Morrey spaces for Schr\"odinger 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 Schr\"odinger 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 H\"older class $RH_s$ 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 Schr\"odinger operator $\mathcal L$ is defined by $\mathcal I_{\alpha}={\mathcal L}^{-{\alpha}/2}$. In this article, the author introduces the Morrey space $L^{p,\kappa}_{\rho,\infty}(\mathbb H^n)$ and weak Morrey space $WL^{p,\kappa}_{\rho,\infty}(\mathbb H^n)$ 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 $L^{p,\kappa}_{\rho,\infty}(\mathbb H^n)$ to $L^{q,\kappa}_{\rho,\infty}(\mathbb H^n)$ 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 $L^{1,\kappa}_{\rho,\infty}(\mathbb H^n)$ to $WL^{q,\kappa}_{\rho,\infty}(\mathbb H^n)$ 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 $\kappa\geq 1-{(\alpha p)}/Q$, the author also introduces the spaces $\mathrm{BMO}_{\rho,\infty}(\mathbb H^n)$ and $\mathcal{C}^{\beta}_{\rho,\infty}(\mathbb H^n)$, $\beta\in(0,1]$ associated with $\mathcal L$.