Atomic Characterizations of Hardy Spaces Associated to Schrödinger Type Operators

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 $A$ is symmetric and 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 $p(\cdot ):{\mathbb{R}}^n\to (0,1]$ 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 $H_L^{p(\cdot )}({\mathbb{R}}^n)$.

arXiv admin note: text overlap with arXiv:1811.10768