Uniform Boundedness for Approximations of the Identity with Nondoubling Measures
Let μ be a nonnegative Radon measure on â„Âd which satisfies the growth condition that there exist constants C0>0 and n∈(0,d] such that for all x∈â„Âd 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 e...
| Main Authors: | , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
SpringerOpen
2007-10-01
|
| Series: | Journal of Inequalities and Applications |
| Online Access: | http://dx.doi.org/10.1155/2007/19574 |
| Summary: | Let μ be a nonnegative Radon measure on â„Âd which satisfies the growth condition that there exist constants C0>0 and n∈(0,d] such that for all x∈â„Âd 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). |
|---|---|
| ISSN: | 1025-5834 1029-242X |