Atomic Characterizations of Hardy Spaces Associated to Schr\"{o}dinger Type Operators

Abstract

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 $A$ is symmetric and 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 $p(\cdot):\ \mathbb{R}^n\to(0,\,1]$ be a variable exponent function satisfying the globally $\log$-H\"{o}lder continuous condition. The authors introduce the variable Hardy space $H_L^{p(\cdot)}(\mathbb{R}^n)$ associated to $L$ and establish its atomic characterization. The atoms here are closer to the atoms of variable Hardy space $H^{p(\cdot)}(\mathbb{R}^n)$ in spirit, which further implies that $H^{p(\cdot)}(\mathbb{R}^n)$ is continuously embedded in $H_L^{p(\cdot)}(\mathbb{R}^n)$.