  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.
