The space H¹ for nondoubling measures in terms of a grand maximal operator

Let μ be a Radon measure on ℝd, which may be nondoubling. The only condition that μ must satisfy is the size condition μ(B(x, r)) ≤ C rn, for some fixed 0 < n ≤ d. Recently, some spaces of type BMO(μ) and H1(μ) were introduced by the author. These new spaces have properties similar to those of the classical spaces BMO and H1 defined for doubling measures, and they have proved to be useful for studying the Lp(μ) boundedness of Calderón-Zygmund operators without assuming doubling conditions. In this paper a characterization of the new atomic Hardy space H1(μ) in terms of a maximal operator MΦ is given. It is shown that f belongs to H1(μ) if and only if f ε L1(μ), ∫ f dμ = 0 and MΦ f ε L1(μ), as in the usual doubling situation.