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Bellman function for extremal problems in BMO


Authors: Paata Ivanisvili, Nikolay N. Osipov, Dmitriy M. Stolyarov, Vasily I. Vasyunin and Pavel B. Zatitskiy
Journal: Trans. Amer. Math. Soc. 368 (2016), 3415-3468
MSC (2010): Primary 42A05, 42B35, 49K20
DOI: https://doi.org/10.1090/tran/6460
Published electronically: September 15, 2015
MathSciNet review: 3451882
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Abstract: We develop a general method for obtaining sharp integral estimates on BMO. Each such estimate gives rise to a Bellman function, and we show that for a large class of integral functionals, this function is a solution of a homogeneous Monge-Ampère boundary-value problem on a parabolic plane domain. Furthermore, we elaborate an essentially geometric algorithm for solving this boundary-value problem. This algorithm produces the exact Bellman function of the problem along with the optimizers in the inequalities being proved. The method presented subsumes several previous Bellman-function results for BMO, including the sharp John-Nirenberg inequality and sharp estimates of $ L^p$-norms of BMO functions.


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Additional Information

Paata Ivanisvili
Affiliation: Department of Mathematics, Michigan State University, East Lansing, Michigan 48824
Email: ivanishvili.paata@gmail.com

Nikolay N. Osipov
Affiliation: St. Petersburg Department of Steklov Mathematical Institute, Russian Academy of Sciences, Fontanka 27, St. Petersburg, Russia
Email: nicknick@pdmi.ras.ru

Dmitriy M. Stolyarov
Affiliation: Chebyshev Laboratory, SPbU, 14th Line 29B, Vasilyevsky Island, St. Petersburg, Russia – and – St. Petersburg Department of Steklov Mathematical Institute, Russian Academy of Sciences, Fontanka 27, St. Petersburg, Russia
Email: dms@pdmi.ras.ru

Vasily I. Vasyunin
Affiliation: St. Petersburg Department of Steklov Mathematical Institute, Russian Academy of Sciences, Fontanka 27, St. Petersburg, Russia
Email: vasyunin@pdmi.ras.ru

Pavel B. Zatitskiy
Affiliation: Chebyshev Laboratory, SPbU, 14th Line 29B, Vasilyevsky Island, St. Petersburg, Russia – and – St. Petersburg Department of Steklov Mathematical Institute, Russian Academy of Sciences, Fontanka 27, St. Petersburg, Russia
Email: paxa239@yandex.ru

DOI: https://doi.org/10.1090/tran/6460
Received by editor(s): December 25, 2013
Received by editor(s) in revised form: March 17, 2014
Published electronically: September 15, 2015
Additional Notes: The first author was supported by Chebyshev Laboratory of SPbU (RF Government grant No. 11.G34.31.0026)
The second author was supported by Chebyshev Laboratory of SPbU (RF Government grant No. 11.G34.31.0026), by RFBR (grant No. 11-01-00526), and by a Rokhlin grant.
The third author was supported by Chebyshev Laboratory of SPbU (RF Government grant No. 11.G34.31.0026) and by RFBR (grant No. 11-01-00526).
The fourth author was supported by RFBR (grant No. 11-01-00584).
The fifth author was supported by Chebyshev Laboratory of SPbU (RF Government grant No. 11.G34.31.0026) and by a Rokhlin grant.
Article copyright: © Copyright 2015 American Mathematical Society