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2016 On Weak$^*$-convergence in the Localized Hardy Spaces $H^1_\rho(\mathcal{X})$ and its Application
Dinh Thanh Duc, Ha Duy Hung, Luong Dang Ky
Taiwanese J. Math. 20(4): 897-907 (2016). DOI: 10.11650/tjm.20.2016.7020

Abstract

Let $(\mathcal{X}, d, \mu)$ be a complete RD-space. Let $\rho$ be an admissible function on $\mathcal{X}$, which means that $\rho$ is a positive function on $\mathcal{X}$ and there exist positive constants $C_0$ and $k_0$ such that, for any $x, y \in \mathcal{X}$,\[ \rho(y) \leq C_0 [\rho(x)]^{1/(1+k_0)} [\rho(x) + d(x, y)]^{k_0/(1+k_0)}.\]In this paper, we define a space $\operatorname{VMO}_\rho(\mathcal{X})$ and show that it is the predual of the localized Hardy space $H^1_\rho(\mathcal{X})$ introduced by Yang and Zhou [14]. Then we prove a version of the classical theorem of Jones and Journé [7] on weak$^*$-convergence in $H^1_\rho(\mathcal{X})$. As an application, we give an atomic characterization of $H^1_\rho(\mathcal{X})$.

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Dinh Thanh Duc. Ha Duy Hung. Luong Dang Ky. "On Weak$^*$-convergence in the Localized Hardy Spaces $H^1_\rho(\mathcal{X})$ and its Application." Taiwanese J. Math. 20 (4) 897 - 907, 2016. https://doi.org/10.11650/tjm.20.2016.7020

Information

Published: 2016
First available in Project Euclid: 1 July 2017

zbMATH: 1357.42022
MathSciNet: MR3535679
Digital Object Identifier: 10.11650/tjm.20.2016.7020

Subjects:
Primary: 42B35

Rights: Copyright © 2016 The Mathematical Society of the Republic of China

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Vol.20 • No. 4 • 2016
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