Revista Matemática Iberoamericana

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

Guoen Hu , Dachun Yang , and Dongyong Yang

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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 ${\mathop\mathrm{rblo}(\mu)}$.

Article information

Rev. Mat. Iberoamericana, Volume 25, Number 2 (2009), 595-667.

First available in Project Euclid: 13 October 2009

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 42B35: Function spaces arising in harmonic analysis
Secondary: 42B25: Maximal functions, Littlewood-Paley theory 42B30: $H^p$-spaces 47A30: Norms (inequalities, more than one norm, etc.) 43A99: None of the above, but in this section

non-doubling measure approximation of the identity maximal operator John-Nirenberg inequality duality cube of generation $g$-function RBMO$(\mu)$ rbmo$(\mu)$ RBLO$(\mu)$ rblo$(\mu)$ $H^1(\mu)$ $h_{\rm atb}^{1,\fz}(\mu)$


Hu, Guoen; Yang, Dachun; Yang, Dongyong. $h^1$, bmo, blo and Littlewood-Paley $g$-functions with non-doubling measures. Rev. Mat. Iberoamericana 25 (2009), no. 2, 595--667.

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