Journal of Inequalities and Applications 
Volume 2006 (2006), Article ID 15063, 13 pages
doi:10.1155/JIA/2006/15063

Outer measures and weak type (1,1) estimates of Hardy-Littlewood maximal operators

Yutaka Terasawa

 Department of Mathematics, Hokkaido University, Sapporo 060-0810, Japan
Received 14 June 2004; Revised 10 July 2004; Accepted 14 October 2004

Abstract

We will introduce the k times modified centered and uncentered Hardy-Littlewood maximal operators on nonhomogeneous spaces for k>0. We will prove that the k times modified centered Hardy-Littlewood maximal operator is weak type (1,1) bounded with constant 1 when k≥2 if the Radon measure of the space has “continuity” in some sense. In the proof, we will use the outer measure associated with the Radon measure. We will also prove other results of Hardy-Littlewood maximal operators on homogeneous spaces and on the real line by using outer measures.