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On weak$$^*$$-convergence in the localized Hardy spaces $$H^1_\rho(\mathcal{X})$$ and its application. (English) Zbl 1357.42022
Summary: 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 $$x, y \in \mathcal{X}$$, $\rho(y) C_0 [\rho(x)]^{1/(1+k_0)} [\rho(x) + d(x, y)]^{k_0/(1+k_0)}.$ In this paper, we define a space $$\mathrm{VMO}_\rho(\mathcal{X})$$ and show that it is the predual of the localized Hardy space $$H^1_\rho(\mathcal{X})$$ introduced by D. Yang and Y. Zhou [Trans. Am. Math. Soc. 363, No. 3, 1197–1239 (2011; Zbl 1217.42044)]. Then we prove a version of the classical theorem of P. W. Jones and J.-L. Journé [Proc. Am. Math. Soc. 120, No. 1, 137–138 (1994; Zbl 0814.42011)] on weak$$^*$$-convergence in $$H^1_\rho(\mathcal{X})$$. As an application, we give an atomic characterization of $$H^1_\rho(\mathcal{X})$$.

MSC:
 42B35 Function spaces arising in harmonic analysis 46E30 Spaces of measurable functions ($$L^p$$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
Keywords:
$$H^1$$; BMO; VMO; spaces of homogeneous type
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