Fractional maximal operator in hyperbolic spaces

Abstract

In this article, we introduce the fractional maximal operator on the Hyperbolic space, a non-doubling measure space, and study its weighted boundedness. Motivated by the weighted boundedness of the Hardy-Littlewood maximal studied by Antezana and Ombrosi in [1], we give conditions for the weak type and strong type estimate for fractional maximal. Also, we present examples of weights for which the fractional maximal operator satisfies weak type $(p,q)$ inequality but strong type $(p,q)$ inequality fails.