Volume 53, Issue 1
Boundedness Characterization of Maximal Commutators on Orlicz Spaces in the Dunkl Setting

J. Math. Study, 53 (2020), pp. 45-65.

Published online: 2020-03

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• Abstract

On the real line, the Dunkl operators

$$D_{\nu}(f)(x):=\frac{d f(x)}{dx} + (2\nu+1) \frac{f(x) - f(-x)}{2x}, ~~ \quad\forall \, x \in \mathbb{R}, ~ \forall \, \nu \ge -\tfrac{1}{2}$$

are differential-difference operators associated with the reflection group $\mathbb{Z}_2$ on $\mathbb{R}$, and on the $\mathbb{R}^d$ the Dunkl operators $\big\{D_{k,j}\big\}_{j=1}^{d}$ are the differential-difference operators associated with the reflection group $\mathbb{Z}_2^d$ on $\mathbb{R}^{d}$.

In this paper, in the setting $\mathbb{R}$ we show that $b \in BMO(\mathbb{R},dm_{\nu})$ if and only if the maximal commutator $M_{b,\nu}$ is bounded on Orlicz spaces $L_{\Phi}(\mathbb{R},dm_{\nu})$. Also in the setting $\mathbb{R}^{d}$ we show that $b \in BMO(\mathbb{R}^{d},h_{k}^{2}(x) dx)$ if and only if the maximal commutator $M_{b,k}$ is bounded on Orlicz spaces $L_{\Phi}(\mathbb{R}^{d},h_{k}^{2}(x) dx)$.

• Keywords

Maximal operator, Orlicz space, Dunkl operator, commutator, BMO.

42B20, 42B25, 42B35

vagif@guliyev.com (Vagif S. Guliyev)

fmuslumova@gmail.com (Fatma A. Muslumova)

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@Article{JMS-53-45, author = {Vagif S. and Guliyev and vagif@guliyev.com and 6783 and Department of Mathematics, Dumlupinar University, Kutahya, Turkey and Vagif S. Guliyev and Yagub Y. and Mammadov and yagubmammadov@yahoo.com and 6784 and Department of Informatics, Nakhchivan State University, Nakhchivan, Azerbaijan and Yagub Y. Mammadov and Fatma A. and Muslumova and fmuslumova@gmail.com and 6785 and Department of Informatics, Nakhchivan State University, Nakhchivan, Azerbaijan and Fatma A. Muslumova}, title = {Boundedness Characterization of Maximal Commutators on Orlicz Spaces in the Dunkl Setting}, journal = {Journal of Mathematical Study}, year = {2020}, volume = {53}, number = {1}, pages = {45--65}, abstract = {

On the real line, the Dunkl operators

$$D_{\nu}(f)(x):=\frac{d f(x)}{dx} + (2\nu+1) \frac{f(x) - f(-x)}{2x}, ~~ \quad\forall \, x \in \mathbb{R}, ~ \forall \, \nu \ge -\tfrac{1}{2}$$

are differential-difference operators associated with the reflection group $\mathbb{Z}_2$ on $\mathbb{R}$, and on the $\mathbb{R}^d$ the Dunkl operators $\big\{D_{k,j}\big\}_{j=1}^{d}$ are the differential-difference operators associated with the reflection group $\mathbb{Z}_2^d$ on $\mathbb{R}^{d}$.

In this paper, in the setting $\mathbb{R}$ we show that $b \in BMO(\mathbb{R},dm_{\nu})$ if and only if the maximal commutator $M_{b,\nu}$ is bounded on Orlicz spaces $L_{\Phi}(\mathbb{R},dm_{\nu})$. Also in the setting $\mathbb{R}^{d}$ we show that $b \in BMO(\mathbb{R}^{d},h_{k}^{2}(x) dx)$ if and only if the maximal commutator $M_{b,k}$ is bounded on Orlicz spaces $L_{\Phi}(\mathbb{R}^{d},h_{k}^{2}(x) dx)$.

}, issn = {2617-8702}, doi = {https://doi.org/10.4208/jms.v53n1.20.03}, url = {http://global-sci.org/intro/article_detail/jms/15207.html} }
TY - JOUR T1 - Boundedness Characterization of Maximal Commutators on Orlicz Spaces in the Dunkl Setting AU - Guliyev , Vagif S. AU - Mammadov , Yagub Y. AU - Muslumova , Fatma A. JO - Journal of Mathematical Study VL - 1 SP - 45 EP - 65 PY - 2020 DA - 2020/03 SN - 53 DO - http://doi.org/10.4208/jms.v53n1.20.03 UR - https://global-sci.org/intro/article_detail/jms/15207.html KW - Maximal operator, Orlicz space, Dunkl operator, commutator, BMO. AB -

On the real line, the Dunkl operators

$$D_{\nu}(f)(x):=\frac{d f(x)}{dx} + (2\nu+1) \frac{f(x) - f(-x)}{2x}, ~~ \quad\forall \, x \in \mathbb{R}, ~ \forall \, \nu \ge -\tfrac{1}{2}$$

are differential-difference operators associated with the reflection group $\mathbb{Z}_2$ on $\mathbb{R}$, and on the $\mathbb{R}^d$ the Dunkl operators $\big\{D_{k,j}\big\}_{j=1}^{d}$ are the differential-difference operators associated with the reflection group $\mathbb{Z}_2^d$ on $\mathbb{R}^{d}$.

In this paper, in the setting $\mathbb{R}$ we show that $b \in BMO(\mathbb{R},dm_{\nu})$ if and only if the maximal commutator $M_{b,\nu}$ is bounded on Orlicz spaces $L_{\Phi}(\mathbb{R},dm_{\nu})$. Also in the setting $\mathbb{R}^{d}$ we show that $b \in BMO(\mathbb{R}^{d},h_{k}^{2}(x) dx)$ if and only if the maximal commutator $M_{b,k}$ is bounded on Orlicz spaces $L_{\Phi}(\mathbb{R}^{d},h_{k}^{2}(x) dx)$.

Vagif S. Guliyev, Yagub Y. Mammadov & Fatma A. Muslumova. (2020). Boundedness Characterization of Maximal Commutators on Orlicz Spaces in the Dunkl Setting. Journal of Mathematical Study. 53 (1). 45-65. doi:10.4208/jms.v53n1.20.03
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