Fractional Bloom boundedness of commutators in spaces of homogeneous type

We aim to characterise boundedness of commutators $[b,T]$ of singular integrals $T$. Boundedness is studied between weighted Lebesgue spaces $L^p(X)$ and $L^q(X)$, $p\leq q$, when the underlying space $X$ is a space of homogeneous type. Commutator theory in spaces of homogeneous type already exist in literature, in particular boundedness results in the setting $p=q$. The purpose here is to extend the earlier results to the setting of $p< q$. Our methods extend those of Duong et al. and Hyt\"onen et al. A novelty here is that in order to show the lower bound of the commutator norm, we demonstrate that the approximate weak factorisation of Hyt\"onen can be used when the underlying setting is a space of homogeneous type and not only in the Euclidean setting. The strength of the approximate weak factorisation is that (when compared to the so-called median method) it readily allows complex-valued $b$ in addition to real-valued ones. However, the median method has been previously successfully applied to iterated commutators and thus has its own strengths. We also present a proof based on that method.