Atomic characterizations of Hardy spaces associated to Schrödinger type operators

Abstract

‎In this article‎, ‎the authors consider the Schrödinger type‎ ‎operator $L:=-{\rm div}(A\nabla)+V$ on $\mathbb{R}^n$ with $n\geq 3$‎, ‎where the matrix $A$ is symmetric and satisfies‎ ‎the uniformly elliptic condition and the nonnegative potential‎ ‎$V$ belongs to the reverse Hölder 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 $\log$-Hölder 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 ‎$H_L^{p(\cdot)}(\mathbb{R}^n)$‎.