Vector-valued extensions of operators related to the Ornstein-Uhlenbeck semigroup
type: Item
instance of: scholarly article
title: Vector-valued extensions of operators related to the Ornstein-Uhlenbeck semigroup
zbMATH Open document ID: 1069.47049
DOI: 10.1007/BF02788780
author: José Luis Torrea
author: E. Harboure
author: B. Viviani
published in: Journal d'Analyse Mathématique
publication date: 2004-09-27T00:00:00Z
review text: The authors consider some operators on functions taking values on a Banach space \(X\). In this setting, the authors investigate the Riesz transforms, the Littlewood-Paley \(g\)-functions, some maximal operators like the maximal operators of Ornstein-Uhlenbeck and the corresponding Poisson semigroups, and the maximal function with respect to the Gaussian measure. Necessary and sufficient conditions on a Banach space \(X\) are found in order that the above operators be of weak type \((1,1)\) or strong type \((p,p)\) in the range \(1 < p < \infty\). These conditions on \(X\) are the same as those required for the corresponding harmonic operator: (1) the UMD-property (which means that the Hilbert transform is a bounded operator in \(L^p_X\) for some \(p\in(1,\infty)\)); (2) Lusin cotype 2 (which means boundedness of the Littlewood-Paley \(g\)-operator for functions defined in the torus); (3) the Hardy-Littlewood property (which means boundedness of the classical Hardy-Littlewood operator). Some results are new even for the classical harmonic case.
reviewed by: Michael Perelmuter
Mathematics Subject Classification ID: 47D07
Mathematics Subject Classification ID: 60J35
Mathematics Subject Classification ID: 42B20
Mathematics Subject Classification ID: 42B25
zbMATH DE Number: 2103487
zbMATH Keywords: Riesz transforms
zbMATH Keywords: Littlewood-Paley \(g\)-functions
zbMATH Keywords: maximal operators
zbMATH Keywords: UMD-property
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02788780
OpenAlex ID: W2013167527
description: scientific article; zbMATH DE number 2103487