STRONG DIFFERENTIAL SUBORDINATES FOR NONCOMMUTATIVE SUBMARTINGALES

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 pro...

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Published in:The Annals of probability 2019-09, Vol.47 (5), p.3108-3142
Main Authors: Jiao, Yong, Osȩkowski, Adam, Wu, Lian
Format: Article
Language:English
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Summary: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 < ∞ 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-λ inequalities.
ISSN:0091-1798
2168-894X
DOI:10.1214/18-AOP1334