Some Estimates of Schr\"{o}dinger Type Operators on Variable Lebesgue and Hardy Spaces

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