Let be a nonnegative Radon measure on which satisfies the growth condition that there exist constants and such that for all and , , where is the open ball centered at and having radius . In this paper, we introduce a local atomic Hardy space , a local BMO-type space and a local BLO-type space in the spirit of Goldberg and establish some useful characterizations for these spaces. Especially, we prove that the space satisfies a John-Nirenberg inequality and its predual is . We also establish some useful properties of and improve the known characterization theorems of 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 -function of Tolsa is bounded from to , and that is bounded from to .
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Guoen Hu, Dachun Yang, Dongyong Yang, , bmo, blo and Littlewood-Paley -functions with non-doubling measures. Rev. Mat. Iberoam. 25 (2009), no. 2, pp. 595–667DOI 10.4171/RMI/577