Improvements on Sawyer type estimates for generalized maximal functions

Abstract

In this paper we prove mixed inequalities for the maximal operator (Formula presented.), for general Young functions Φ with certain additional properties, improving and generalizing some previous estimates for the Hardy–Littlewood maximal operator proved by E. Sawyer. We show that given (Formula presented.), if (Formula presented.) are weights belonging to the A1-Muckenhoupt class and Φ is a Young function as above, then the inequality (Formula presented.) holds for every positive t. A motivation for studying these type of estimates is to find an alternative way to prove the boundedness properties of (Formula presented.). Moreover, it is well-known that for the particular case (Formula presented.) with (Formula presented.) these maximal functions control, in some sense, certain operators in harmonic analysis.