Weak boundedness of Calderón–Zygmund operators on noncommutative L1-spaces

Abstract

In 2008, J. Parcet showed the (1, 1) weak-boundedness of Calderón-Zygmund operators acting on functions taking values in a von Neumann algebra. We propose a simplified version of his proof using the same tools: Cuculescu's projections and a pseudo-localisation theorem. This will unable us to recover the L p-boundedness of Calderón-Zygmund operators with Hilbert valued kernels acting on operator valued functions for 1 < p < ∞ and an L p-pseudo-localisation result of P. Hytönen.

Weak type operator Lipschitz and commutator estimates for commuting tuples

“…The result is ultimate for the functions of 1 variable: it is optimal within the category of symmetric spaces and it implies all other known estimates on perturbations of commutators and Lipschitz functions obtained before [7], [8], [11], [12], [13], [18], [19], [24], [30]. The key ingredient of the proof in [9] is a new connection with non-commutative Calderón-Zygmund theory and in particular with the main result from Parcet's fundamental paper [27] (see also the recent paper by Cadilhac [6] for a substantially shorter proof).…”

“…The following theorem in particular gives a sufficient condition for such an operator to act from L 1 to L 1,∞ . Its proof was improved/shortened very recently by Cadilhac [6]. [27]).…”

Noncommutative differential transforms for averaging operators

“…The first obstacle is that it is easy to see that the kernel associated with T does not enjoy any regularity while the methods applied in [33,31,2] depend heavily on Lipschitz's smoothness condition (see also [12] for the weaker smoothness condition). On the other hand, Cadilhac's decomposition, compared to Parcet's one [33], admits a big advantage that the off-diagonal term that appears in the good functions vanishes (see Remark 2.4); recall that in order to deal with this term, Parcet had to develop a so-called pseudo-localization principle which constituted a major part of his long paper [33] (see also [2,Theorem 2.1] for a simplified proof of this principle). However, for the present operator, the difficulty lies in that it is unclear at all that whether there is a desired pseudo-localization principle.…”

“…[15,Lemma 3.13]) played a crucial role in estimating both good and bad functions appeared in the noncommutative Calderón-Zygmund decomposition. We also refer the reader to [2,13,31,12] for more related results on noncommutative Calderón-Zygmund decomposition and weak type (1, 1) estimates.…”

Algebraic Calderón-Zygmund theory

“…On the contrary, no endpoint estimate for p = 1 is possible using vector-valued methods. The original argument in [49] -also in a recent simpler form [4]-combines noncommutative martingales with a pseudolocalization principle for classical Calderón-Zygmund operators. More precisely, a quantification of how much L 2 -mass of a singular integral is concentrated around the support of the function on which it acts.…”