- journal article metadata
European Mathematical Society Publishing House
2016-09-19 17:05:45
Revista Matemática Iberoamericana
Rev. Mat. Iberoamericana
RMI
0213-2230
2235-0616
General
10.4171/RMI
http://www.ems-ph.org/doi/10.4171/RMI
subscribers
European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society (from 2012)
25
2009
2
$h^1$, bmo, blo and Littlewood-Paley $g$-functions with non-doubling measures
Guoen
Hu
Zhengzhou Information Science and Technology Insti, ZHENGZHOU, CHINA
Dachun
Yang
Beijing Normal University, BEIJING, CHINA
Dongyong
Yang
Beijing Normal University, BEIJING, CHINA
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$(\mu)$, $H^1(\mu)$, $h_{\rm atb}^{1,\fz}(\mu)$
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 ${\mathop\mathrm{rblo}(\mu)}$.
Fourier analysis
Abstract harmonic analysis
Operator theory
General
595
667
10.4171/RMI/577
http://www.ems-ph.org/doi/10.4171/RMI/577