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Endpoint estimates of generalized homogeneous Littlewood–Paley g-functions over non-homogeneous metric measure spaces

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Abstract

Let (X, d, μ) be a metric measure space satisfying both the geometrically doubling and the upper doubling conditions. Under the weak reverse doubling condition, the authors prove that the generalized homogeneous Littlewood–Paley g-function ġ r (r ∈ [2,∞)) is bounded from Hardy space H 1(μ) into L 1(μ). Moreover, the authors show that, if f ∈ RBMO(μ), then [ġ r (f)]r is either infinite everywhere or finite almost everywhere, and in the latter case, [ġ r (f)]r belongs to RBLO(μ) with the norm no more than ‖f‖RBMO(μ) multiplied by a positive constant which is independent of f. As a corollary, the authors obtain the boundedness of ġ r from RBMO(μ) into RBLO(μ). The vector valued Calderón–Zygmund theory over (X, d, μ) is also established with details in this paper.

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Authors and Affiliations

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

    Xing Fu & Ji Man Zhao

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  1. Xing Fu
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  2. Ji Man Zhao
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Correspondence to Ji Man Zhao.

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Supported by National Natural Science Foundation of China (Grant No. 11471040) and the Fundamental Research Funds for the Central Universities (Grant No. 2014KJJCA10)

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Fu, X., Zhao, J.M. Endpoint estimates of generalized homogeneous Littlewood–Paley g-functions over non-homogeneous metric measure spaces. Acta. Math. Sin.-English Ser. 32, 1035–1074 (2016). https://doi.org/10.1007/s10114-016-5059-5

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