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2021 Sharp maximal Lp-bounds for continuous martingales and their differential subordinates
Adam Osękowski, Yahui Zuo
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Electron. J. Probab. 26: 1-22 (2021). DOI: 10.1214/21-EJP596


Suppose that X, Y are Hilbert-space-valued continuous-path martingales such that Y is differentially subordinate to X. The paper contains the proof of sharp estimates between p-th moments of Y and the maximal function of X for 0<p<1. The proof rests on Burkholder’s method and exploits a certain special function of three variables, enjoying appropriate size and concavity requirements. The analysis reveals an unexpected phase transition between the cases 0<p<12 and 12p<1. The latter case is relatively simple: the special function is essentially quadratic and the best constant is equal to 2p. The analysis of the former case is much more intricate and involves the study of a non-linear ordinary differential equation.


The authors would like to thank an anonymous Referee for the careful reading of the first version of the paper and several helpful suggestions, which improved the presentation. A. Osękowski was supported by Narodowe Centrum Nauki (Poland), grant DEC-2014/14/E/ST1/00532, Y. Zuo was supported by China Scholarship Council.


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Adam Osękowski. Yahui Zuo. "Sharp maximal Lp-bounds for continuous martingales and their differential subordinates." Electron. J. Probab. 26 1 - 22, 2021.


Received: 18 April 2020; Accepted: 1 February 2021; Published: 2021
First available in Project Euclid: 23 March 2021

Digital Object Identifier: 10.1214/21-EJP596

Primary: 60G44

Keywords: Differential subordination , martingale , maximal inequality , stochastic integral


Vol.26 • 2021
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