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Bellman VS. Beurling: sharp estimates of uniform convexity for $ L^p$ spaces


Authors: P. B. Zatitskiy, P. Ivanisvili and D. M. Stolyarov
Translated by: the authors
Original publication: Algebra i Analiz, tom 27 (2015), nomer 2.
Journal: St. Petersburg Math. J. 27 (2016), 333-343
MSC (2010): Primary 42B20, 42B35, 47A30
Published electronically: January 29, 2016
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Abstract: The classical Hanner inequalities are obtained by the Bellman function method. These inequalities give sharp estimates for the moduli of convexity of Lebesgue spaces, initially due to Clarkson and Beurling. Easy ideas from differential geometry make it possible to find the Bellman function by using neither ``magic guesses'' nor bulky calculations.


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Additional Information

P. B. Zatitskiy
Affiliation: St. Petersburg Branch, Steklov Mathematical Institute, Russian Academy of Sciences, Fontanka 27, 191023 St. Petersburg; P. L. Chebyshev Research Laboratory, St. Petersburg State University
Email: paxa239@yandex.ru

P. Ivanisvili
Affiliation: Department of Mathematics, Michigan State University, East Lansing, Michigan 48823
Email: ivanishvili.paata@gmail.com

D. M. Stolyarov
Affiliation: St. Petersburg Branch, Steklov Mathematical Institute, Russian Academy of Sciences, Fontanka 27, 191023 St. Petersburg; P. L. Chebyshev Research Laboratory, St. Petersburg State University
Email: dms@pdmi.ras.ru

DOI: https://doi.org/10.1090/spmj/1390
Keywords: Bellman function, uniform convexity, torsion
Received by editor(s): September 21, 2014
Published electronically: January 29, 2016
Additional Notes: The work of the first author was supported by the Chebyshev Laboratory (Department of Mathematics and Mechanics, St. Petersburg State University), RF Government grant 11.G34.31.0026, by JSC “Gazprom Neft”, by President of Russia grant for young researchers MK-6133.2013.1, by the RFBR (grant 13-01-12422 ofi_m2, 14-01-00373_A), and by SPbSU (thematic project 6.38.223.2014).
This paper was completed during a visit of the second author to the Hausdorff Research Institute for Mathematics (HIM) in the framework of the Trimester Program “Harmonic Analysis and Partial Differential Equations”. He thanks HIM for the hospitality.
The work of the third author was supported by the Chebyshev Laboratory (Department of Mathematics and Mechanics, St. Petersburg State University) RF Government grant 11.G34.31.0026, by JSC “Gazprom Neft”, and by RFBR grant no. 11-01-00526.
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