Atomic characterizations of Hardy spaces associated to Schrödinger type operators. (English) Zbl 1414.42027

Zhang, Junqiang; Liu, Zongguang

Adv. Oper. Theory 4, No. 3, 604-624 (2019).

Summary: In this article, the authors consider the Schrödinger type operator \(L:=-\operatorname{div}(A\nabla)+V\) 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 \(H_L^{p(\cdot)}(\mathbb{R}^n)\).

MSC:

42B30	\(H^p\)-spaces
42B35	Function spaces arising in harmonic analysis
35J10	Schrödinger operator, Schrödinger equation

Keywords:

Hardy space; Schrödinger-type operator; variable exponent; atom