Riesz transforms for a non-symmetric Ornstein--Uhlenbeck semigroup
type: Item
instance of: scholarly article
title: Riesz transforms for a non-symmetric Ornstein--Uhlenbeck semigroup
zbMATH Open document ID: 1163.47039
author: Giancarlo Mauceri
author: Luana Noselli
published in: Semigroup Forum
publication date: 2009-03-20T00:00:00Z
review text: Let \(({\mathcal H}_t)_{t\geq 0}\) be the Ornstein--Uhlenbeck semigroup on \(\mathbb{R}^d\) with covariance matrix \(Q\) and drift \(B\), where \(B\) is any real matrix whose eigenvalues have a negative real part, and let \(\gamma_\infty\) be the invariant measure. The corresponding Riesz transform \({\mathcal R}_\alpha\) of order \(\alpha\), where \(\alpha\) is a \(d\)-dimensional multi-index, is defined by \[ {\mathcal R}_\alpha f:= (1/\Gamma(|\alpha|/2))\,\partial^\alpha\int^\infty_0 t^{|\alpha|/2- 1}{\mathcal H}_t f\,dt \] for \(f\in L^p_0(\gamma_\infty)\), where \(L^p_0(\gamma_\infty)\) is the subspace of all functions with \(\gamma_\infty\)-mean \(0\) in \(L^p(\gamma_\infty)\). For \(1<p<\infty\), the Riesz transforms of any order are bounded as mappings from \(L^p_0(\gamma_\infty)\) to \(L^p(\gamma_\infty)\). If \(Q= I\) and \(B= -\lambda(I+ R)\), where \(\lambda> 0\) and \(R\) is a skew adjoint matrix generating a one-parameter group of periodic rotations, then the first-order Riesz transforms are of weak type 1 with respect to \(\gamma_\infty\), i.e., there exists a constant \(C\) such that \[ \gamma_\infty\{x\in \mathbb{R}^d:|{\mathcal R}_\alpha f(x)|> s\}\leq C\| f\|_1/s \] for each a with \(|\alpha|= 1\), every \(f\in L^2_0(\gamma_\infty)\), and all \(s>0\).
reviewed by: Heinrich Hering
Mathematics Subject Classification ID: 47D62
Mathematics Subject Classification ID: 44A15
Mathematics Subject Classification ID: 60B15
Mathematics Subject Classification ID: 60E10
Mathematics Subject Classification ID: 60G15
zbMATH DE Number: 5530524
zbMATH Keywords: Gaussian measure
zbMATH Keywords: Ornstein-Uhlenbeck semigroup
zbMATH Keywords: singular integrals
zbMATH Keywords: Riesz transforms
MaRDI profile type: Publication
full work available at URL: s00233-007-9028-2
OpenAlex ID: W2019688209
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DOI: 10.1007/S00233-007-9028-2
description: scientific article; zbMATH DE number 5530524