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

Some embeddings into the Morrey and modified Morrey spaces associated with the Dunkl operator

Emin V. Guliyev and Yagub Y. Mammadov

Institute of Mathematics and Mechanics, Azerbaijan National Academy of Sciences, Baku, Azerbaijan

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, 0λ<2α+2 and modified Morrey space (modified D-Morrey space) L̃p,λ,α associated with the Dunkl operator on . As applications we get boundedness of the fractional maximal operator Mβ, 0β<2α+2, associated with the Dunkl operator (fractional D-maximal operator) from the spaces Lp,λ,α to L() for p=(2α+2-λ)/β and from the spaces L̃p,λ,α() to L() for (2α+2-λ)/βp(2α+2)/β.