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学术报告:Recent advacnes on the theory of noncommutative differentially subordinate martingales

发布时间:2018年10月29日 浏览次数:发布者:mathky

报告人:焦勇教授

        中南大学

报告题目:Recent advacnes on the theory of noncommutative differentially subordinate martingales

报告时间:2018年10月31日上午10:30

报告地点:海韵实验楼108

报告摘要:In this talk we will report some new advances regarding the theory of noncommutative differentially subordinate martingales. The classical differential subordination of martingales, introduced by Burkholder in the eighties, is generalized to the noncommutative setting. Working under this domination, we establish the strong-type inequalities with the constants of optimal order as $p\to 1$ and $p\to \infty$, and the corresponding endpoint weak-type (1,1) estimate. In contrast to the classical case, we need to introduce two different versions of noncommutative differential subordination, depending on the range of the exponents. For the $L^p$-estimate, $2\leq p<\infty$, a certain weaker version is sufficient; on the other hand, the strong-type $(p,p)$ inequality, $1<p<2$, and the weak-type (1,1) estimate require a stronger version. We also introduce a notion of strong differential subordination of noncommutative semimartingales, establish the maximal weak-type (1,1) inequality under the additional assumption that the dominating process is a submartingale, and show the corresponding strong-type $(p,p)$ estimate for $1<p<\infty$ under the assumption that the dominating process is a nonnegative submartingale. Finally, we give some estimates of square functions for noncommutative differentially subordinate martingales. This is accomplished by combining several techniques, including interpolation flavor method, Doob-Meyer decomposition, a significant extension of the maximal weak-type estimate of Cuculescu and a Gundy-type decomposition of an arbitrary noncommutative submartingale.

This is several joint work with Adam Osekowski, Lian Wu, Narcisse R. and Dejian Zhou.

报告人简介:中南大学“升华学者”特聘教授,博士生导师;湖南省杰青,国家优秀青年基金获得者。2009年6月博士毕业,同时获得了武汉大学和法国弗朗什孔泰大学颁发的理学双博士学位;目前,以第一或通讯作者在《ProbabilityTheory and Related Fields》、《JFA》、《Transactions ofAmerican Math Society》、《JournalofLondon Math Society》、《IndianaUniversity MathematicsJournal》、《Proceedingsof American Math Society》、《Journalof Operator Theory》、《QuarterlyJournalof Mathematics》、《StudiaMath》、《Journalof Mathematical Analysis and Applications》、《Nonlinear Analysis, Theory methods and Applications》、《Science China Math》、《Acta MathematicaSinica》等国内外刊物上发表学术论文近30篇。

联系人:张文教授

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