An invitation to harmonic analysis associated with semigroups of operators

TL;DR

This paper introduces harmonic analysis techniques linked to semigroups of operators, aiming to develop a noncommutative Calderón-Zygmund theory applicable in settings with limited metric information.

Contribution

It proposes a new approach using semigroups of operators to extend harmonic analysis and Calderón-Zygmund theory into noncommutative and metric-limited contexts.

Findings

Applicable to noncommutative von Neumann algebras

Provides a framework for harmonic analysis without classical metric assumptions

Useful in both noncommutative and classical settings

Abstract

This article is an introduction to our recent work in harmonic analysis associated with semigroups of operators, in the effort of finding a noncommutative Calder\'on-Zygmund theory for von Neumann algebras. The classical CZ theory has been traditionally developed on metric measure spaces satisfying additional regularity properties. In the lack of such metrics -or with very little information on the metric- Markov semigroups of operators appear to be the right substitutes of classical metric/geometric tools in harmonic analysis. Our approach is particularly useful in the noncommutative setting but it is also valid in classical/commutative frameworks.