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Vector-valued q-variational inequalities for averaging operators and the Hilbert transform

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Abstract

Recently, the authors have established \(L^p\)-boundedness of vector-valued q-variational inequalities for averaging operators which take values in the Banach space satisfying the martingale cotype q property in Hong and Ma (Math Z 286(1–2):89–120, 2017). In this paper, we prove that the martingale cotype q property is also necessary for the vector-valued q-variational inequalities, which was a question left open in the previous paper. Moreover, we also prove that the UMD property and the martingale cotype q property can be characterized in terms of vector valued q-variational inequalities for the Hilbert transform.

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Acknowledgements

Guixiang Hong is partially supported by the NSF of China (Grant Nos. 11601396, 11501169, 11431011). Tao Ma is supported by NSF of China (Nos. 11671308, 11431011).

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Authors and Affiliations

  1. School of Mathematics and Statistics, Wuhan University, Wuhan, 430072, China

    Guixiang Hong, Wei Liu & Tao Ma

  2. Hubei Key Laboratory of Computational Science, Wuhan University, Wuhan, 430072, China

    Guixiang Hong

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  1. Guixiang Hong
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  2. Wei Liu
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  3. Tao Ma
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Correspondence to Wei Liu.

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Hong, G., Liu, W. & Ma, T. Vector-valued q-variational inequalities for averaging operators and the Hilbert transform. Arch. Math. 115, 423–433 (2020). https://doi.org/10.1007/s00013-020-01472-1

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