An introduction to pointwise sparse domination

TL;DR

This paper provides a comprehensive introduction to sparse domination, a technique in Harmonic Analysis that simplifies proving weighted inequalities for various operators, with applications to singular integrals.

Contribution

It offers a self-contained exposition of the Lerner-Ombrosi theorem on pointwise sparse domination and demonstrates its applications to different operator families.

Findings

Unified approach to weighted norm inequalities

Application to singular integral operators

Clarification of dyadic Harmonic Analysis techniques

Abstract

The goal of this expository paper is to give a self-contained introduction to sparse domination. This is a method relying on techniques from dyadic Harmonic Analysis which has received a lot of attention in recent years. Essentially, it allows for a unified approach to proving weighted norm inequalities for a large variety of operators. In this work, we will introduce the basic ideas of dyadic Harmonic Analysis, which we use to build up to the main result we discuss on pointwise sparse domination, which is the Lerner-Ombrosi theorem. We also give applications of this theorem to some families of operators, mainly relating to singular integral operators. The text has been structured so as to motivate the introduction of new ideas through the lens of solving specific problems in Harmonic Analysis.