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The space \(H^1\) for nondoubling measures in terms of a grand maximal operator. (English) Zbl 1021.42010

Let \(\mu\) be a Radon measure in \(\mathbb{R}^{d}\) which may be nondoubling, but should satisfy the growth condition, \(\mu(B(x,r))\leq Cr^{n}\) for all \(r\) and \(x\in\text{supp}(\mu)\) and some fixed \(0<n\leq d\).
The main result in this paper is that one can characterize the atomic block Hardy space \(H^{1,\infty}_{\text{atb}}(\mu)\) in terms of a grand maximal operator \(M_{\Phi}\), as in the doubling case:
A function \(f\) belongs to \(H^{1,\infty}_{\text{atb}}(\mu)\) if and only if \(f\in L^{1}(\mu)\), \(\int fd\mu=0\) and \(M_{\Phi}f\in L^{1}(\mu)\).

MSC:

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