Noncommutative Good-$λ$ Inequalities

Abstract

We propose a novel approach in noncommutative probability, which can be regarded as an analogue of good-λ inequalities from the classical case due to Burkholder and Gundy (Acta Math 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 -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-λ 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 → ∞.

Asymmetric Burkholder inequalities in noncommutative symmetric spaces

“…In fact, assume that the inequality (17) holds true. Then, combining inequality (17) with the estimate established in Step 1, we deduce that…”

“…where (e n ) n≥1 are the standard unit vectors in ℓ ∞ . Applying a noncommutative good-λ approach (we refer to [17] for more information about this method), Jiao, Zanin and Zhou [22] generalized (5) to the symmetric operator spaces: if the symmetric Banach function space E lies in Int(L p , L q ) for 2 < p < q < ∞ and if…”

Strong differential subordinates for noncommutative submartingales

“…Let us stress here that we do not assume that r, s are M N -measurable. One of the main results of [17] is the following. THEOREM 4.14.…”

“…Then, We turn our attention to the upper bound for b N p . We will exploit the noncommutative good-λ inequalities developed by Jiao et al in [17,18]. Let us briefly recall the framework.…”

“…In the classical case this extrapolation technique was introduced by Burkholder in [8], and it has turned out to be very powerful in a number of problems arising in harmonic analysis and probability. The noncommutative counterpart of this method, recently obtained by the authors in [17], allows us to obtain a submartingale version of the noncommutative Doob's inequality (see Lemma 4.3 and Theorem 4.15, or Remark 4.17(i)). This estimate, combined with a certain novel L p bound for submartingale differences (see Theorem 4.12), yields the moment inequality for the strong subordinates in the range p > 2.…”