Some Estimates of Schrödinger Type Operators on Variable Lebesgue and Hardy Spaces

Abstract

In this article, the authors consider the Schrödinger type operator $L:=-\mathrm{div}(A\nabla )+V$ 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 Hölder class $R{H}_q({\mathbb{R}}^n)$ with $q\in (n/2,\infty )$. Let $p(\cdot ):{\mathbb{R}}^n\to (0,\infty )$ be a variable exponent function satisfying the globally $\log$-Hölder continuous condition. When $p(\cdot ):{\mathbb{R}}^n\to (1,\infty )$, the authors prove that the operators $V{L}^{-1}$, $V^{1/2}\nabla L^{-1}$ and ${\nabla }^2{L}^{-1}$ 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 $V{L}^{-1}$, $V^{1/2}\nabla L^{-1}$ and ${\nabla }^2{L}^{-1}$ are bounded from $H_L^{p(\cdot )}({\mathbb{R}}^n)$ to $L^{p(\cdot )}({\mathbb{R}}^n)$.