An invitation to harmonic analysis associated with semigroups of operators
type: Item
instance of: scholarly article
zbMATH Open document ID: 1305.42013
author: Marius Junge
author: Tao Mei
author: Javier Parcet
publication date: 2014-08-12T00:00:00Z
full work available at URL: https://arxiv.org/abs/1304.4922
Mathematics Subject Classification ID: 42B15
Mathematics Subject Classification ID: 42B20
Mathematics Subject Classification ID: 46L51
Mathematics Subject Classification ID: 47C15
Mathematics Subject Classification ID: 47D06
zbMATH DE Number: 6329357
zbMATH Keywords: Fourier multipliers
zbMATH Keywords: von Neumann algebras
zbMATH Keywords: non-commutative \(L^p\) spaces
zbMATH Keywords: Markov semigroups
zbMATH Keywords: discrete groups
zbMATH Keywords: quantum metric spaces
zbMATH Keywords: singular integral operators
zbMATH Keywords: BMO
zbMATH Keywords: Riesz transforms
MaRDI profile type: Publication
arXiv ID: 1304.4922
arXiv classification: math.CA
arXiv classification: math.FA
arXiv classification: math.OA
title: An invitation to harmonic analysis associated with semigroups of operators
review text: The authors of the paper under review are currently developing an ambitious generalisation of Calderón-Zygmund theory of singular integral operators in non-commutative settings. Their theory deals with operators acting on \(L^p\) spaces naturally associated with a semifinite von Neumann algebra \(\mathcal{M}\). If \(\mathcal{M} = L^{\infty}(\mathbb{T})\), then one recovers the traditional setting of Fourier multipliers acting on \(L^{p}(\mathbb{T})\). Other classical commutative settings, over locally compact abelian groups or Riemannian manifolds, can also be recovered. The scope of Junge-Mei-Parcet's theory, however, is much wider, and includes, in particular, non-commutative discrete groups. The paper under review is a survey of their theory, in the state it had reached by 2013.NEWLINENEWLINEThe central idea is to use a Markov semigroup \((e^{tL})_{t \geq 0}\) as a replacement for the geometric averaging operators used in commutative settings. This allows the authors to define spaces of bounded mean oscillations by using the semigroup to define a notion of mean, analogues of Fourier multipliers using the functional calculus of \(L\), analogues of Riesz transforms using Bakry's carré du champs (a bilinear form defined in terms of \(L\)), and analogues of Hardy spaces. Under additional assumptions on \((e^{tL})_{t \geq 0}\) that are satisfied, for instance, when \(L\) is the Laplace Beltrami operator on a complete Riemannian manifold with nonnegative Ricci curvature, they obtain a satisfying abstract theory. This theory shows that their Hardy and BMO spaces are appropriate end points for interpolation, and includes \(L^p\) boundedness results for Riesz transforms and a large class of analogues of Fourier multipliers. Junge-Mei-Parcet's theory can be seen as a generalisation of Stein's Littlewood-Paley theory associated with diffusion semigroups, Cowling's abstract spectral multiplier theory, and Bakry-Emery's carré du champs theory. It uses both an analytic and a p…
reviewed by: Pierre Portal
description: scientific article; zbMATH DE number 6329357