报告人：Yong Jiao（Central South University）
时间：2018-12-17 14:00-15:00
地点：Room 1303, Sciences Building No. 1
Abstract: We propose a novel approach in noncommutative probability, which can be regarded as an
analogue of good-$\lambda$ inequalities from the classical case due to Burkholder and Gundy (Acta
Math {\bf 124}: 249-304,1970). This resolves a longstanding open problem in noncommutative realm.
Using this technique, we present new proofs of noncommutative Burkholder-Gundy inequalities, Stein's
inequality, Doob's inequality and $L^p$-bound for martingale transforms. The approach also allows us
to investigate the noncommutative analogues of decoupling techniques and, in particular, to obtain new
estimates for noncommutative martingales with tangent difference sequences and sums of tangent positive
operators. These in turn yield an enhanced version of Doob's maximal inequality for adapted sequences
and a sharp estimate for a certain class of Schur multipliers. We also present two fully new applications of
good-$\lambda$ approach to noncommutative harmonic analysis, including inequalities for differentially
subordinate operators motivated by the classical $L^p$-bound for the Hilbert transform and the $L^p$-
estimate for the $j$-th Riesz transform on group von Neumann algebras. We emphasize that all the constants
obtained in this paper are of optimal orders. This is a joint work with Adam Osekowski and Lian Wu.
