Spectral multipliers for the Ornstein-Uhlenbeck semigroup
type: Item
instance of: scholarly article
title: Spectral multipliers for the Ornstein-Uhlenbeck semigroup
zbMATH Open document ID: 0939.42007
DOI: 10.1007/BF02791138
author: José García-Cuerva
author: Giancarlo Mauceri
author: Peter Sjögren
author: José Luis Torrea
published in: Journal d'Analyse Mathématique
publication date: 2000-07-03T00:00:00Z
review text: The authors study the possibility to introduce functional calculus generated by the Laplacian corresponding to the multidimensional space with Gaussian measure. Though they prove, that the general result of \textit{S. Meda} [Proc. Am. Math. Soc. 110, No. 3, 639-647 (1990; Zbl 0760.42007)] cannot be applied to the case. They succeed in constructing the multipliers of Laplace transform type associated to the Laplacian. The boundednes on \(L^p\) for \(1<p<\infty\) and \((1,1)\)-boundedness of these operators are also proved. The authors also study the question of representability of the operators as principal value (or a weaker analog of the principal value) of a singular integral. Namely, if \(m\) is of Laplace transform type, then there exists a bounded function \(\alpha\) on \((0,\infty)\), such that \(m(L)f(x)=\lim_{\varepsilon\rightarrow 0+} (\alpha(\varepsilon)f(x)+\int_{|x-y|>\varepsilon} K(x,y)f(y)dy),\) for almost all \(x\), whenever \(f\) is a test function.
Mathematics Subject Classification ID: 42B15
Mathematics Subject Classification ID: 42B25
Mathematics Subject Classification ID: 42B20
Mathematics Subject Classification ID: 47D03
Mathematics Subject Classification ID: 47A60
zbMATH DE Number: 1369965
zbMATH Keywords: Hermite semigroup
zbMATH Keywords: functional calculus of an operator
zbMATH Keywords: weak type (1,1)
zbMATH Keywords: maximal function
zbMATH Keywords: kernel of an operator
zbMATH Keywords: Laplace transform type function
reviewed by: Maria Roginskaya
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02791138
OpenAlex ID: W2032464775
description: scientific article; zbMATH DE number 1369965