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

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

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.

**1.**D. L. Burkholder. Boundary value problems and sharp inequalities for martingale transforms.*Annals of Probability*, Vol. 12 (1984), no. 3, pp. 647-702. MR**744226 (86b:60080)****2.**C. Fefferman. Characterizations of bounded mean oscillations.*Bull. Amer. Math. Soc.*, Vol. 77 (1971), pp. 587-588. MR**0280994 (43:6713)****3.**F. John, L. Nirenberg. On functions of bounded mean oscillation.*Comm. Pure Appl. Math.*, Vol. 14 (1961), pp. 415-426. MR**0131498 (24:A1348)****4.**A. Korenovskii. The connection between mean oscillations and exact exponents of summability of functions.*Math. USSR-Sb.*, Vol. 71 (1992), no. 2, pp. 561-567. MR**1099524 (92b:26019)****5.**A. Melas. The Bellman functions of dyadic-like maximal operators and related inequalities. Advances in Mathematics, Vol. 192 (2005), no. 2, pp. 310-340. MR**2128702 (2005k:42052)****6.**F. Nazarov, S. Treil. The hunt for Bellman function: Applications to estimates of singular integral operators and to other classical problems in harmonic analysis.*Algebra i Analiz*, Vol. 8 (1997), no. 5, pp. 32-162. MR**1428988 (99d:42026)****7.**F. Nazarov, S. Treil, A. Volberg. Bellman function in stochastic control and harmonic analysis in*Systems, Approximation, Singular integral operators, and related topics*, ed. A Borichev, N. Nikolski,*Operator Theory: Advances and Applications*, Vol. 129, 2001, pp. 393-424, Birkhauser Verlag. MR**1882704 (2003b:49024)****8.**F. Nazarov, S. Treil, A. Volberg. The Bellman functions and two-weight inequalities for Haar multipliers. 1995, Preprint, MSU, pp. 1-25.**9.**F. Nazarov, S. Treil, A. Volberg. The Bellman functions and two-weight inequalities for Haar multipliers.*Journal of the American Mathematical Society*, Vol. 12 (1999), no. 4, pp. 909-928. MR**1685781 (2000k:42009)****10.**L. Slavin. Bellman function and BMO. Ph.D. thesis, Michigan State University, ProQuest LLC, Ann Arbor, MI, 2004. MR**2706427****11.**L. Slavin, A. Volberg. The -function and the exponential integral. Topics in harmonic analysis and ergodic theory, pp. 215-228,*Contemp. Math.*, Vol. 444, Amer. Math. Soc., Providence, RI, 2007. MR**2423630 (2009c:26048)****12.**T. Tao. Bellman function and the John-Nirenberg inequality, Preprint, http://www. math.ucla.edu/˜tao/˜preprints/harmonic.html.**13.**V. Vasyunin. The sharp constant in the reverse Hölder inequality for Muckenhoupt weights.*Algebra i Analiz*, Vol. 15 (2003), no. 1, pp. 73-117. MR**1979718 (2004h:42017)****14.**V. Vasyunin. The sharp constant in the John-Nirenberg inequality, Preprint POMI no. 20, 2003. http://www.pdmi.ras.ru/preprint/index.html**15.**V. Vasyunin, A. Volberg. The Bellman functions for a certain two weight inequality: The case study.*Algebra i Analiz*, Vol. 18 (2006), No. 2. MR**2244935 (2007k:47053)****16.**A. Volberg. Bellman approach to some problems in harmonic analysis.*Équations aux Dérivées Partielles*, Exposé n. XX, 2002.

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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:
https://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.