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

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.)
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