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

### 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
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