## Taiwanese Journal of Mathematics

### On Weak$^*$-convergence in the Localized Hardy Spaces $H^1_\rho(\mathcal{X})$ and its Application

#### 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 $H^1_\rho(\mathcal{X})$.

#### Article information

Source
Taiwanese J. Math., Volume 20, Number 4 (2016), 897-907.

Dates
First available in Project Euclid: 1 July 2017

https://projecteuclid.org/euclid.twjm/1498874496

Digital Object Identifier
doi:10.11650/tjm.20.2016.7020

Mathematical Reviews number (MathSciNet)
MR3535679

Zentralblatt MATH identifier
1357.42022

Subjects
Primary: 42B35: Function spaces arising in harmonic analysis

#### Citation

Duc, Dinh Thanh; Hung, Ha Duy; Ky, Luong Dang. On Weak$^*$-convergence in the Localized Hardy Spaces $H^1_\rho(\mathcal{X})$ and its Application. Taiwanese J. Math. 20 (2016), no. 4, 897--907. doi:10.11650/tjm.20.2016.7020. https://projecteuclid.org/euclid.twjm/1498874496

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