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New Doob’s Maximal Inequalities for Martingales

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Abstract

Let 1 ≤ q ≤ ∞, b be a slowly varying function and let Φ: [0, ∞) → [0, ∞) be an increasing convex function with Φ(0) = 0 and \(\mathop {\lim }\limits_{r \to \infty } \,\,\Phi \left( r \right) = \infty \). In this paper, we present a new class of Doob’s maximal inequality on Orlicz-Lorentz-Karamata spaces LΦ,q,b. The results are new, even for the Lorentz-Karamata spaces with Φ(t) = tp, the Orlicz-Lorentz spaces with b ≡ 1, and weak Orlicz-Karamata spaces with q = ∞ in the framework of. Moreover, we obtain some even stronger qualitative results that can remove the Δ2-condition of Liu, Hou and Wang (Sci China Math, 2010, 53(4): 905–916).

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Authors and Affiliations

  1. School of Mathematics and Computing Science, Hunan University of Science and Technology, Xiangtan, 411201, China

    Zhiwei Hao  (郝志伟) & Libo Li  (李丽波)

Authors
  1. Zhiwei Hao  (郝志伟)
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  2. Libo Li  (李丽波)
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Corresponding author

Correspondence to Libo Li  (李丽波).

Additional information

This project was supported by the National Natural Science Foundation of China (11801001, 12101223), the Scientific Research Fund of Hunan Provincial Education Department (20C0780) and the Natural Science Foundation of Hunan Province (2022JJ40145, 2022JJ40146).

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Hao, Z., Li, L. New Doob’s Maximal Inequalities for Martingales. Acta Math Sci 43, 531–538 (2023). https://doi.org/10.1007/s10473-023-0204-6

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  • DOI: https://doi.org/10.1007/s10473-023-0204-6

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