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.
- D. L. Burkholder, Boundary value problems and sharp inequalities for martingale transforms, Ann. Probab. 12 (1984), no. 3, 647–702. MR 744226
- Charles Fefferman, Characterizations of bounded mean oscillation, Bull. Amer. Math. Soc. 77 (1971), 587–588. MR 280994, DOI 10.1090/S0002-9904-1971-12763-5
- F. John and L. Nirenberg, On functions of bounded mean oscillation, Comm. Pure Appl. Math. 14 (1961), 415–426. MR 131498, DOI 10.1002/cpa.3160140317
- A. A. Korenovskiĭ, The connection between mean oscillations and exact exponents of summability of functions, Mat. Sb. 181 (1990), no. 12, 1721–1727 (Russian); English transl., Math. USSR-Sb. 71 (1992), no. 2, 561–567. MR 1099524, DOI 10.1070/SM1992v071n02ABEH001409
- Antonios D. Melas, The Bellman functions of dyadic-like maximal operators and related inequalities, Adv. Math. 192 (2005), no. 2, 310–340. MR 2128702, DOI 10.1016/j.aim.2004.04.013
- F. L. Nazarov and S. R. Treĭl′, The hunt for a Bellman function: applications to estimates for singular integral operators and to other classical problems of harmonic analysis, Algebra i Analiz 8 (1996), no. 5, 32–162 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 8 (1997), no. 5, 721–824. MR 1428988
- F. Nazarov, S. Treil, and A. Volberg, Bellman function in stochastic control and harmonic analysis, Systems, approximation, singular integral operators, and related topics (Bordeaux, 2000) Oper. Theory Adv. Appl., vol. 129, Birkhäuser, Basel, 2001, pp. 393–423. MR 1882704
- F. Nazarov, S. Treil, A. Volberg. The Bellman functions and two-weight inequalities for Haar multipliers. 1995, Preprint, MSU, pp. 1–25.
- F. Nazarov, S. Treil, and A. Volberg, The Bellman functions and two-weight inequalities for Haar multipliers, J. Amer. Math. Soc. 12 (1999), no. 4, 909–928. MR 1685781, DOI 10.1090/S0894-0347-99-00310-0
- Leonid Slavin, Bellman function and BMO, ProQuest LLC, Ann Arbor, MI, 2004. Thesis (Ph.D.)–Michigan State University. MR 2706427
- Leonid Slavin and Alexander Volberg, The $s$-function and the exponential integral, Topics in harmonic analysis and ergodic theory, Contemp. Math., vol. 444, Amer. Math. Soc., Providence, RI, 2007, pp. 215–228. MR 2423630, DOI 10.1090/conm/444/08582
- T. Tao. Bellman function and the John–Nirenberg inequality, Preprint, http://www. math.ucla.edu/˜tao/˜preprints/harmonic.html.
- V. I. Vasyunin, The exact constant in the inverse Hölder inequality for Muckenhoupt weights, Algebra i Analiz 15 (2003), no. 1, 73–117 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 15 (2004), no. 1, 49–79. MR 1979718, DOI 10.1090/S1061-0022-03-00802-1
- V. Vasyunin. The sharp constant in the John–Nirenberg inequality, Preprint POMI no. 20, 2003. http://www.pdmi.ras.ru/preprint/index.html
- V. Vasyunin and A. Vol′berg, The Bellman function for the simplest two-weight inequality: an investigation of a particular case, Algebra i Analiz 18 (2006), no. 2, 24–56 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 18 (2007), no. 2, 201–222. MR 2244935, DOI 10.1090/S1061-0022-07-00953-3
- A. Volberg. Bellman approach to some problems in harmonic analysis. Équations aux Dérivées Partielles, Exposé n. XX, 2002.
- L. Slavin
- Affiliation: Department of Mathematical Sciences, University of Cincinnati, Cincinnati, Ohio 45221-0025
- MR Author ID: 121075
- ORCID: 0000-0002-9502-8852
- Email: email@example.com
- V. Vasyunin
- Affiliation: St. Petersburg Department of the V. A. Steklov Mathematical Institute, Russian Academy of Sciences, St. Petersburg, Russia
- Email: firstname.lastname@example.org
- 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)
- © Copyright 2011
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
- 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
- MathSciNet review: 2792983