Endpoint theory for the compactness of commutators

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.