Atomic Hardy-type spaces between \( H^1\) and \( L^1\) on metric spaces with non-doubling measures
type: Item
instance of: scholarly article
title: Atomic Hardy-type spaces between \( H^1\) and \( L^1\) on metric spaces with non-doubling measures
zbMATH Open document ID: 1266.42061
author: Li Guang Liu
author: Dongyong Yang
author: Dachun Yang
publication date: 2013-03-18T00:00:00Z
review text: Let \(({\mathcal Y}, d, d\lambda)\) be one of the following three metric spaces with non-doubling measures: (1) \((\mathbb{R^N}, |\cdot|, \mu)\) with \( |\cdot|\) the Euclidean distance and \(\mu\) a non-negative Radon measure on \(\mathbb{R^N}\), satisfying a polynomial growth condition. (2) \((\mathbb{R^N}, |\cdot|, d\gamma)\) the Gauss measure metric space. (3) the space \((S,d,\rho)\) where \(S=\mathbb{R^N}\times \mathbb{R^+}\) is the affine group endowed with the product \((x,a)\cdot(x',a')=(x+ax',aa')\), \(d\) is the left-invariant Riemannian metric and \(\rho\) is the right Haar measure on \(S\). The paper deals with the study of some properties of the atomic Hardy spaces \(\{X_s({\mathcal Y})\}_{0<s\leq \infty}\) and the BMO-type spaces \(\{\text{BMO}({\mathcal Y},s)\}_{0<s< \infty}\). In particular, it is proved that the dual space of \(X_s({\mathcal Y})\) is \(\text{BMO}({\mathcal Y},s)\) when \(s\in (0,\infty)\), \(X_s({\mathcal Y})=H^1({\mathcal Y})\) when \(s\in (0,1]\), and \(X_{\infty}({\mathcal Y})=L^1\) or \(L_0^1({\mathcal Y})\), subspace of functions in \(L^1({\mathcal Y})\) with integral 0. As an application, it is shown that a sublinear operator \(T\), which is bounded from \(L^1({\mathcal Y})\) to \(L^{1,\infty}({\mathcal Y})\), is bounded from \(X_s({\mathcal Y})\) to \(L^1({\mathcal Y})\) if and only if \(T\) maps all \((p,s)\)-atoms with \(p\in (1,\infty)\) into uniformly bounded elements of \(L^1({\mathcal Y})\) . Interpolation results in this context are also proved. These results about the boundedness of linear and sublinear operators are applied to the Calderón-Zygmund operators on \((\mathbb{R^N}, |\cdot|, \mu)\), the imaginary powers of the Ornstein-Uhlenbeck operator on \((\mathbb{R^N}, |\cdot|, d\gamma)\) and the spectral operator associated with the spectral multiplier on \((S,d,\rho)\). All these results are generalization of the corresponding results obtained for Euclidean spaces by several authors such as \textit{C. Sweezy} [Proc…
reviewed by: Santiago Boza
Mathematics Subject Classification ID: 42B35
Mathematics Subject Classification ID: 42B30
Mathematics Subject Classification ID: 42B20
zbMATH DE Number: 6145916
zbMATH Keywords: atomic Hardy-type space
zbMATH Keywords: BMO-type space
zbMATH Keywords: Non-doubling measure
zbMATH Keywords: polynomial growth
zbMATH Keywords: Gauss measure
zbMATH Keywords: \(ax+b\)-group
MaRDI profile type: Publication
full work available at URL: s10114-011-9118-7
OpenAlex ID: W2060183721
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DOI: 10.1007/S10114-011-9118-7
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published in: Acta Mathematica Sinica, English Series
description: scientific article; zbMATH DE number 6145916