Functional calculus for the Laguerre operator
type: Item
instance of: scholarly article
title: Functional calculus for the Laguerre operator
zbMATH Open document ID: 1071.47020
published in: Mathematische Zeitschrift
publication date: 2005-04-14T00:00:00Z
review text: In this paper, the author studies the \(L^p\) boundedness of spectral multipliers associated to the multidimensional Laguerre operator \({\mathcal L}_\alpha\) on the Euclidean space. It is known that the analysis of the Laguerre operator can be interpreted as the analysis of the Ornstein-Uhlenbeck operator acting on polyradial functions. Using this relationship, the author can reduce the multiplier problem to the study of the boundedness of certain auxiliary operators which involve the purely imaginary powers of \({\mathcal L}_\alpha\). The main result is that if \(M\) is a bounded holomorphic function in the sector \(S= \{z\in\mathbb{C}:|\arg z|<\arcsin|2/p- 1|\}\), satisfying a suitable Hörmander type condition on the boundary, then the spectral operator \(M({\mathcal L}_\alpha)\) is bounded on \(L^q\) (with respect to the Laguerre measure) for a range of values \(q\) which depends on \(p\). The author also shows that holomorphy in the sector \(S\) is a necessary condition for the boundedness of the spectral multiplier whose norm is invariant under dilations.
reviewed by: Xuan Thinh Duong
Mathematics Subject Classification ID: 47A60
Mathematics Subject Classification ID: 42B15
Mathematics Subject Classification ID: 47D03
zbMATH DE Number: 2156069
zbMATH Keywords: \(L^p\) boundedness of spectral multipliers
zbMATH Keywords: multidimensional Laguerre operator
author: Emanuela Sasso
MaRDI profile type: Publication
full work available at URL: s00209-004-0729-x
OpenAlex ID: W1968443500
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DOI: 10.1007/S00209-004-0729-X
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description: scientific article; zbMATH DE number 2156069