Abstract and Applied Analysis
Volume 2010 (2010), Article ID 976493, 10 pages
Research Article

Necessary and Sufficient Conditions for the Boundedness of Dunkl-Type Fractional Maximal Operator in the Dunkl-Type Morrey Spaces

1Institute of Mathematics and Mechanics, AZ 1141 Baku, Azerbaijan
2Department of Mathematics, Nigde University, 51100 Nigde, Turkey
3Nakhchivan Teacher-Training Institute, AZ 7003 Nakhchivan, Azerbaijan

Received 15 April 2010; Revised 13 June 2010; Accepted 23 June 2010

Academic Editor: Dumitru Baleanu

Copyright © 2010 Emin Guliyev et al. 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.


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 the boundedness of the Dunkl-type fractional maximal operator Mβ in the Dunkl-type Morrey space Lp,λ,α(), 0λ<2α+2. We obtain necessary and sufficient conditions on the parameters for the boundedness Mβ, 0β<2α+2 from the spaces Lp,λ,α() to the spaces Lq,λ,α(), 1<pq<, and from the spaces L1,λ,α() to the weak spaces WLq,λ,α(), 1<q<. As an application of this result, we get the boundedness of Mβ from the Dunkl-type Besov-Morrey spaces Bpθ,λ,αs() to the spaces Bqθ,λ,αs(), 1<pq<, 0λ<2α+2, 1/p-1/q=β/(2α+2-λ), 1θ, and 0<s<1.