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Some estimates of Schrödinger type operators on variable Lebesgue and Hardy spaces
Banach Journal of Mathematical Analysis ( IF 1.1 ) Pub Date : 2020-01-01 , DOI: 10.1007/s43037-019-00020-6
Junqiang Zhang , Zongguang Liu

In this article, the authors consider the Schr\"{o}dinger type operator $L:=-{\rm div}(A\nabla)+V$ on $\mathbb{R}^n$ with $n\geq 3$, where the matrix $A$ satisfies uniformly elliptic condition and the nonnegative potential $V$ belongs to the reverse H\"{o}lder class $RH_q(\mathbb{R}^n)$ with $q\in(n/2,\,\infty)$. Let $p(\cdot):\ \mathbb{R}^n\to(0,\,\infty)$ be a variable exponent function satisfying the globally $\log$-H\"{o}lder continuous condition. When $p(\cdot):\ \mathbb{R}^n\to(1,\,\infty)$, the authors prove that the operators $VL^{-1}$, $V^{1/2}\nabla L^{-1}$ and $\nabla^2L^{-1}$ are bounded on variable Lebesgue space $L^{p(\cdot)}(\mathbb{R}^n)$. When $p(\cdot):\ \mathbb{R}^n\to(0,\,1]$, the authors introduce the variable Hardy space $H_L^{p(\cdot)}(\mathbb{R}^n)$, associated to $L$, and show that $VL^{-1}$, $V^{1/2}\nabla L^{-1}$ and $\nabla^2L^{-1}$ are bounded from $H_L^{p(\cdot)}(\mathbb{R}^n)$ to $L^{p(\cdot)}(\mathbb{R}^n)$.

中文翻译:

变量 Lebesgue 和 Hardy 空间上薛定谔类型算子的一些估计

在本文中,作者考虑了 $\mathbb{R}^n$ 上的 Schr\"{o}dinger 类型运算符 $L:=-{\rm div}(A\nabla)+V$ with $n\geq 3$,其中矩阵 $A$ 满足一致椭圆条件,非负势 $V$ 属于反向 H\"{o}lder 类 $RH_q(\mathbb{R}^n)$ 和 $q\in( n/2,\,\infty)$。令 $p(\cdot):\ \mathbb{R}^n\to(0,\,\infty)$ 为满足全局 $\log$-H\"{o}lder 连续条件的变指数函数。当 $p(\cdot):\ \mathbb{R}^n\to(1,\,\infty)$ 时,作者证明了算子 $VL^{-1}$, $V^{1/2 }\nabla L^{-1}$ 和 $\nabla^2L^{-1}$ 在变量 Lebesgue 空间 $L^{p(\cdot)}(\mathbb{R}^n)$ 上有界。当$p(\cdot):\ \mathbb{R}^n\to(0,\,1]$,作者引入了变量哈代空间$H_L^{p(\cdot)}(\mathbb{R}^ n)$,与 $L$ 相关联,并表明 $VL^{-1}$,
更新日期:2020-01-01
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