On the best constant in the estimate related to 𝐻¹-𝐵𝑀𝑂 duality

Osękowski, Adam

doi:10.1090/proc/15234

On the best constant in the estimate related to $H^1-BMO$ duality

Author: Adam Osękowski
Journal: Proc. Amer. Math. Soc. 149 (2021), 333-343
MSC (2010): Primary 42A05, 42B35, 49K20
DOI: https://doi.org/10.1090/proc/15234
Published electronically: October 20, 2020
MathSciNet review: 4172609
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $I\subset \mathbb {R}$ be an interval and let $f$, $\varphi$ be arbitrary elements of $H^1(I)$ and $BMO(I)$, respectively, with $\int _I\varphi =0$. The paper contains the proof of the estimate \begin{equation*} \int _I f\varphi \leq \sqrt {2}\|f\|_{H^1(I)}\|\varphi \|_{BMO(I)}, \end{equation*} and it is shown that $\sqrt {2}$ cannot be replaced by a smaller universal constant. The argument rests on the existence of a special function enjoying appropriate size and concavity requirements.

References

Richard Bellman, Dynamic programming, Princeton Landmarks in Mathematics, Princeton University Press, Princeton, NJ, 2010. Reprint of the 1957 edition; With a new introduction by Stuart Dreyfus. MR 2641641
Colin Bennett, Ronald A. DeVore, and Robert Sharpley, Weak-$L^{\infty }$ and BMO, Ann. of Math. (2) 113 (1981), no. 3, 601–611. MR 621018, DOI https://doi.org/10.2307/2006999
Ch’ing Sung Chou, Sur certaines généralisations de l’inégalité de Fefferman, Seminar on probability, XVIII, Lecture Notes in Math., vol. 1059, Springer, Berlin, 1984, pp. 219–222 (French). MR 770963, DOI https://doi.org/10.1007/BFb0100046
Tahir Choulli, Christophe Stricker, and Leszek Krawczyk, On Fefferman and Burkholder-Davis-Gundy inequalities for $\scr E$-martingales, Probab. Theory Related Fields 113 (1999), no. 4, 571–597. MR 1717531, DOI https://doi.org/10.1007/s004400050218
Charles Fefferman, Characterizations of bounded mean oscillation, Bull. Amer. Math. Soc. 77 (1971), 587–588. MR 280994, DOI https://doi.org/10.1090/S0002-9904-1971-12763-5
Adriano M. Garsia, Martingale inequalities: Seminar notes on recent progress, W. A. Benjamin, Inc., Reading, Mass.-London-Amsterdam, 1973. Mathematics Lecture Notes Series. MR 0448538
R. K. Getoor and M. J. Sharpe, Conformal martingales, Invent. Math. 16 (1972), 271–308. MR 305473, DOI https://doi.org/10.1007/BF01425714
Loukas Grafakos, Classical Fourier analysis, 2nd ed., Graduate Texts in Mathematics, vol. 249, Springer, New York, 2008. MR 2445437
F. John and L. Nirenberg, On functions of bounded mean oscillation, Comm. Pure Appl. Math. 14 (1961), 415–426. MR 131498, DOI https://doi.org/10.1002/cpa.3160140317
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, The Bellman functions and two-weight inequalities for Haar multipliers, J. Amer. Math. Soc. 12 (1999), no. 4, 909–928. MR 1685781, DOI https://doi.org/10.1090/S0894-0347-99-00310-0
Maurizio Pratelli, Sur certains espaces de martingales localement de carré intégrable, Séminaire de Probabilités, X (Seconde partie: Théorie des intégrales stochastiques, Univ. Strasbourg, Strasbourg, année universitaire 1974/1975), Springer, Berlin, 1976, pp. 401–413. Lecture Notes in Math., Vol. 511. MR 0438467
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 https://doi.org/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 https://doi.org/10.1512/iumj.2012.61.4651
Leonid Slavin and Alexander Volberg, Bellman function and the $H^1$-$\rm BMO$ duality, Harmonic analysis, partial differential equations, and related topics, Contemp. Math., vol. 428, Amer. Math. Soc., Providence, RI, 2007, pp. 113–126. MR 2322382, DOI https://doi.org/10.1090/conm/428/08216
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 https://doi.org/10.1112/plms/pdt063