The space $H^1$ for nondoubling measures in terms of a grand maximal operator

Abstract

Let μ be a Radon measure on R d , which may be nondoubling. The only condition that μ must satisfy is the size condition μ(B(x,r)) < C r n , for some fixed 0 < n < d. Recently, some spaces of type BMO(μ) and H 1 (μ) were introduced by the author. These new spaces have properties similar to those of the classical spaces BMO and H 1 defined for doubling measures, and they have proved to be useful for studying the L P (μ) boundedness of Calderon-Zygmund operators without assuming doubling conditions. In this paper a characterization of the new atomic Hardy space H 1 (p) in terms of a maximal operator M Φ is given. It is shown that f belongs to H 1 (μ) if and only if f E L 1 (μ), ∫ f dμ = 0 and M Φ f f ∈ L 1 (μ), as in the usual doubling situation.