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Noncommutative maximal inequalities associated with convex functions


Authors: Turdebek N. Bekjan, Zeqian Chen and Adam Osȩkowski
Journal: Trans. Amer. Math. Soc. 369 (2017), 409-427
MSC (2010): Primary 46L53, 46L51
DOI: https://doi.org/10.1090/tran/6663
Published electronically: February 24, 2016
MathSciNet review: 3557778
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Abstract: We prove several noncommutative maximal inequalities associated with convex functions, including a Doob type inequality for a convex function of maximal operators on noncommutative martingales, and noncommutative Dunford-Schwartz and Stein maximal ergodic inequalities for a convex function of positive and symmetric positive contractions. The key ingredient in our proofs is a Marcinkiewicz type interpolation theorem for a convex function of maximal operators in the noncommutative setting, which we establish in this paper. These generalize the results of Junge and Xu in the $ L^p$ case to the case of convex functions.


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Additional Information

Turdebek N. Bekjan
Affiliation: College of Mathematics and Systems Science, Xinjiang University, Urumqi 830046, People’s Republic of China
Email: bek@xju.edu.cn

Zeqian Chen
Affiliation: Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, West District 30, Xiao-Hong-Shan, Wuhan 430071, People’s Republic of China
Email: zqchen@wipm.ac.cn

Adam Osȩkowski
Affiliation: Department of Mathematics, Informatics and Mechanics, University of Warsaw, Banacha 2, 02-097 Warsaw, Poland
Email: ados@mimuw.edu.pl

DOI: https://doi.org/10.1090/tran/6663
Keywords: Noncommutative martingale, noncommutative maximal operator, convex function, maximal ergodic inequality, interpolation.
Received by editor(s): June 2, 2014
Received by editor(s) in revised form: December 30, 2014
Published electronically: February 24, 2016
Additional Notes: The first author was partially supported by NSFC grant No. 11371304
The second author was partially supported by NSFC grant No. 11171338 and No. 11431011
The third author was supported in part by MNiSW Grant N N201 364436
Article copyright: © Copyright 2016 American Mathematical Society