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

We consider the generalized shift operator, associated with the Dunkl operator Λ ð ›¼ ( ð ‘“ ) ( ð ‘¥ ) = ( ð ‘‘ / ð ‘‘ ð ‘¥ ) ð ‘“ ( ð ‘¥ ) + ( ( 2 ð ›¼ + 1 ) / ð ‘¥ ) ( ( ð ‘“ ( ð ‘¥ ) − ð ‘“ ( − ð ‘¥ ) ) / 2 ) , ð ›¼ > − 1 / 2 . We study the boundedness of the Dunkl-type fractional maximal operator ð ‘€ ð ›½ in the Dunkl-type Morrey space ð ¿ ð ‘ , 𠜆 , ð ›¼ ( â„ ) , 0 ≤ 𠜆 < 2 ð ›¼ + 2 . We obtain necessary and sufficient conditions on the parameters for the boundedness ð ‘€ ð ›½ , 0 ≤ ð ›½ < 2 ð ›¼ + 2 from the spaces ð ¿ ð ‘ , 𠜆 , ð ›¼ ( â„ ) to the spaces ð ¿ ð ‘ž , 𠜆 , ð ›¼ ( â„ ) , 1 < ð ‘ â‰¤ ð ‘ž < ∞ , and from the spaces ð ¿ 1 , 𠜆 , ð ›¼ ( â„ ) to the weak spaces ð ‘Š ð ¿ ð ‘ž , 𠜆 , ð ›¼ ( â„ ) , 1 < ð ‘ž < ∞ . As an application of this result, we get the boundedness of ð ‘€ ð ›½ from the Dunkl-type Besov-Morrey spaces ð µ ð ‘ ð ‘ ð œƒ , 𠜆 , ð ›¼ ( â„ ) to the spaces ð µ ð ‘ ð ‘ž 𠜃 , 𠜆 , ð ›¼ ( â„ ) , 1 < ð ‘ â‰¤ ð ‘ž < ∞ , 0 ≤ 𠜆 < 2 ð ›¼ + 2 , 1 / ð ‘ âˆ’ 1 / ð ‘ž = ð ›½ / ( 2 ð ›¼ + 2 − 𠜆 ) , 1 ≤ 𠜃 ≤ ∞ , and 0 < ð ‘ < 1 .