\(h^1\), bmo, blo and Littlewood-Paley \(g\)-functions with non-doubling measures. (English) Zbl 1179.42018

Hu, Guoen; Yang, Dachun; Yang, Dongyong

Rev. Mat. Iberoam. 25, No. 2, 595-667 (2009).

Let \(\mu\) be a non-doubling 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 \(r> 0\), \[ \mu(B(x, r)) \leq C_{0}r^n, \] where \(B(x, r)\) is the open ball centered at \(x\) and having radius \(r\).
In this paper the authors introduce a local atomic Hardy space \(h_{atb}^{1, \infty}(\mu)\), a local BMO-type space \(rbmo(\mu)\) and a local BLO-type space \(rblo(\mu)\), and establish characterizations for these spaces. In particular, the authors prove that the space \(rbmo(\mu)\) satisfies a John-Nirenberg inequality and its predual is \(h_{atb}^{1, \infty}(\mu)\).
The authors also improve the known characterization theorems of \(RBLO(\mu)\) in terms of the natural maximal function by removing the assumption on the regularity condition, and obtain a characterization of \(rblo(\mu)\) by a local maximal operator.
Moreover, the relations of these local spaces with the correspnding spaces: the Hardy space \(H^1(\mu)\), the BMO-type space \(RBMO(\mu)\) and the BLO-type space \(RBLO(\mu)\), are presented.
As applications, the authors prove that the inhomogeneous Littlewood-Paley \(g\)-function \(g(f)\) is bounded from \(h_{atb}^{1, \infty}(\mu)\) to \(L^1(\mu)\) , and that \([g(f)]^2\) is bounded from \(rbmo(\mu)\) to \(rblo(\mu)\). As a collorary, the authors obtain the boundedness of the Littlewoo-Paley \(g\)-function from \(rbmo(\mu)\) to \(rblo(\mu)\).

Reviewer: Koichi Saka (Akita)

MSC:

42B35	Function spaces arising in harmonic analysis
42B25	Maximal functions, Littlewood-Paley theory
42B30	\(H^p\)-spaces

Keywords:

non-doubling measure; maximal function; John-Nirenberg inequality; Littlewood-Paley \(g\)-function; Hardy space; BMO

References:

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