Brownian martingales and some new results on interpolation of Hardy spaces
type: Item
instance of: scholarly article
title: Brownian martingales and some new results on interpolation of Hardy spaces
zbMATH Open document ID: 0946.60035
DOI: 10.1023/A:1008604106453
author: Quanhua Xu
published in: Potential Analysis
publication date: 2000-10-22T00:00:00Z
review text: The following result is proved: If \(f\in H_1(\mathbb{R}^n) +L_\infty (\mathbb{R}^n)\), then there exist two linear operators \(U\) and \(V\) such that \(Uf=Nf\), \(U\) is contractive from \(H_1(\mathbb{R}^n)\) to \(L_1(\mathbb{R}^n)\) and also from \(L_\infty (\mathbb{R}^n)\) to \(L_\infty (\mathbb{R}^n)\) and, moreover, \(V(Nf)=f\) and \(V\) is bounded from \(L_1(\mathbb{R}^n)\) to \(H_1(\mathbb{R}^n)\) and from \(L_\infty(\mathbb{R}^n)\) to \(L_\infty (\mathbb{R}^n)\), where \(Nf\) denotes the nontangential maximal function of \(f\). From this it easily follows that the couple \((H_1(\mathbb{R}^n)\), \(L_\infty (\mathbb{R}^n))\) is a Calderón couple. These results are proved with the help of Brownian motion and Hardy spaces containing continuous martingales. Similar results are proved for the couple \((L_p(\mathbb{R}^n)\), BMO\((\mathbb{R}^n))\) \((1<p<\infty)\).
reviewed by: Ferenc Weisz
Mathematics Subject Classification ID: 60G44
Mathematics Subject Classification ID: 46M15
Mathematics Subject Classification ID: 46E10
zbMATH DE Number: 1385173
zbMATH Keywords: classical and martingale Hardy spaces
zbMATH Keywords: Brownian motion
zbMATH Keywords: interpolation
zbMATH Keywords: Calderón couple
MaRDI profile type: Publication
full work available at URL: a:1008604106453
OpenAlex ID: W28757063
description: scientific article; zbMATH DE number 1385173