Norm inequalities for maximal operators

Abstract

In this paper, we introduce a family of one-dimensional maximal operators mathscr{M}_{kappa ,m}, kappa geq 0 and min mathbb{N}setminus {0}, which includes the Hardy–Littlewood maximal operator as a special case (kappa =0, m=1). We establish the weak type (1,1) and the strong type (p,p) inequalities for mathscr{M}_{kappa ,m}, p>1. To do so, we prove a technical covering lemma for a finite collection of intervals.