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

Yang, Dachun; Yang, Dongyong

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

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 .

Copyright

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.

Journal of Function Spaces

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

Abstract

Copyright

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