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 . 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 and . The latter case is relatively simple: the special function is essentially quadratic and the best constant is equal to . 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.
"Sharp maximal -bounds for continuous martingales and their differential subordinates." Electron. J. Probab. 26 1 - 22, 2021. https://doi.org/10.1214/21-EJP596