Noncommutative differential transforms for averaging operators

TL;DR

This paper investigates the properties of noncommutative differential transforms related to averaging operators, establishing key boundedness estimates and extending the understanding of their behavior on noncommutative Lp-spaces.

Contribution

It provides the first comprehensive analysis of the mapping properties of noncommutative differential transforms, including weak and strong type estimates, using advanced harmonic analysis techniques.

Findings

Established weak type (1,1) estimates.

Proved (L∞, BMO) estimates.

Derived all strong type (p,p) estimates through interpolation.

Abstract

In this paper, we complete the study of mapping properties for a family of operators evaluating the difference between differentiation operators and conditional expectations acting on noncommutative $L_{p}$-spaces. To be more precise, we establish the weak type $(1,1)$ and $(L_{\infty},\mathrm{BMO})$ estimates of this difference. Consequently, in conjunction with interpolation and duality, we obtain all strong type $(p,p)$ estimates. This allows us to obtain a quick application to noncommutative differential transforms for averaging operators.