Bourgain's method for K-closedness in the semicommmutative setting

TL;DR

This paper extends Bourgain's classical harmonic analysis method to the semicommutative setting, utilizing recent Calderón-Zygmund decomposition advances, to recover and expand results on noncommutative Hardy and Sobolev spaces.

Contribution

It introduces a semicommutative adaptation of Bourgain's method, enabling new interpolation results for noncommutative Sobolev spaces on the torus.

Findings

Recovered Pisier's K-closedness of noncommutative Hardy spaces.

Established new interpolation results for noncommutative Sobolev spaces.

Extended Bourgain's method to the semicommutative setting.

Abstract

In the early 1990s, J.Bourgain proved a general result $K$-closedness result in the context of classical harmonic analysis. In this paper, we extend Bourgain's method to the semicommutative setting, making use of the recent semicommutative Calder\'on-Zygmund decomposition introduced by L.Cadilhac, JM.Conde-Alonso and J.Parcet. As an application, we recover Pisier's result about $K$-closedness of noncommutative Hardy spaces on the torus, and we also establish new interpolation results for noncommutative Sobolev spaces on the torus.