# Algebraic Calder\'on-Zygmund theory

## TL;DR

This paper develops a new algebraic approach to Calderón-Zygmund theory applicable to general measure spaces without regular metrics, including noncommutative and fractal settings, expanding classical harmonic analysis.

## Contribution

It introduces an abstract 'Markov metric' framework that extends Calderón-Zygmund theory to arbitrary von Neumann algebras and other irregular spaces.

## Key findings

- Constructed a new 'Markov metric' governing BMO spaces.
- Established endpoint inequalities for Calderón-Zygmund operators.
- Extended theory to noncommutative and fractal spaces.

## Abstract

Calder\'on-Zygmund theory has been traditionally developed on metric measure spaces satisfying additional regularity properties. In the lack of good metrics, we introduce a new approach for general measure spaces which admit a Markov semigroup satisfying purely algebraic assumptions. We shall construct an abstract form of "Markov metric" governing the Markov process and the naturally associated BMO spaces, which interpolate with the Lp-scale and admit endpoint inequalities for Calder\'on-Zygmund operators. Motivated by noncommutative harmonic analysis, this approach gives the first form of Calder\'on-Zygmund theory for arbitrary von Neumann algebras, but is also valid in classical settings like Riemannian manifolds with nonnegative Ricci curvature or doubling/nondoubling spaces. Other less standard commutative scenarios like fractals or abstract probability spaces are also included. Among our applications in the noncommutative setting, we improve recent results for quantum Euclidean spaces and group von Neumann algebras, respectively linked to noncommutative geometry and geometric group theory.