The Annals of Probability

Strong differential subordinates for noncommutative submartingales

Yong Jiao, Adam Osȩkowski, and Lian Wu

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Abstract

We introduce a notion of strong differential subordination of noncommutative semimartingales, extending Burkholder’s definition from the classical case (Ann. Probab. 22 (1994) 995–1025). Then we establish the maximal weak-type $(1,1)$ inequality under the additional assumption that the dominating process is a submartingale. The proof rests on a significant extension of the maximal weak-type estimate of Cuculescu and a Gundy-type decomposition of an arbitrary noncommutative submartingale. We also show the corresponding strong-type $(p,p)$ estimate for $1<p<\infty $ under the assumption that the dominating process is a nonnegative submartingale. This is accomplished by combining several techniques, including interpolation-flavor method, Doob–Meyer decomposition and noncommutative analogue of good-$\lambda$ inequalities.

Article information

Source
Ann. Probab., Volume 47, Number 5 (2019), 3108-3142.

Dates
Received: May 2018
First available in Project Euclid: 22 October 2019

Permanent link to this document
https://projecteuclid.org/euclid.aop/1571731446

Digital Object Identifier
doi:10.1214/18-AOP1334

Mathematical Reviews number (MathSciNet)
MR4021246

Subjects
Primary: 46L53: Noncommutative probability and statistics 60G42: Martingales with discrete parameter
Secondary: 46L52: Noncommutative function spaces 60G50: Sums of independent random variables; random walks

Keywords
Noncommutative submartingale strong differential subordination weak-type inequality strong-type inequality

Citation

Jiao, Yong; Osȩkowski, Adam; Wu, Lian. Strong differential subordinates for noncommutative submartingales. Ann. Probab. 47 (2019), no. 5, 3108--3142. doi:10.1214/18-AOP1334. https://projecteuclid.org/euclid.aop/1571731446


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