On the Relationship Between AK-Stability and BMO-Regularity

Abstract

Let (X, Y) be a couple of Banach lattices of measurable functions on $$ \mathbb{T} $$ × Ω having the Fatou property and satisfying a certain condition (∗), which makes it possible to consistently introduce the Hardy-type subspaces of X and Y. We show that the bounded AK-stability property and the BMO-regularity property are equivalent for such couples. If either the lattice XY′ is Banach, or both lattices X2 and Y2 are Banach, or Y = Lp with p ∈ {1, 2, ∞}, then the AK-stability property and the BMO-regularity property are also equivalent for such couples (X, Y).