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Some remarks to the corona theorem


Authors: S. V. Kislyakov and D. V. Rutsky
Translated by: S. Kislyakov
Original publication: Algebra i Analiz, tom 24 (2012), nomer 2.
Journal: St. Petersburg Math. J. 24 (2013), 313-326
MSC (2010): Primary 30H80
DOI: https://doi.org/10.1090/S1061-0022-2013-01240-4
Published electronically: January 22, 2013
MathSciNet review: 3013331
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Abstract: With the help of a fixed point theorem, in §1 it is shown that the so-called $ L^{\infty }$- 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 $ \ell ^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^{\infty }$-estimates for solutions of corona problems.


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Additional Information

S. V. Kislyakov
Affiliation: St. Petersburg Branch, Steklov Mathematical Institute, Russian Academy of Sciences Fontanka 27, Saint Petersburg 191023, Russia
Email: skis@pdmi.ras.ru

D. V. Rutsky
Affiliation: St. Petersburg Branch, Steklov Mathematical Institute, Russian Academy of Sciences Fontanka 27, Saint Petersburg 191023, Russia
Email: rutsky@pdmi.ras.ru

DOI: https://doi.org/10.1090/S1061-0022-2013-01240-4
Keywords: Corona theorem, fixed point theorem, analytic partition of unity
Received by editor(s): November 1, 2011
Published electronically: January 22, 2013
Article copyright: © Copyright 2013 American Mathematical Society

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