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.

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