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

Let μ be a Radon measure in d which may be nondoubling, but should satisfy the growth condition, μ(B(x,r))Cr n for all r and xsupp(μ) and some fixed 0<nd.

The main result in this paper is that one can characterize the atomic block Hardy space H atb 1, (μ) in terms of a grand maximal operator M Φ , as in the doubling case:

A function f belongs to H atb 1, (μ) if and only if fL 1 (μ), fdμ=0 and M Φ fL 1 (μ).

42B30H p -spaces (Fourier analysis)
42B20Singular and oscillatory integrals, several variables
42B25Maximal functions, Littlewood-Paley theory