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

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.