Isomorphisms and several characterizations of Musielak-Orlicz-Hardy spaces associated with some Schrödinger operators

Sibei Yang

Czechoslovak Mathematical Journal (2015)

  • Volume: 65, Issue: 3, page 747-779
  • ISSN: 0011-4642

Abstract

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Let L : = - Δ + V be a Schrödinger operator on ℝ n with n ≥ 3 and V ≥ 0 satisfying Δ - 1 V ∈ L ∞ ( ℝ n ) . Assume that ϕ : ℝ n × [ 0 , ∞ ) → [ 0 , ∞ ) is a function such that ϕ ( x , · ) is an Orlicz function, ϕ ( · , t ) ∈ 𝔸 ∞ ( ℝ n ) (the class of uniformly Muckenhoupt weights). Let w be an L -harmonic function on ℝ n with 0 < C 1 ≤ w ≤ C 2 , where C 1 and C 2 are positive constants. In this article, the author proves that the mapping H ϕ , L ( ℝ n ) ∋ f ↦ w f ∈ H ϕ ( ℝ n ) is an isomorphism from the Musielak-Orlicz-Hardy space associated with L , H ϕ , L ( ℝ n ) , to the Musielak-Orlicz-Hardy space H ϕ ( ℝ n ) under some assumptions on ϕ . As applications, the author further obtains the atomic and molecular characterizations of the space H ϕ , L ( ℝ n ) associated with w , and proves that the operator ( - Δ ) - 1 / 2 L 1 / 2 is an isomorphism of the spaces H ϕ , L ( ℝ n ) and H ϕ ( ℝ n ) . All these results are new even when ϕ ( x , t ) : = t p , for all x ∈ ℝ n and t ∈ [ 0 , ∞ ) , with p ∈ ( n / ( n + μ 0 ) , 1 ) and some μ 0 ∈ ( 0 , 1 ] .

How to cite

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Yang, Sibei. "Isomorphisms and several characterizations of Musielak-Orlicz-Hardy spaces associated with some Schrödinger operators." Czechoslovak Mathematical Journal 65.3 (2015): 747-779. <http://eudml.org/doc/271829>.

@article{Yang2015,
abstract = {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 $\varphi \colon \mathbb \{R\}^n\times [0,\infty )\rightarrow [0,\infty )$ 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 $\mu _0\in (0,1]$.},
author = {Yang, Sibei},
journal = {Czechoslovak Mathematical Journal},
keywords = {Musielak-Orlicz-Hardy space; Schrödinger operator; $L$-harmonic function; isomorphism of Hardy space; atom; molecule},
language = {eng},
number = {3},
pages = {747-779},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Isomorphisms and several characterizations of Musielak-Orlicz-Hardy spaces associated with some Schrödinger operators},
url = {http://eudml.org/doc/271829},
volume = {65},
year = {2015},
}

TY - JOUR
AU - Yang, Sibei
TI - Isomorphisms and several characterizations of Musielak-Orlicz-Hardy spaces associated with some Schrödinger operators
JO - Czechoslovak Mathematical Journal
PY - 2015
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 65
IS - 3
SP - 747
EP - 779
AB - 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 $\varphi \colon \mathbb {R}^n\times [0,\infty )\rightarrow [0,\infty )$ 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 $\mu _0\in (0,1]$.
LA - eng
KW - Musielak-Orlicz-Hardy space; Schrödinger operator; $L$-harmonic function; isomorphism of Hardy space; atom; molecule
UR - http://eudml.org/doc/271829
ER -

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