Pseudo-localization of singular integrals and noncommutative Calderon-Zygmund theory

TL;DR

This paper establishes weak type (1,1) boundedness for Calderon-Zygmund operators on operator-valued functions using a novel noncommutative Calderon-Zygmund decomposition and a new pseudo-localization principle, advancing noncommutative harmonic analysis.

Contribution

Introduces a noncommutative Calderon-Zygmund decomposition and a pseudo-localization principle, addressing longstanding open problems in noncommutative Calderon-Zygmund theory.

Findings

Proves weak type (1,1) boundedness for operator-valued Calderon-Zygmund operators.

Develops a noncommutative Calderon-Zygmund decomposition technique.

Introduces a pseudo-localization principle for singular integrals.

Abstract

In this paper we obtain the weak type (1,1) boundedness of Calderon-Zygmund operators acting over operator-valued functions. Our main tools for its solution are a noncommutative form of Calderon-Zygmund decomposition in conjunction with a pseudo-localization principle for singular integrals, which is new even in the classical setting and of independent interest. Perhaps because of the hidden role of pseudo-localization and almost orthogonality, this problem has remained open for quite some time. We also consider Calderon-Zygmund operators associated to certain operator-valued kernels.