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

Abstract EN

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

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