Noncommutative Good-$\lambda$ Inequalities

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 {\bf124}: 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 $L^p$-bounds for martingale transforms; all the constants obtained are of optimal orders. 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 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 estimate for the $j$-th Riesz transform on group von Neumann algebras with constants of optimal orders as $p\to\infty.$