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
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.
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).