Journal of Inequalities and Applications 
Volume 2007 (2007), Article ID 19574, 25 pages
doi:10.1155/2007/19574
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

Recommended by Shusen Ding

Abstract

Let μ be a nonnegative Radon measure on ℝd which satisfies the growth condition that there exist constants C0>0 and n∈(0,d] such that for all x∈ℝd 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 ℳ.s (homogeneous) and ℳs (inhomogeneous) associated with a given approximation of the identity S, and prove that ℳ.s is bounded from H1(μ) to L1(μ) and ℳs 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).