Remarks on BMO-regularity and AK-stability

This paper concerns BMO-regularity and AK-stability for couples $(X,Y)$ of quasi-Banach lattices of measurable functions on the measure space $(\mathbb T,m)\times(\Omega,\mu)$, where $(\mathbb T,m)$ is the unit circle with Lebesgue measure. In an earlier work S. Kislyakov introduced a weaker version of BMO-regularity and conjectured that it is the same as the “strong” one in the case of couples of lattices having the Fatou property. Here we prove that these properties are indeed equivalent, thus verifying that BMO-regularity for couples is a self-dual property stable under division by a lattice. We also study another refinement of the AK-stability property and develop some techniques that allow us to slightly enlarge the class of weighted $l^p$-valued lattices for which AK-stability implies BMO-regularity. Finally, we discuss some points that might be relevant to the yet unanswered question about the relationship between AK-stability and BMO-regularity in general. Bibl. – 15 titles.