On K-closedness, BMO-regularity and real interpolation of Hardy-type spaces

Original · EN

Let (X, Y) be a suitable couple of quasi-Banach lattices of measurable functions on T × Ω, and let (Xₐ, Yₐ) be the couple of the corresponding Hardy-type spaces. It has long been suspected that the BMO-regularity property of (X, Y) is not only sufficient for the K-closedness of (Xₐ, Yₐ) in (X, Y) but also necessary. We establish the equivalence of these two properties for a general couple of Banach lattices having the Fatou property when Ω is a discrete measurable space, and also for couples (X, Y) where X is allowed to be quasi-Banach but Y is assumed to be p-convex with some p > 1 (here Ω is arbitrary). We show under certain mild restrictions that the "good interpolation" formula (Xₐ, Hq)θ, ₚ = [(X, Lq)θ, ₚ]ₐ holds true if and only if X is BMO-regular.