Journal of Function Spaces

Journal of Function Spaces / 2009 / Article

Open Access

Volume 7 |Article ID 284849 | 21 pages | https://doi.org/10.1155/2009/284849

Endpoint estimates for homogeneous Littlewood-Paley g-functions with non-doubling measures

Academic Editor: Hans Triebel
Received01 Apr 2008

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, when ℝd is not an initial cube which implies µ(ℝd) = ∞, the authors prove that the homogeneous Littlewood-Paley g-function of Tolsa is bounded from the Hardy space H1 (µ) to L1(µ), and furthermore, that if f ∈ RBMO (µ), then [ġ(f )]2 is either infinite everywhere or finite almost everywhere, and in the latter case, [ġ(f)]2 belongs to RBLO (µ) with norm no more than CfRBMO(μ)2, where C0 is independent of f .

Copyright © 2009 Hindawi Publishing Corporation. 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.


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