Advances in Operator Theory

Atomic characterizations of Hardy spaces associated to Schrödinger type operators

Junqiang Zhang and Zongguang Liu

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‎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 ‎$H_L^{p(\cdot)}(\mathbb{R}^n)$‎.

Article information

Adv. Oper. Theory, Volume 4, Number 3 (2019), 604-624.

Received: 27 November 2018
Accepted: 19 December 2018
First available in Project Euclid: 2 March 2019

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 42B30: $H^p$-spaces
Secondary: 42B35‎ ‎35J10

Hardy space‎ Schrödinger type operator ‎‎variable exponent ‎ ‎atom‎


Zhang, Junqiang; Liu, Zongguang. Atomic characterizations of Hardy spaces associated to Schrödinger type operators. Adv. Oper. Theory 4 (2019), no. 3, 604--624. doi:10.15352/aot.1811-1440.

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