BMO-regularity in lattices of measurable functions on spaces of homogeneous type

Let $X$ be a lattice of measurable functions on a space of homogeneous type $(S, \nu )$ (for example, $S = \mathbb{R}^n$ with Lebesgue measure). Suppose that $X$ has the Fatou property. Let $T$ be either a Calderón-Zygmund singular integral operator with a singularity nondegenerate in a certain sense, or the Hardy-Littlewood maximal operator. It is proved that $T$ is bounded on the lattice $\bigl (X^\alpha \mathrm {L}_1^{1 - \alpha }\bigr )^\beta$ for some $\beta \in (0, 1)$ and sufficiently small $\alpha \in (0, 1)$ if and only if $X$ has the following simple property: for every $f \in X$ there exists a majorant $g \in X$ such that $\log g \in \mathrm {BMO}$ with proper control on the norms. This property is called $\mathrm {BMO}$-regularity. For the reader's convenience, a self-contained exposition of the $\mathrm {BMO}$-regularity theory is developed in the new generality, as well as some refinements of the main results.