Description of weak-type BMO-regularity

The weak-type BMO-regularity property for couples of quasi-Banach lattices of measurable functions was recently introduced as a suitable substitute for the usual BMO-regularity in connection with characterization of the $K$-closedness of Hardy-type spaces on the unit circle and stability for the real interpolation. It was characterized in terms of the BMO-regularity of couples $\left((X, Y)_{\alpha, p}, (X, Y)_{\beta, q}\right)$, $0 < \alpha < \beta < 1$, of the real interpolation spaces. In the present note, a natural characterization of this property similar to that of BMO-regularity for couples of Banach lattices $(X, Y)$ in terms of the BMO-regularity of $X' Y$ is extended to couples of lattices of measurable functions on homogeneous type spaces. We also derive equivalent conditions corresponding to the limit case where $\alpha = 0$.