Journal of Inequalities and Applications
Volume 2006 (2006), Article ID 15063, 13 pages

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

Copyright © 2006 Yutaka Terasawa. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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 k2 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.