Illinois Journal of Mathematics

Weighted local Hardy spaces associated to Schrödinger operators

Hua Zhu and Lin Tang

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In this paper, we characterize the weighted local Hardy spaces $h^{p}_{\rho}(\omega)$ related to the critical radius function $\rho$ and weights $\omega\in A_{\infty}^{\rho,\infty}(\mathbb{R}^{n})$ which locally behave as Muckenhoupt’s weights and actually include them, by the local vertical maximal function, the local nontangential maximal function and the atomic decomposition. Then, we establish the equivalence of the weighted local Hardy space $h^{1}_{\rho}(\omega)$ and the weighted Hardy space $H^{1}_{\mathcal{L}}(\omega)$ associated to Schrödinger operators $\mathcal{L}$ with $\omega\in A_{1}^{\rho,\infty}(\mathbb{R}^{n})$. By the atomic characterization, we also prove the existence of finite atomic decompositions associated with $h^{p}_{\rho}(\omega)$. Furthermore, we establish boundedness in $h^{p}_{\rho}(\omega)$ of quasi-Banach-valued sublinear operators.

Article information

Illinois J. Math., Volume 60, Number 3-4 (2016), 687-738.

Received: 16 May 2016
Revised: 6 June 2017
First available in Project Euclid: 22 September 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 42B30: $H^p$-spaces
Secondary: 42B25: Maximal functions, Littlewood-Paley theory 42B20: Singular and oscillatory integrals (Calderón-Zygmund, etc.)


Zhu, Hua; Tang, Lin. Weighted local Hardy spaces associated to Schrödinger operators. Illinois J. Math. 60 (2016), no. 3-4, 687--738. doi:10.1215/ijm/1506067287.

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