The authors first give a detailed proof on the coincidence between atomic Hardy spaces of Coifman and Weiss on a space of homogeneous type with those Hardy spaces on the same underlying space with the original distance replaced by the measure distance. Then the authors present some general criteria which guarantee the boundedness of considered linear operators from a Hardy space to some Lebesgue space or Hardy space, provided that it maps all atoms into uniformly bounded elements of that Lebesgue space or Hardy space. Third, the authors obtain the boundedness in Hardy spaces of singular integrals with kernels only having weak regularity by characterizing these Hardy spaces with a new kind of molecules, which is deeply related to the kernels of considered singular integrals. Finally, as an application, the authors obtain the boundedness in Hardy spaces of Monge-Ampère singular integral operators.

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Let µ be a nonnegative Radon measure on R d which satisfies the growth condition that there exist constants C 0 > 0 and n ∈ (0, d] such that for all x ∈ R d and r > 0, µ(B(x, r)) ≤ C 0 r n , where B(x, r) is the open ball centered at x and having radius r. In this paper, we introduce a local atomic Hardy space h 1, ∞ atb (µ), a local BMO-type space rbmo (µ) and a local BLO-type space rblo (µ) in the spirit of Goldberg and establish some useful characterizations for these spaces. Especially, we prove that the space rbmo (µ) satisfies a John-Nirenberg inequality and its predual is h 1, ∞ atb (µ). We also establish some useful properties of RBLO (µ) and improve the known characterization theorems of RBLO (µ) in terms of the natural maximal function by removing the assumption on the regularity condition. Moreover, the relations of these local spaces with known corresponding function spaces are also presented. As applications, we prove that the inhomogeneous Littlewood-Paley g-function g(f) of Tolsa is bounded from h 1, ∞ atb (µ) to L 1 (µ), and that [g(f)] 2 is bounded from rbmo (µ) to rblo (µ).

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