Abstract and Applied Analysis
Volume 2010 (2010), Article ID 291345, 10 pages
doi:10.1155/2010/291345
Research Article

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

Emin V. Guliyev1 and Yagub Y. Mammadov1,2

1Institute of Mathematics and Mechanics, Azerbaijan National Academy of Sciences, Baku, Azerbaijan
2Nakhchivan Teacher-Training Institute, Azerbaijan

Received 29 October 2009; Revised 5 February 2010; Accepted 2 March 2010

Academic Editor: Ağacik Zafer

Copyright © 2010 Emin V. Guliyev and Yagub Y. Mammadov. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We consider the generalized shift operator, associated with the Dunkl operator Λα(f)(x)=(d/dx)f(x)+((2α+1)/x)((f(x)-f(-x))/2), α>-1/2. We study some embeddings into the Morrey space (D-Morrey space) Lp,λ,α, 0≤λ<2α+2 and modified Morrey space (modified D-Morrey space) L̃p,λ,α associated with the Dunkl operator on ℝ. As applications we get boundedness of the fractional maximal operator Mβ, 0≤β<2α+2, associated with the Dunkl operator (fractional D-maximal operator) from the spaces Lp,λ,α to L∞(ℝ) for p=(2α+2-λ)/β and from the spaces L̃p,λ,α(ℝ) to L∞(ℝ) for (2α+2-λ)/β≤p≤(2α+2)/β.