Hardy-Littlewood maximal operator on variable Lebesgue spaces with respect to a probability measure

Abstract

In this paper we are going to prove that the Hardy-Litllewood maximal operators on variable Lebesgue spaces L ^p(·)(µ) with respect to a probability Borel measure µ, are weak type and strong type for two conditions of regularity on the exponent function p(·); following [2] and [3]. Also following [2] we extend some important properties of this operator to a probability Borel measure µ, the key to extend these results is using the Besicovitch covering lemma instead of the Calderón-Zygmund decomposition.