  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.
