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June, 2009 $h^1$, bmo, blo and Littlewood-Paley $g$-functions with non-doubling measures
Guoen Hu , Dachun Yang , Dongyong Yang
Rev. Mat. Iberoamericana 25(2): 595-667 (June, 2009).


Let $\mu$ be a nonnegative Radon measure on ${\mathbb R}^d$ which satisfies the growth condition that there exist constants $C_0 > 0$ and $n\in(0,d]$ such that for all $x\in{\mathbb R}^d$ and $r > 0$, $\mu(B(x,\,r)) \le C_0 r^n$, 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 ${h_{\rm atb}^{1,\infty}(\mu)}$, a local BMO-type space ${\mathop\mathrm{rbmo}(\mu)}$ and a local BLO-type space ${\mathop\mathrm{rblo}(\mu)}$ in the spirit of Goldberg and establish some useful characterizations for these spaces. Especially, we prove that the space ${\mathop\mathrm{rbmo}(\mu)}$ satisfies a John-Nirenberg inequality and its predual is ${h_{\rm atb}^{1,\infty}(\mu)}$. We also establish some useful properties of ${\mathop\mathrm{RBLO}\,(\mu)}$ and improve the known characterization theorems of ${\mathop\mathrm{RBLO}(\mu)}$ 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 ${h_{\rm atb}^{1,\infty}(\mu)}$ to ${L^1(\mu)}$, and that $[g(f)]^2$ is bounded from ${\mathop\mathrm{rbmo}(\mu)}$ to ${\mathop\mathrm{rblo}(\mu)}$.


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Guoen Hu . Dachun Yang . Dongyong Yang . "$h^1$, bmo, blo and Littlewood-Paley $g$-functions with non-doubling measures." Rev. Mat. Iberoamericana 25 (2) 595 - 667, June, 2009.


Published: June, 2009
First available in Project Euclid: 13 October 2009

zbMATH: 1179.42018
MathSciNet: MR2569548

Primary: 42B35
Secondary: 42B25 , 42B30 , 43A99 , 47A30

Keywords: $g$-function , $H^1(\mu)$ , $h_{\rm atb}^{1,\fz}(\mu)$ , approximation of the identity , cube of generation , Duality , John-Nirenberg inequality , Maximal operator , Non-doubling measure , RBLO$(\mu)$ , RBLO$(\mu)$ , RBMO$(\mu)$ , RBMO$(\mu)$

Rights: Copyright © 2009 Departamento de Matemáticas, Universidad Autónoma de Madrid

Vol.25 • No. 2 • June, 2009
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