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Inequalities for Noncommutative Weakly Dominated Martingales and Applications

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Abstract

Motivated by the results from the classical probability theory, we introduce the concepts of tangency and weak domination of noncommutative martingales. Then we establish the weak-type and strong-type estimates arising in this context. The proof rests on a novel Gundy-type decomposition which is of independent interest. We also show the corresponding square function inequalities under the assumption of the weak domination, which extends Pisier and Xu’s Burkholder–Gundy inequalities. The results strengthen and extend the very recent works on noncommutative differentially subordinate martingales, which in turn, give rise to a new application in harmonic analysis: a weak-type estimate (along with a completely bounded version) for the directional Hilbert transform associated with quantum tori.

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Acknowledgements

The authors would like to thank an anonymous referee for the very careful and thorough reading of the first version of the paper, and for several helpful suggestions.

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Authors and Affiliations

  1. School of Mathematics and Statistics, HNP-LAMA, Central South University, Changsha, 410083, People’s Republic of China

    Yong Jiao, Lian Wu & Yahui Zuo

  2. Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, Banacha 2, 02-097, Warsaw, Poland

    Adam Osękowski

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  1. Yong Jiao
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Correspondence to Yahui Zuo.

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Communicated by J. Ding.

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Yong Jiao is supported by the NSFC (Nos. 12125109, 11961131003). Adam Osȩkowski is supported by Narodowe Centrum Nauki (Poland), Grant 2018/30/Q/ST1/00072. Lian Wu is supported by the NSFC (No. 11971484). Yahui Zuo is supported by China Postdoctoral Science Foundation (No. 2021M703646), Changsha Municipal Nature Science Foundation (No. kq2202079).

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Jiao, Y., Osękowski, A., Wu, L. et al. Inequalities for Noncommutative Weakly Dominated Martingales and Applications. Commun. Math. Phys. 396, 787–816 (2022). https://doi.org/10.1007/s00220-022-04477-9

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