The Hardy-Littlewood maximal type operators between Banach function spaces

Abstract

We investigate variants of the maximal operator and show their applications to study boundedness of the classical Hardy-Littlewood maximal operator between weighted Banach function spaces which satisfy certain geometrical lattice conditions. We prove inequalities for rearrangement of the maximal operators generated by rearrangement invariant spaces. Applying this to the Lorentz spaces, we give new sufficient conditions for the boundedness of the Hardy-Littlewood maximal operator between weighted Lp-spaces with different weights. We also prove that under some mild hypotheses these conditions are also necessary.