Let X be a lattice of measurable functions on a space of homogeneous type (S, ν) (for example, S = R n with Lebesgue measure). Suppose that X has the Fatou property. Let T be either a Calderón-Zygmund singular integral operator with a singularity nondegenerate in a certain sense, or the Hardy-Littlewood maximal operator. It is proved that T is bounded on the lattice X α L 1−α 1 β for some β ∈ (0, 1) and sufficiently small α ∈ (0, 1) if and only if X has the following simple property: for every f ∈ X there exists a majorant g ∈ X such that log g ∈ BMO with proper control on the norms. This property is called BMO-regularity. For the reader's convenience, a self-contained exposition of the BMO-regularity theory is developed in the new generality, as well as some refinements of the main results.

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Let (X, Y ) be a couple of Banach lattices of measurable functions on 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 X 2 and Y 2 are Banach, or Y = L p with p ∈ {1, 2, ∞}, then the AK-stability property and the BMO-regularity property are also equivalent for such couples (X, Y ). Bibliography: 17 titles.

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With the help of a fixed point theorem, in §1 it is shown that the so-called L ∞ -and L p -corona problems are equivalent in the general situation. This equivalence extends to the case where L p is replaced by a more or less arbitrary Banach lattice of measurable functions on the circle. In §2, the corona theorem for 2 -valued analytic functions is exploited to give a new proof for the existence of an analytic partition of unity subordinate to a weight with logarithm in BMO. In §3, simple observations are presented that make it possible to pass from one sequence space to another in L ∞ -estimates for solutions of corona problems.2010 Mathematics Subject Classification. Primary 30H80.