$h^1$, bmo, blo and Littlewood-Paley $g$-functions with non-doubling measures

Let μ be a nonnegative Radon measure on Rd which satisfies the growth condition that there exist constants C0 > 0 and n ∈ (0, d] such that for all x ∈ Rd and r > 0, μ(B(x, r)) ≤ C0r, 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 h1,∞ 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 h1,∞ 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 h1,∞ atb (μ) to L 1(μ), and that [g(f)]2 is bounded from rbmo (μ) to rblo (μ).