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Sharp results in the integral-form John-Nirenberg inequality

Authors: L. Slavin and V. Vasyunin
Journal: Trans. Amer. Math. Soc. 363 (2011), 4135-4169
MSC (2010): Primary 42A05, 42B35, 49K20
Published electronically: March 9, 2011
MathSciNet review: 2792983
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Abstract: We consider the strong form of the John-Nirenberg inequality for the $ L^2$-based BMO. We construct explicit Bellman functions for the inequality in the continuous and dyadic settings and obtain the sharp constant, as well as the precise bound on the inequality's range of validity, both previously unknown. The results for the two cases are substantially different. The paper not only gives another instance in the short list of such explicit calculations, but also presents the Bellman function method as a sequence of clear steps, adaptable to a wide variety of applications.

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Additional Information

L. Slavin
Affiliation: Department of Mathematical Sciences, University of Cincinnati, Cincinnati, Ohio 45221-0025

V. Vasyunin
Affiliation: St. Petersburg Department of the V. A. Steklov Mathematical Institute, Russian Academy of Sciences, St. Petersburg, Russia

Keywords: Bellman function method, John–Nirenberg inequality, BMO
Received by editor(s): June 18, 2008
Received by editor(s) in revised form: May 16, 2009
Published electronically: March 9, 2011
Additional Notes: The second author’s research was supported in part by RFBR (grant no. 08-01-00723-a)
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.