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 Zap. Nauchn. Sem. POMI, 2010, Volume 376, Pages 116–166 (Mi znsl3621)

Remarks on BMO-regularity and AK-stability

D. V. Rutsky

St. Petersburg Department of V. A. Steklov Institute of Mathematics, Russian Academy of Sciences, St. Petersburg, Russia

Abstract: This paper concerns BMO-regularity and AK-stability for couples $(X,Y)$ of quasi-Banach lattices of measurable functions on the measure space $(\mathbb T,m)\times(\Omega,\mu)$, where $(\mathbb T,m)$ is the unit circle with Lebesgue measure. In an earlier work S. Kislyakov introduced a weaker version of BMO-regularity and conjectured that it is the same as the “strong” one in the case of couples of lattices having the Fatou property. Here we prove that these properties are indeed equivalent, thus verifying that BMO-regularity for couples is a self-dual property stable under division by a lattice. We also study another refinement of the AK-stability property and develop some techniques that allow us to slightly enlarge the class of weighted $l^p$-valued lattices for which AK-stability implies BMO-regularity. Finally, we discuss some points that might be relevant to the yet unanswered question about the relationship between AK-stability and BMO-regularity in general. Bibl. – 15 titles.

Key words and phrases: BMO-regularity, Hardy-type spaces, AK-stability, K-closedness, interpolation, Ky Fan–Kakutani fixed point theorem.

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English version:
Journal of Mathematical Sciences (New York), 2011, 172:2, 243–269

UDC: 517.982.1+517.538

Citation: D. V. Rutsky, “Remarks on BMO-regularity and AK-stability”, Investigations on linear operators and function theory. Part 38, Zap. Nauchn. Sem. POMI, 376, POMI, St. Petersburg, 2010, 116–166; J. Math. Sci. (N. Y.), 172:2 (2011), 243–269

Citation in format AMSBIB
\Bibitem{Rut10} \by D.~V.~Rutsky \paper Remarks on BMO-regularity and AK-stability \inbook Investigations on linear operators and function theory. Part~38 \serial Zap. Nauchn. Sem. POMI \yr 2010 \vol 376 \pages 116--166 \publ POMI \publaddr St.~Petersburg \mathnet{http://mi.mathnet.ru/znsl3621} \transl \jour J. Math. Sci. (N. Y.) \yr 2011 \vol 172 \issue 2 \pages 243--269 \crossref{https://doi.org/10.1007/s10958-010-0196-3} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-78651321165} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. D. V. Rutsky, “$\mathrm{BMO}$-regularity for lattices of measurable functions on spaces of homogeneous type”, St. Petersburg Math. J., 23:2 (2012), 381–412
2. D. V. Rutsky, “On the relationship between $\mathrm{AK}$-stability and $\mathrm{BMO}$-regularity”, J. Math. Sci. (N. Y.), 202:4 (2014), 601–612
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