Two remarks on the relationship between BMO-regularity and analytic stability of interpolation for lattices of measurable functions

Abstract

We study in this paper Hardy-type spaces on a measure space ( $$ \mathbb{T} $$ , m) × (Ω, µ), where ( $$ \mathbb{T} $$ , m) is the unit circle with Lebesgue measure. There is a characterization of analytic stability for real interpolation of weighted Hardy spaces on $$ \mathbb{T} $$ × Ω, a complete proof of which was present in the literature only for the case where µ is a point mass. Here this gap is filled, and a proof of the general case is presented. In a previous work by Kislyakov, certain results concerning BMO-regular lattices on ( $$ \mathbb{T} $$ × Ω, m × µ) were proved under the assumption that the measure µ is discrete. Here this extraneous assumption is lifted. Bibliography: 9 titles.