Noncommutative differential transforms for averaging operators

Abstract

In this paper, we complete the study of mapping properties about the difference associated with dyadic differentiation average and dyadic martingale on noncommutative Lp-spaces. To be more precise, we establish the weak type (1, 1) and (L∞, BMO) estimates of this difference. Consequently, in conjunction with interpolation and duality, we obtain the corresponding all strong type (p, p) estimates. As an important application, we obtain the weak type (1, 1) and strong type (p, p) estimates of noncommutative differential transforms. The main difficulties lie in the fact that the kernel associated with considering operator does not enjoy any regularity, while the necessary regularity assumption is required to prove such endpoint estimates for the operator-valued Calderón-Zygmund singular integrals. Moreover, the almost orthogonality principle used in [15] is no longer applicable in the present case.

Noncommutative maximal ergodic inequalities for amenable groups

“…Remark 6.8. Our method is also useful for the study of some other more general or stronger form of ergodic theorems, for instance, the square estimates (A n − A n−1 ) n≥0 and differential transforms similar to [11,21], maximal and individual ergodic theorems for group actions on noncommutative L p -spaces with a fixed p ∈ (1, ∞) as in [10]. Also our previous results for noncommutative L pspaces with p ∈ (1, ∞) holds true as well for general non-tracial von Neumann algebras up to standard adaptations as in [14,9].…”

“…We will then show that these geometric data are sufficient to yield nice martingales and derive suitable local estimates even in the general abstract setting, still following a scheme similar to that of [11,21]. It is worth mentioning that the Calderón-Zygmund arguments and local estimates in [11,21] work with doubling metric measure spaces, so new adaptations are bound to be necessary in our general abstract setting.…”

“…We will then show that these geometric data are sufficient to yield nice martingales and derive suitable local estimates even in the general abstract setting, still following a scheme similar to that of [11,21]. It is worth mentioning that the Calderón-Zygmund arguments and local estimates in [11,21] work with doubling metric measure spaces, so new adaptations are bound to be necessary in our general abstract setting. Our argument will rely on a Calderón-Zygmund decomposition associated with non-regular martingales given in [4], where we replace again the Euclidean geometric behaviors in [4] by purely group-theoretical ones.…”

“…But the geometric requirements underlying the proof seem too strong to hold in any amenable group, and those dominations by martingales are not powerful enough to study other ergodic estimates such as variational inequalities. This leads the experts to find other related strategies; in particular, instead of dominating A n by martingales averages, Hong and Xu [11,21] considered the mappings (1.2)…”

“…Let us summarize some key ideas and strategies around the study of this operator D in our new setting. In the Euclidean case, the boundedness of D associated with dyadic martingales was proved by Hong and Xu [11,21] by exploiting the idea that D is very close to being a Calderón-Zygmund operator. Its kernel does not enjoy smoothness properties but it possesses an (even better) pseudo-localizing behavior due to the fact that the D n 's satisfy local estimates (see Subsection 5.1) which boil down to geometric consideration on the dyadic filtration.…”