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September 2019 Strong differential subordinates for noncommutative submartingales
Yong Jiao, Adam Osȩkowski, Lian Wu
Ann. Probab. 47(5): 3108-3142 (September 2019). DOI: 10.1214/18-AOP1334

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.

Citation

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Yong Jiao. Adam Osȩkowski. Lian Wu. "Strong differential subordinates for noncommutative submartingales." Ann. Probab. 47 (5) 3108 - 3142, September 2019. https://doi.org/10.1214/18-AOP1334

Information

Received: 1 May 2018; Published: September 2019
First available in Project Euclid: 22 October 2019

zbMATH: 07145312
MathSciNet: MR4021246
Digital Object Identifier: 10.1214/18-AOP1334

Subjects:
Primary: 46L53 , 60G42
Secondary: 46L52 , 60G50

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

Rights: Copyright © 2019 Institute of Mathematical Statistics

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Vol.47 • No. 5 • September 2019
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