Norm inequalities for maximal operators

Abstract

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

Keywords

• Covering lemma
• Generalized Fourier transform
• Maximal operators
• Weak and strong type inequalities