On fractional maximal function and fractional integrals associated with the Dunkl operator on the real line

In this paper we obtain necessary and sufficient conditions on the parameters for the boundedness of the Dunkl-type fractional maximal operator M β , and the Dunkl-type fractional integral operator I β from the spaces L p , α ( R ) to the spaces L q , α ( R ) , 1 < p < q < ∞ , and from the spaces L 1 , α ( R ) to the weak spaces WL q , α ( R ) , 1 < q < ∞ . In the case p = 2 α + 2 β , we prove that the operator M β is bounded from the space L p , α ( R ) to the space L ∞ , α ( R ) , and the Dunkl-type modified fractional integral operator I ˜ β is bounded from the space L p , α ( R ) to the Dunkl-type BMO space BMO α ( R ) . By this results we get boundedness of the operators M β and I β from the Dunkl-type Besov spaces B p θ , α s ( R ) to the spaces B q θ , α s ( R ) , 1 < p < q < ∞ , 1 / p − 1 / q = β / ( 2 α + 2 ) , 1 ⩽ θ ⩽ ∞ and 0 < s < 1 .