## On BMO-regular couples of lattices of measurable functions

### Volume 159 / 2003

#### 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 (Bedlewo, September 22–29, 2002).