Let $\mathcal L=-\Delta_{\mathbb H^n}+V$ be a Schrodinger 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 Holder 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 Schrodinger 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| 0}$.

中文翻译：

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具有特定非负势的薛定谔算符的莫雷空间、海森堡群上的 Littlewood-Paley 和 Lusin 函数

令 $\mathcal L=-\Delta_{\mathbb H^n}+V$ 是海森堡群 $\mathbb H^n$ 上的薛定谔算子，其中 $\Delta_{\mathbb H^n}$ 是次拉普拉斯算子在 $\mathbb H^n$ 上，非负势 $V$ 属于具有 $q\geq Q/2$ 的反向持有者类 $RH_q$。这里$Q=2n+2$是$\mathbb H^n$的齐次维数。假设 $\{e^{-s\mathcal L}\}_{s>0}$ 是 $\mathcal L$ 产生的热半群。分别定义了 Littlewood-Paley 函数 $\mathfrak{g}_{\mathcal L}$ 和与薛定谔算子关联的 Lusin 面积积分 $\mathcal{S}_{\mathcal L}$ , 由 \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*} 和 \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| 0}$。