An abstract theory of singular operators

TL;DR

This paper introduces a unified abstract framework for singular operators on measure spaces, demonstrating they can be dominated by sparse operators, leading to improved bounds and results across various operator classes.

Contribution

It presents a new abstract class of operators unifying several classical operators and proves they can be dominated by sparse operators, enhancing existing estimates.

Findings

Operators can be dominated by sparse operators with explicit constants

Unified framework encompasses Calderón-Zygmund, maximal functions, and martingale transforms

Improves bounds and results in harmonic analysis

Abstract

We introduce a class of operators on abstract measure spaces, which unifies the Calder\'on-Zygmund operators on spaces of homogeneous type, the maximal functions and the martingale transforms. We prove that such operators can be dominated by simple sparse operators with a definite form of the domination constant. Applying these estimates, we improve several results obtained by different authors in recent years.