Morrey spaces for Schrödinger 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 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 $R{H}_q$ with $q\ge Q/2$. 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 Schrödinger 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_{Γ(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*} Γ(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 Schrödinger 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 $\{e^{-s\sqrt{\mathcal{L}}}{\}}_{s>0}$.

28 pages. arXiv admin note: text overlap with arXiv:1802.08550, arXiv:1907.03573