\(L^{p}\)-Fourier multipliers for the Dunkl operator on the real line.
type: Item
instance of: scholarly article
title: \(L^{p}\)-Fourier multipliers for the Dunkl operator on the real line.
zbMATH Open document ID: 1045.43003
author: Fethi Soltani
published in: Journal of Functional Analysis
publication date: 2004-06-11T00:00:00Z
review text: The Dunkl transform \({\mathcal F}_\alpha\), \(\alpha\geq -1/2\), is a generalization of the Fourier transform \({\mathcal F}\), which corresponds to \(\alpha=-1/2\). Starting with the Dunkl operator: \[ (\Lambda_\alpha f)(x)={d\over dx}\,f(x) + {2\alpha+1\over x}\bigg[{f(x)-f(-x)\over 2}\bigg], \] one considers the Dunkl kernel \(E_\alpha(\lambda x)\), \(\lambda\in {\mathbb C}\,\), which is the unique solution of the equation \(\Lambda_\alpha f(x)=\lambda f(x)\) with \(f(0)=1\). An explicit expression of \(E_\alpha(\lambda x)\) can be given, using a series with the gamma function, and \(E_{-1/2}(\lambda x)=\text{ e}^{\lambda x}\). The Dunkl transform is then defined, for \(\lambda\in {\mathbb R}\), by: \[ \big({\mathcal F}_\alpha f\big)(\lambda)= \int_{\mathbb R} E_\alpha (-i\lambda x) f(x)\,d\mu_\alpha(x), \] where \(d\mu_\alpha(x)= \big(2^{\alpha+1} \Gamma(\alpha+1)\big)^{-1}| x|^{2\alpha+1} dx\). An inversion formula and a Plancherel theorem are available for the Dunkl transform [\textit{M. F. E. de Jeu}, Invent. Math. 113, No. 1, 147--162 (1993; Zbl 0789.33007)]. The aim of this paper is to prove the analogue of Hörmander's theorems on Fourier multipliers for the Dunkl transform. For this, the author needs weighted Sobolev spaces. His results allow the author to give some examples of multipliers. In the last part, the author investigates the \(L^p-L^q\) boundedness of multipliers (\(q>p\)).
reviewed by: Daniel Li
Mathematics Subject Classification ID: 43A15
Mathematics Subject Classification ID: 42A38
Mathematics Subject Classification ID: 42A45
Mathematics Subject Classification ID: 42B10
Mathematics Subject Classification ID: 43A32
Mathematics Subject Classification ID: 44A15
zbMATH DE Number: 2073657
zbMATH Keywords: Dunkl operator
zbMATH Keywords: Dunkl transform
zbMATH Keywords: multiplier
zbMATH Keywords: weighted Sobolev spaces
MaRDI profile type: Publication
full work available at URL: j.jfa.2003.11.009
OpenAlex ID: W2086404182
cites work: \(\mathbb{L}_ p\) Fourier multipliers on Riemannian symmetric spaces of the noncompact type
cites work: The Calogero-Sutherland model and generalized classical polynomials
cites work: Q4112938
cites work: Fourier Multipliers for L p on Chébli-Trimèche Hypergroups
cites work: A unification of Knizhnik-Zamolodchikov and Dunkl operators via affine Hecke algebras
cites work: Q2548398
cites work: \(L^ p\) estimates for strongly singular integrals on spaces of homogeneous type
cites work: Differential-Difference Operators Associated to Reflection Groups
cites work: Integral Kernels with Reflection Group Invariance
cites work: Q3915429
cites work: The Dunkl transform
cites work: A Weak-Type Estimate for Fourier-Bessel Multipliers
cites work: Estimates for translation invariant operators in \(L^p\) spaces
cites work: Exact operator solution of the Calogero-Sutherland model
cites work: Harmonic analysis for certain representations of graded Hecke algebras
cites work: Generalized Fock spaces and Weyl relations for the Dunkl kernel on the real line
cites work: Q5607484
cites work: Q4169288
cites work: Paley-Wiener Theorems for the Dunkl Transform and Dunkl Translation Operators
DOI: 10.1016/J.JFA.2003.11.009
Recommended article: Relating transplantation and multipliers for Dunkl and Hankel transforms
Recommended article: The multiplier of the interval [−1, 1] for the Dunkl transform of arbitrary order on the real line
Recommended article: Local restriction theorem and maximal Bochner-Riesz operators for the Dunkl transforms
Recommended article: On the Dunkl convolution equations in spaces of type
Recommended article: Linear and bilinear multiplier operators for the Dunkl transform
Recommended article: Q5714603
Recommended article: Inversion of the Dunkl intertwining operator and its dual using Dunkl wavelets.
Recommended article: Q5376294
Recommended article: Littlewood--Paley operators associated with the Dunkl operator on \(\mathbb R\).
Recommended article: A multiplier theorem for Herz-type Hardy spaces associated with the Dunkl transform
description: scientific article; zbMATH DE number 2073657