$ and ${\mathcal{C}}_{\rho ,\infty }^{\beta }({\mathbb{H}}^n)$, $\beta \in (0,1]$ associated with $\mathcal{L}$.

29 pages. arXiv admin note: substantial text overlap with arXiv:1802.08550