Commutators on Spaces of Homogeneous Type in Generalized Block Spaces

In this paper, we are interested in studying a generalized block space (denoted as \(\textbf{B}^{p,r}_\varphi \) ${\mathbf{B}}_{\phi }^{p,r}$ ) on a space of homogeneous type. We show that this space is the predual of certain generalized Morrey–Lorentz space. By duality, we obtain the \(\textbf{B}^{p,r}_\varphi \) ${\mathbf{B}}_{\phi }^{p,r}$ -bound of operators of Calderón–Zygmund type. In addition, we prove a weak Hardy factorization in terms of commutators of integral operator of Calderón–Zygmund type in block spaces. Thanks to the Hardy factorization result, we obtain a characterization of functions in \(\textrm{BMO}\) $\text{BMO}$ via the boundedness of commutators of homogeneous linear Calderón–Zygmund operators in the generalized block space (resp. the generalized Morrey–Lorentz space). Finally, we study a compactness characterization of commutators of Calderón–Zygmund type in generalized Morrey–Lorentz spaces.