On the weak\(*\) convergence in \(H_{\mathcal{F}}^1(\mathbb{R}^n)\)
type: Item
instance of: scholarly article
title: On the weak\(*\) convergence in \(H_{\mathcal{F}}^1(\mathbb{R}^n)\)
zbMATH Open document ID: 1384.46020
author: Chin-Cheng Lin
published in: Journal of Mathematical Analysis and Applications
publication date: 2017-08-03T00:00:00Z
review text: The main result of this paper consists of proving the \(\mu\)-almost everywhere convergence of a sequence of functions in the Hardy space \(H^1_{\mathcal F}\) defined by \textit{Y. Ding} and \textit{C.-C. Lin} [Tohoku Math. J. (2) 57, No. 2, 147--170 (2005; Zbl 1085.42014)] as the predual of \(\mathrm{BMO}_{\mathcal F}\), that is, the space of bounded mean oscillations associated to a family of bounded convex sets \({\mathcal F}=\{S(x,t) \subset \mathbb{R}^N: x\in \mathbb{R}^N, \;t>0\} \) and a Borel measure \(\mu\) satisfying a doubling property with respect the parameter \(t>0\). For this purpose, the predual \(\mathrm{VMO}_{\mathcal F}(\mathbb{R}^N)\) of \(H^1_{\mathcal F}\) is previously identified, which is the closure of \(C_c(\mathbb{R}^N)\cap \mathrm{Lip}(\mathbb{R}^N)\) with respect the seminorm \(\| \cdot \|_{\mathrm{BMO}_{\mathcal F}}\).
reviewed by: Santiago Boza
Mathematics Subject Classification ID: 46E30
Mathematics Subject Classification ID: 42B30
zbMATH DE Number: 6756246
zbMATH Keywords: \(A_p\) weights
zbMATH Keywords: BMO
zbMATH Keywords: Hardy spaces
zbMATH Keywords: Monge-Ampère equation
zbMATH Keywords: VMO
MaRDI profile type: Publication
full work available at URL: j.jmaa.2017.06.005
OpenAlex ID: W2625180549
cites work: Balls and quasi-metrics: A space of homogeneous type modeling the real analysis related to the Monge-Ampère equation
cites work: Some regularity properties of solutions of Monge Ampère equation
cites work: Boundary regularity of maps with convex potentials
cites work: Real analysis related to the Monge-Ampère equation
cites work: Properties of the solutions of the linearized Monge-Ampere equation
cites work: Extensions of Hardy spaces and their use in analysis
cites work: Compensated compactness and Hardy spaces
cites work: Hardy spaces associated to the sections
cites work: On Weak Convergence in H 1 ( R d )
cites work: Lipschitz functions on spaces of homogeneous type
DOI: 10.1016/J.JMAA.2017.06.005
description: scientific article; zbMATH DE number 6756246