# Sparse and weighted estimates for generalized H\"ormander operators and commutators

## TL;DR

This paper develops sparse domination techniques for generalized H"ormander operators and their commutators, leading to improved quantitative estimates and endpoint results in harmonic analysis.

## Contribution

It extends sparse domination results to iterated commutators and generalized kernels, providing new estimates and endpoint bounds.

## Key findings

- Established pointwise sparse domination for generalized H"ormander operators.
- Extended sparse domination to iterated commutators.
- Derived new quantitative and endpoint estimates for these operators.

## Abstract

In this paper we obtain a pointwise sparse domination for generalized H\"ormander operators and also for iterated commutators with those operators. As a particular case of our result we obtain a extension of the sparse domination for commutators obtained by A. Lerner, S. Ombrosi and the second author to iterated commutators. Relying upon that sparse domination a number of quantitative estimates such as Coifman-Fefferman estimates, strong type estimates, and endpoint estimates that improve and complete results in papers by M. Lorente, J.M. Martell, C. P\'erez, M.S. Riveros and A. de la Torre are obtained. We also provide a new local decay estimate and we also extend results in a paper due to J.M. Martell, C. Perez and R. Trujillo-Gonzalez to kernels satisfying generalized H\"ormander conditions. Among other applications, as a particular case of our result for endpoint estimates, we extend the endpoint estimate obtained in a work by A. Lerner, S. Ombrosi and the second author to iterated commutators.