Journal of Inequalities and Applications
Volume 2007 (2007), Article ID 19574, 25 pages
Research Article

Uniform Boundedness for Approximations of the Identity with Nondoubling Measures

Dachun Yang and Dongyong Yang

School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China

Received 15 May 2007; Accepted 19 August 2007

Academic Editor: Shusen Ding

Copyright © 2007 Dachun Yang and Dongyong Yang. 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.


Let μ be a nonnegative Radon measure on d which satisfies the growth condition that there exist constants C0>0 and n(0,d] such that for all xd and r>0, μ(B(x,r))C0rn, where B(x,r) is the open ball centered at x and having radius r. In this paper, the authors establish the uniform boundedness for approximations of the identity introduced by Tolsa in the Hardy space H1(μ) and the BLO-type space RBLO (μ). Moreover, the authors also introduce maximal operators .s (homogeneous) and s (inhomogeneous) associated with a given approximation of the identity S, and prove that .s is bounded from H1(μ) to L1(μ) and s is bounded from the local atomic Hardy space hatb1,(μ) to L1(μ). These results are proved to play key roles in establishing relations between H1(μ) and hatb1,(μ), BMO-type spaces RBMO (μ) and rbmo (μ) as well as RBLO (μ) and rblo (μ), and also in characterizing rbmo (μ) and rblo (μ).