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

Abstract

Let $\mathcal L=-Δ_{\mathbb H^n}+V$ be a Schrödinger operator on the Heisenberg group $\mathbb H^n$, where $Δ_{\mathbb H^n}$ is the sublaplacian on $\mathbb H^n$ and the nonnegative potential $V$ belongs to the reverse Hölder class $RH_s$ with $s\in[Q/2,\infty)$. Here $Q=2n+2$ is the homogeneous dimension of $\mathbb H^n$. For given $α\in(0,Q)$, the fractional integral operator associated with the Schrödinger operator $\mathcal L$ is defined by $\mathcal I_α={\mathcal L}^{-α/2}$. In this article, the author introduces the Morrey space $L^{p,κ}_{ρ,\infty}(\mathbb H^n)$ and weak Morrey space $WL^{p,κ}_{ρ,\infty}(\mathbb H^n)$ associated with $\mathcal L$, where $(p,κ)\in[1,\infty)\times[0,1)$ and $ρ(\cdot)$ 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 $\mathcal I_α={\mathcal L}^{-α/2}$ is bounded from $L^{p,κ}_{ρ,\infty}(\mathbb H^n)$ to $L^{q,κ}_{ρ,\infty}(\mathbb H^n)$ with $0<α<Q$, $1<p<Q/α$, $0<κ<1-{(αp)}/Q$ and $1/q=1/p-α/{Q(1-κ)}$, and bounded from $L^{1,κ}_{ρ,\infty}(\mathbb H^n)$ to $WL^{q,κ}_{ρ,\infty}(\mathbb H^n)$ with $0<α<Q$, $0<κ<1-α/Q$ and $1/q=1-α/{Q(1-κ)}$. Moreover, in order to deal with the extreme cases $κ\geq 1-{(αp)}/Q$, the author also introduces the spaces $\mathrm{BMO}_{ρ,\infty}(\mathbb H^n)$ and $\mathcal{C}^β_{ρ,\infty}(\mathbb H^n)$, $β\in(0,1]$ associated with $\mathcal L$.