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 -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

Email:
leonid.slavin@uc.edu

**V. Vasyunin**

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

Email:
vasyunin@pdmi.ras.ru

DOI:
http://dx.doi.org/10.1090/S0002-9947-2011-05112-3

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.