Dunkl spectral multipliers with values in UMD lattices
type: Item
instance of: scholarly article
title: Dunkl spectral multipliers with values in UMD lattices
zbMATH Open document ID: 1371.42011
author: Christoph Kriegler
author: Luc Deleaval
published in: Journal of Functional Analysis
publication date: 2017-01-17T00:00:00Z
review text: The spectral multiplier problem is a basic harmonic analytic question: for which functions \(m \in L^{\infty}(\mathbb{R})\) and which \(p \in [1,\infty)\), does the operator \(m(\Delta)\) extends from \(L^{2}(\mathbb{R}^d)\cap L^{p}(\mathbb{R}^d)\) to a bounded operator on \(L^{p}(\mathbb{R}^d)\)? The Hörmander-Mihlin Fourier multiplier theorem provides satisfying sufficient conditions, and has been generalised in many directions. One can replace \(\mathbb{R}^{d}\) by another metric measure space, one can replace \(\Delta\) by another (usually differential) operator, and one can replace \(L^p\) by another function space. In any of these directions, one can find good motivation for generalisation, either coming from applications to PDE and/or geometry, or from harmonic analysis itself (such as the desire to obtain simpler and/or more flexible proofs). The paper under review deals with two levels of generalisation that have rarely been combined before: replacing \(\Delta\) by a Dunkl Laplacian (and weighting the Lebesgue measure accordingly), and replacing \(L^p\) by a Bochner space \(L^{p}(\mathbb{R}^d;X)\) where \(X\) is a Banach space. Dunkl Laplacians are differential-difference operators that arise from deformations of the directional derivatives through the action of a reflection group associated with a root system. They naturally occur in harmonic analysis on certain Lie groups. In the other direction of generalisation, the action of spectral multipliers such as \(m(\Delta \otimes I_{X})\) on Bochner spaces \(L^{p}(\mathbb{R}^d;X)\) has been extensively studied to better understand the interactions between harmonic analysis and probability. A typical example of such interactions is the fact that Hörmander-Mihlin's theorem extends to \(X\)-valued functions if and only if \(X\) has the Unconditional Martingale Differences property (UMD). See the books [\textit{T. Hytönen} et al., Analysis in Banach spaces. Volume I. Martingales and Littlewood-Paley theory. Cham: …
reviewed by: Pierre Portal
Mathematics Subject Classification ID: 42B15
Mathematics Subject Classification ID: 42B25
Mathematics Subject Classification ID: 47A60
zbMATH DE Number: 6675463
zbMATH Keywords: spectral multiplier theorems
zbMATH Keywords: UMD valued \(L^p\) spaces
zbMATH Keywords: Dunkl Laplacian
MaRDI profile type: Publication
full work available at URL: j.jfa.2016.12.013
OpenAlex ID: W2565159635
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Dunkl spectral multipliers with values in UMD lattices (DOI)
DOI: 10.1016/J.JFA.2016.12.013

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