We consider the generalized shift operator, associated with the Dunkl operatorΛα(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+2and modified Morrey space (modifiedD-Morrey space)L̃p,λ,αassociated with the Dunkl operator onℝ. As applications we get boundedness of the fractional maximal operatorMβ,0≤β<2α+2, associated with the Dunkl operator (fractionalD-maximal operator) from the

more »

... nd from the spacesL̃p,λ,α(ℝ)toL∞(ℝ)for(2α+2-λ)/β≤p≤(2α+2)/β.