The John–Nirenberg constant of $\mathrm {BMO}^p$, $p> 2$
HTML articles powered by AMS MathViewer
- by
L. Slavin and V. Vasyunin
Translated by: The Authors - St. Petersburg Math. J. 28 (2017), 181-196
- DOI: https://doi.org/10.1090/spmj/1445
- Published electronically: February 15, 2017
- PDF | Request permission
Abstract:
This paper is a continuation of earlier work by the first author who determined the John–Nirenberg constant of $\mathrm {BMO}^p\big ((0,1)\big )$ for the range $1\le p\le 2$. Here, that constant is computed for $p>2$. As before, the main results rely on Bellman functions for the $L^p$ norms of the logarithms of $A_\infty$ weights, but for $p>2$ these functions turn out to have a significantly more complicated structure than for $1\le p\le 2$.References
- John B. Garnett and Peter W. Jones, The distance in BMO to $L^{\infty }$, Ann. of Math. (2) 108 (1978), no. 2, 373–393. MR 506992, DOI 10.2307/1971171
- Paata Ivanishvili, Nikolay N. Osipov, Dmitriy M. Stolyarov, Vasily I. Vasyunin, and Pavel B. Zatitskiy, On Bellman function for extremal problems in BMO, C. R. Math. Acad. Sci. Paris 350 (2012), no. 11-12, 561–564 (English, with English and French summaries). MR 2956143, DOI 10.1016/j.crma.2012.06.011
- Paata Ivanisvili, Nikolay N. Osipov, Dmitriy M. Stolyarov, Vasily I. Vasyunin, and Pavel B. Zatitskiy, Bellman function for extremal problems in BMO, Trans. Amer. Math. Soc. 368 (2016), no. 5, 3415–3468. MR 3451882, DOI 10.1090/tran/6460
- Paata Ivanisvili, Nikolay N. Osipov, Dmitriy M. Stolyarov, Vasily I. Vasyunin, and Pavel B. Zatitskiy, Sharp estimates of integral functionals on classes of functions with small mean oscillation, C. R. Math. Acad. Sci. Paris 353 (2015), no. 12, 1081–1085 (English, with English and French summaries). MR 3427912, DOI 10.1016/j.crma.2015.07.016
- 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
- Andrei K. Lerner, The John-Nirenberg inequality with sharp constants, C. R. Math. Acad. Sci. Paris 351 (2013), no. 11-12, 463–466 (English, with English and French summaries). MR 3090130, DOI 10.1016/j.crma.2013.07.007
- L. Slavin, The John–Nirenberg constant of $\mathrm {BMO}^p$, $1\le p\le 2$ (submitted), arXiv:1506.04969.
- L. Slavin and V. Vasyunin, Sharp results in the integral-form John-Nirenberg inequality, Trans. Amer. Math. Soc. 363 (2011), no. 8, 4135–4169. MR 2792983, DOI 10.1090/S0002-9947-2011-05112-3
- Leonid Slavin and Vasily Vasyunin, Sharp $L^p$ estimates on BMO, Indiana Univ. Math. J. 61 (2012), no. 3, 1051–1110. MR 3071693, DOI 10.1512/iumj.2012.61.4651
- D. M. Stolyarov and P. B. Zatitskiy, Theory of locally concave functions and its applications to sharp estimates of integral functionals, Adv. Math. 291 (2016), 228–273. MR 3459018, DOI 10.1016/j.aim.2015.11.048
- Vasily Vasyunin and Alexander Volberg, Sharp constants in the classical weak form of the John-Nirenberg inequality, Proc. Lond. Math. Soc. (3) 108 (2014), no. 6, 1417–1434. MR 3218314, DOI 10.1112/plms/pdt063
Bibliographic Information
- L. Slavin
- Affiliation: University of Cincinnati, 2815 Commons Way, Cincinnati, Ohio 45221
- MR Author ID: 121075
- ORCID: 0000-0002-9502-8852
- Email: leonid.slavin@uc.edu
- V. Vasyunin
- Affiliation: St. Petersburg Branch, V. A. Steklov Mathematical Institute, Russian Academy of Sciences, Fontanka 27, 191023, St. Petersburg, Russia; St. Petersburg State University, Universitetskii pr. 28, 198504, St. Petersburg, Russia
- Email: vasyunin@pdmi.ras.ru
- Received by editor(s): June 1, 2015
- Published electronically: February 15, 2017
- Additional Notes: The authors were supported by RSF (grant no. 14-41-00010)
- © Copyright 2017 American Mathematical Society
- Journal: St. Petersburg Math. J. 28 (2017), 181-196
- MSC (2010): Primary 42A05, 42B35, 49K20
- DOI: https://doi.org/10.1090/spmj/1445
- MathSciNet review: 3593004