On fractional maximal function and fractional integrals associated with the Dunkl operator on the real line
type: Item
instance of: scholarly article
title: On fractional maximal function and fractional integrals associated with the Dunkl operator on the real line
zbMATH Open document ID: 1162.42006
author: Vagif S. Guliyev
author: Yagub Y. Mammadov
published in: Journal of Mathematical Analysis and Applications
publication date: 2009-04-29T00:00:00Z
review text: The Dunkl operators are differential-difference operators associated with the reflection group \(Z_2\) on \(\mathbb{R}^1\). \textit{F. Soltani} [J. Funct. Anal. 221, No. 1, 205--225 (2005; Zbl 1071.42008)] considered the maximal operator associated with the Dunkl operators. The authors study the fractional maximal function and the fractional integral associated with the Dunkl operators. Let \(\alpha > -1/2\) and \(\mu_{\alpha}\) be the weighted Lebesgue measure on \(\mathbb{R}^1\) given by \( d\mu_{\alpha}(x) = (2^{\alpha + 1} \Gamma (\alpha + 1))^{-1} | x |^{2 \alpha +1}dx, \) and \(L_{p, \alpha}(\mathbb{R}^1)\) is the space of functions \(f\) such that \( \| f \|_{p, \alpha}= \left( \int_{\mathbb{R}^1} | f(x) |^{p} d \mu_{\alpha}(x) \right)^{1/p}< \infty. \) The operators \(\tau_x, x \in \mathbb{R}^1\) are Dunkl translation operators, and the Dunkl-type fractional integral is defined by \[ I_{\beta}f(x) = \int_{\mathbb{R}^1} \tau_x | y |^{\beta - 2\alpha -2}f(y) d \mu_{\alpha}(y) \quad 0 < \beta < 2\alpha +2. \] They prove the following: If \(1<p< \frac{2\alpha +2}{\beta}\), then the condition \(\frac{1}{p} = \frac{1}{q}- \frac{\beta}{2\alpha +2}\) is the necessary and sufficient for the boundedness of \(I_{\beta}\) from \(L_{p, \alpha}\) to \(L_{q, \alpha}\). They also consider the estimates on BMO and the Besov spaces.
Mathematics Subject Classification ID: 42B20
Mathematics Subject Classification ID: 42B25
Mathematics Subject Classification ID: 42-02
zbMATH DE Number: 5549442
zbMATH Keywords: Dunkl operators
zbMATH Keywords: Dunkl-type fractional integral operator
zbMATH Keywords: Dunkl-type BMO
zbMATH Keywords: Dunkl-type Besov spaces
reviewed by: Yasuo Komori-Furuya
MaRDI profile type: Publication
full work available at URL: j.jmaa.2008.11.083
OpenAlex ID: W2007090926
cites work: Dunkl translation and uncentered maximal operator on the real line
cites work: Q3421762
cites work: Further results of integrability for the Dunkl transform
cites work: Besov spaces associated with Dunkl's operator
cites work: On maximal function and fractional integral, associated with the Bessel differential operator
cites work: Differential-Difference Operators Associated to Reflection Groups
cites work: Besov-type spaces for the Dunkl operator on the real line
cites work: Generalized Fock spaces and Weyl relations for the Dunkl kernel on the real line
cites work: \(L^{p}\)-Fourier multipliers for the Dunkl operator on the real line.
cites work: Littlewood--Paley operators associated with the Dunkl operator on \(\mathbb R\).
cites work: Q4213324
cites work: Paley-Wiener Theorems for the Dunkl Transform and Dunkl Translation Operators
cites work: Q5562752
DOI: 10.1016/J.JMAA.2008.11.083
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description: scientific article; zbMATH DE number 5549442