Abstract. We introduce a new "weak" BMO-regularity condition for couples (X, Y ) of lattices of measurable functions on the circle (Definition 3, Section 9), describe it in terms of the lattice X 1/2 (Y ) 1/2 , and prove that this condition still ensures "good" interpolation for the couple (X A , Y A ) of the Hardy-type spaces corresponding to X and Y (Theorem 1, Section 9). Also, we present a neat version of Pisier's approach to interpolation of Hardy-type subspaces (Theorem 2, Section 13). These two main results of the paper are proved in Sections 10-18, where some related material of independent interest is also discussed. Sections 1-8 are devoted to the background and motivations, and also include a short survey of some previously known results concerning BMO-regularity. To a certain extent, the layout of the paper models that of the lecture delivered by the author at the conference in functional analysis in honour of Aleksander Pełczyński (Będlewo, September 22-29, 2002).
The set-up. By a lattice of measurable functions (or simply a lattice)we mean a quasi-Banach space X consisting of measurable functions on some σ-finite measure space (Σ, µ) and satisfying the following condition: if f ∈ X, g is measurable, and |g| ≤ |f |, then g ∈ X and g ≤ C f . When talking of a Banach lattice, we also assume that C = 1. In this paper, for the basic measure space we take the product (T × Ω, m × µ), where m is the normalized Lebesgue measure on the unit circle T, and (Ω, µ) is some σ-finite measure space. For technical reasons (see [6]), we often assume that the measure µ is discrete. This assumption is not too restrictive, however, because still fairly often it can be lifted by approximation by step functions. The case of functions on T is included by taking a point mass for µ.