Weighted weak-type $\mathrm{BMO}$-regularity

Stability for the weak-type $\mathrm{BMO}$-regularity property of a couple $(X, Y)$ under the perturbation $(X (u), Y (v))$ by some weights is considered. An example of weighted Lorentz spaces $\mathrm{L}_{p, q (\cdot)}$ with piecewise constant $q (\cdot)$ shows that in general such stability does not characterize the usual $\mathrm{BMO}$-regularity. On the other hand, for couples of Banach lattices $X$ and $Y$ with the Fatou property such that $(X^r)' Y^r$ is also Banach with some $r > 0$, the simultaneous weak-type $\mathrm{BMO}$-regularity of $(X, Y)$ and $(X (u), Y (v))$ implies that $\log (u / v) \in \mathrm{BMO}$. For couples of $r$-convex lattices with the Fatou property we establish the sufficiency of the weak-type $\mathrm{BMO}$-regularity for the $K$-closedness of the respective Hardy-type spaces without the assumption that the space of the second variable is discrete, generalizing earlier results.