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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.

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