Some remarks to the corona theorem

Abstract

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.

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

“…The proof of Theorem 5 is given in Section 4 below; we establish that under the stated conditions AK-stability implies the bounded AKstability property, which is known to be equivalent to BMO-regularity by [30,Theorem 4]. The idea and many of the details of the argument are essentially identical to the main result of [12]: we use a fixed point theorem in order to derive the existence of a decomposition of the required form from a weaker property. Although this approach has been known to the author at the time the question about the equivalence between AK-stability and bounded AK-stability was raised in [27], the technique developed by that time proved to be inadequate.…”

“…Although this approach has been known to the author at the time the question about the equivalence between AK-stability and bounded AK-stability was raised in [27], the technique developed by that time proved to be inadequate. The Fan-Kakutani fixed point theorem, which was successfully applied before to similar problems (see [17], [27], [28], [12]), seems to be inapplicable to the task at hand, and we have to take advantage of a much more potent Powers's fixed point theorem [26].…”

“…Raising the couple (11) to the power q 2 by [30, Proposition 14] and replacing X with X q 2 and p with 2p q allows us to assume that the couple (X, L 2 ) θ 0 ,p , (X, L 2 ) θ 1 ,p is BMO-regular, and hence it is AK-stable. By [11,Lemma 1.1] and the reiteration theorem this implies that the couple (12) (X, L 2 ) α 0 ,2 , (X, L 2 ) α 1 ,2 is also AK-stable for all θ 0 < α 0 < α 1 < θ 1 , and hence it is BMO-regular by Theorem 6. By a well-known relation (see, e. g., [30,Proposition 11]) we have (13) L 2 = (X, X ′ ) 1 2 ,2 ; plugging this formula into (12) and applying the reiteration theorem allows us to rewrite (12) as (X, X ′ ) β 0 ,2 , (X, X ′ ) β 1 ,2…”

“…By [11,Lemma 1.1] and the reiteration theorem this implies that the couple (12) (X, L 2 ) α 0 ,2 , (X, L 2 ) α 1 ,2 is also AK-stable for all θ 0 < α 0 < α 1 < θ 1 , and hence it is BMO-regular by Theorem 6. By a well-known relation (see, e. g., [30,Proposition 11]) we have (13) L 2 = (X, X ′ ) 1 2 ,2 ; plugging this formula into (12) and applying the reiteration theorem allows us to rewrite (12) as (X, X ′ ) β 0 ,2 , (X, X ′ ) β 1 ,2…”

“…In [12], a technique was developed for a question related to the corona theorem. In the present work essentially the same technique is used to establish the equivalence between AK-stability and the bounded AKstability in the proof of Theorem 5.…”