Norm inequalities for maximal operators

Abstract

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