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

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 \(H_L^{p(\cdot)}(\mathbb{R}^n)\).


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