Holomorphy of spectral multipliers of the Ornstein--Uhlenbeck operator
type: Item
instance of: scholarly article
title: Holomorphy of spectral multipliers of the Ornstein--Uhlenbeck operator
zbMATH Open document ID: 1069.47017
author: Waldemar Hebisch
author: Giancarlo Mauceri
author: Stefano Meda
published in: Journal of Functional Analysis
publication date: 2004-08-16T00:00:00Z
review text: The closure \(\mathcal L\) of the Ornstein--Uhlenbeck operator has spectral resolution \[ {\mathcal L}f=\sum_{n=0}^\infty n P_n f, \] where \(P_n\) is the orthogonal projection onto the linear span of Hermite polynomials of degree \(n\) in \(d\) variables. For a given function \(M\) defined in \( \mathbb R^+\), define the operator \(M({\mathcal L})f=\sum_{n=0}^\infty M(n) P_n f\). Then \(M\) is said to be an \(L^p(\gamma)\) spectral multiplier if \(M({\mathcal L})\) extends to a bounded operator on \(L^p(\gamma)\) (\(\gamma\) being the Gauss measure on \( \mathbb R^d\)). In this paper, the authors study the problem of finding conditions on the \(L^p(\gamma)\) spectral multipliers \(M\) that force \(M\) to extend to a holomorphic function in some sector containing the positive real line. In fact, they prove that if \(1<p<\infty\), \(p\neq 2\), and \(M\) is a continuous function on \( \mathbb R^+\) such that \(\sup_{t>0} | | M(t{\mathcal L})| | _{L^p(\gamma)}<\infty,\) then \(M\) extends to a holomorphic function in the sector \(\{z\in{\mathbb C}: | \arg\;z| <\arcsin | 2/p-1| \}\). This type of result is a continuation of some previous results in [J. Funct. Anal. 183, 413--450 (2001; Zbl 0995.47010)]. Related results for the case \(p=1\) are also presented.
Mathematics Subject Classification ID: 47A60
Mathematics Subject Classification ID: 42B15
Mathematics Subject Classification ID: 47D03
Mathematics Subject Classification ID: 60G15
zbMATH DE Number: 2091740
zbMATH Keywords: Ornstein--Uhlenbeck operator
zbMATH Keywords: functional calculus
zbMATH Keywords: uniform spectral multiplier
reviewed by: María J. Carro
MaRDI profile type: Publication
full work available at URL: j.jfa.2003.08.010
OpenAlex ID: W2078981059
cites work: Q4309937
cites work: On \(L^ p\) spectral multipliers for a solvable Lie group
cites work: L p -multipliers for Noncompact Symmetric Spaces
cites work: Harmonic analysis on semigroups
cites work: Q4881087
cites work: Q4041031
cites work: Q5618011
cites work: The hypercontractive approach to exactly bounding an operator with complex Gaussian kernel
cites work: Spectral multipliers for the Ornstein-Uhlenbeck semigroup
cites work: Functional calculus for the Ornstein-Uhlenbeck operator
cites work: Spectral multipliers on exponential growth solvable Lie groups
cites work: Sub-Laplacians of holomorphic \(L^p\)-type on rank one AN-groups and related solvable groups
cites work: Analyticity of Submarkovian Semigroups
cites work: A General Multiplier Theorem
cites work: Q3909512
DOI: 10.1016/J.JFA.2003.08.010
Recommended article: Functional calculus for the Ornstein-Uhlenbeck operator
Recommended article: Functional calculus for the Laguerre operator
Recommended article: An \(L^1\)-estimate for certain spectral multipliers associated with the Ornstein-Uhlenbeck operator
Recommended article: Q5188820
Recommended article: Multivariate spectral multipliers for systems of Ornstein-Uhlenbeck operators
Recommended article: Endpoint estimates for first-order Riesz transforms associated to the Ornstein-Uhlenbeck operator
Recommended article: Littlewood-Paley and spectral multipliers on weighted \(L^p\) spaces
Recommended article: Spectral multipliers for Markov chains
Recommended article: Q5489494
Recommended article: \(L^{1}\)-spectrum of Banach space valued Ornstein--Uhlenbeck operators
description: scientific article; zbMATH DE number 2091740