Endpoint theory for the compactness of commutators

TL;DR

This paper characterizes the compactness of commutators at the endpoint in Hardy and Lebesgue spaces, introducing a Minkowski-type inequality and factorization theorems for Hardy spaces related to Calderón-Zygmund operators.

Contribution

It introduces the first endpoint compactness results for commutators and establishes a Minkowski-type inequality for weak Lebesgue spaces, along with Hardy space factorization theorems.

Findings

Characterization of relative compactness in weak Lebesgue spaces.

First investigation of endpoint compactness of commutators.

Factorization theorems for Hardy spaces via singular integral operators.

Abstract

In this paper, we establish a Minkowski-type inequality for weak Lebesgue space, which allows us to obtain a characterization of relative compactness in these spaces. Furthermore, we are the first to investigate the compactness results of commutators at the endpoint. The paper provides a comprehensive study of the compactness properties of commutators of Calder\'{o}n-Zygmund operators in Hardy and $L^{1}(\mathbb{R}^n)$ type spaces. Additionally, we provide factorization theorems for Hardy spaces in terms of singular integral operators in the $L^1(\mathbb{R}^n)$ space.