(L-p, L-q) Boundedness of the fractional maximal operator associated with the Dunkl operator on the real line

Abstract

On the real line, the Dunkl operators are differential-difference operators associated with the reflection group Z(2) on R. In this paper, we obtain necessary and sufficient conditions on the parameters for the boundedness of the fractional maximal operator associated with the Dunkl operator on R from the spaces L-p,L-alpha(R) to the spaces L-q,L-alpha(R) and from the spaces L-1,L-alpha(R) to the weak spaces W L-q,L-alpha(R).