The John–Nirenberg constant of 𝐵𝑀𝑂^{𝑝}, 𝑝>2

Slavin, L.; Vasyunin, V.

doi:10.1090/spmj/1445

The John–Nirenberg constant of $\mathrm {BMO}^p$, $p> 2$
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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

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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