## Abstract and Applied Analysis

### Some Embeddings into the Morrey and Modified Morrey Spaces Associated with the Dunkl Operator

#### Abstract

We consider the generalized shift operator, associated with the Dunkl operator ${\Lambda }_{\alpha }(f)(x)=(d/dx)f(x)+((2\alpha +1)/x)((f(x)-f(-x))/2)$, $\alpha \gt-1/2$. We study some embeddings into the Morrey space ($D$-Morrey space) ${L}_{p,\lambda ,\alpha }$, $0\leq \lambda \lt2\alpha +2$ and modified Morrey space (modified $D$-Morrey space) ${\widetilde{L}}_{p,\lambda ,\alpha }$ associated with the Dunkl operator on $\mathbb{R}$. As applications we get boundedness of the fractional maximal operator ${M}_{\beta }$, $0\leq \beta \lt2\alpha +2$, associated with the Dunkl operator (fractional $D$-maximal operator) from the spaces ${L}_{p,\lambda ,\alpha }$ to ${L}_{\infty }(\mathbb{R})$ for $p=(2\alpha +2-\lambda )/\beta$ and from the spaces ${\widetilde{L}}_{p,\lambda ,\alpha }(\mathbb{R})$ to ${L}_{\infty }(\mathbb{R})$ for $(2\alpha +2-\lambda )/\beta \leq p\leq (2\alpha +2)/\beta$.

#### Article information

Source
Abstr. Appl. Anal., Volume 2010 (2010), Article ID 291345, 10 pages.

Dates
First available in Project Euclid: 1 November 2010

https://projecteuclid.org/euclid.aaa/1288620709

Digital Object Identifier
doi:10.1155/2010/291345

Mathematical Reviews number (MathSciNet)
MR2607126

Zentralblatt MATH identifier
1198.46027

#### Citation

Guliyev, Emin V.; Mammadov, Yagub Y. Some Embeddings into the Morrey and Modified Morrey Spaces Associated with the Dunkl Operator. Abstr. Appl. Anal. 2010 (2010), Article ID 291345, 10 pages. doi:10.1155/2010/291345. https://projecteuclid.org/euclid.aaa/1288620709