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
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.
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.