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Keywords:
Musielak-Orlicz-Hardy space; Schrödinger operator; $L$-harmonic function; isomorphism of Hardy space; atom; molecule
Summary:
Let $L:=-\Delta +V$ be a Schrödinger operator on $\mathbb {R}^n$ with $n\ge 3$ and $V\ge 0$ satisfying $\Delta ^{-1} V\in L^\infty (\mathbb {R}^n)$. Assume that $\varphi \colon \mathbb {R}^n\times [0,\infty )\to [0,\infty )$ is a function such that $\varphi (x,\cdot )$ is an Orlicz function, $\varphi (\cdot ,t)\in {\mathbb A}_{\infty }(\mathbb {R}^n)$ (the class of uniformly Muckenhoupt weights). Let $w$ be an $L$-harmonic function on $\mathbb {R}^n$ with $0<C_1\le w\le C_2$, where $C_1$ and $C_2$ are positive constants. In this article, the author proves that the mapping $H_{\varphi ,L}(\mathbb {R}^n)\ni f\mapsto wf\in H_\varphi (\mathbb {R}^n)$ is an isomorphism from the Musielak-Orlicz-Hardy space associated with $L$, $H_{\varphi ,L}(\mathbb {R}^n)$, to the Musielak-Orlicz-Hardy space $H_{\varphi }(\mathbb {R}^n)$ under some assumptions on $\varphi $. As applications, the author further obtains the atomic and molecular characterizations of the space $H_{\varphi ,L}(\mathbb {R}^n)$ associated with $w$, and proves that the operator $(-\Delta )^{-1/2}L^{1/2}$ is an isomorphism of the spaces $H_{\varphi ,L}(\mathbb {R}^n)$ and $H_{\varphi }(\mathbb {R}^n)$. All these results are new even when $\varphi (x,t):=t^p$, for all $x\in \mathbb {R}^n$ and $t\in [0,\infty )$, with $p\in ({n}/{(n+\mu _0)},1)$ and some $\mu _0\in (0,1]$.
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