Morrey spaces for Schr\"odinger operators with certain nonnegative potentials, Littlewood-Paley and Lusin functions 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 $RH_q$ with $q\geq Q/2$. Here $Q=2n+2$ is the homogeneous dimension of $\mathbb H^n$. Assume that $\{e^{-s\mathcal L}\}_{s>0}$ is the heat semigroup generated by $\mathcal L$. The Littlewood-Paley function $\mathfrak{g}_{\mathcal L}$ and the Lusin area integral $\mathcal{S}_{\mathcal L}$ associated with the Schr\"odinger operator $\mathcal L$ are defined, respectively, by \begin{equation*} \mathfrak{g}_{\mathcal L}(f)(u) := \bigg(\int_0^{\infty}\bigg|s\frac{d}{ds} e^{-s\mathcal L}f(u) \bigg|^2\frac{ds}{s}\bigg)^{1/2} \end{equation*} and \begin{equation*} \mathcal{S}_{\mathcal L}(f)(u) := \bigg(\iint_{\Gamma(u)} \bigg|s\frac{d}{ds} e^{-s\mathcal L}f(v) \bigg|^2 \frac{dvds}{s^{Q/2+1}}\bigg)^{1/2}, \end{equation*} where \begin{equation*} \Gamma(u) := \big\{(v,s)\in\mathbb H^n\times(0,\infty): |u^{-1}v| < \sqrt{s\,}\big\}. \end{equation*} In this paper the author first introduces a class of Morrey spaces associated with the Schr\"odinger operator $\mathcal L$ on $\mathbb H^n$. Then by using some pointwise estimates of the kernels related to the nonnegative potential $V$, the author establishes the boundedness properties of these two operators $\mathfrak{g}_{\mathcal L}$ and $\mathcal{S}_{\mathcal L}$ acting on the Morrey spaces. It can be shown that the same conclusions also hold for the operators $\mathfrak{g}_{\sqrt{\mathcal L}}$ and $\mathcal{S}_{\sqrt{\mathcal L}}$ with respect to the Poisson semigroup $\{e^{-s\sqrt{\mathcal L}}\}_{s>0}$.