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

Yang, Dachun; Yang, Dongyong

doi:https://doi.org/10.1155/2009/284849

Journal of Function Spaces

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

Dachun Yang, Dongyong Yang, "Endpoint estimates for homogeneous Littlewood-Paley g-functions with non-doubling measures", Journal of Function Spaces, vol. 7, Article ID 284849, 21 pages, 2009. https://doi.org/10.1155/2009/284849

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

Dachun Yang¹ and Dongyong Yang¹

¹School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, China

Academic Editor: Hans Triebel

Received01 Apr 2008

Abstract

Let µ be a nonnegative Radon measure on ℝ^d which satisfies the growth condition that there exist constants C₀ > 0 and n ∈ (0, d] such that for all x ∈ ℝ^d 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 ℝ^d is not an initial cube which implies µ(ℝ^d) = ∞, 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 C‖f‖RBMO(μ)2, where C≻0 is independent of f .

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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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