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
HTML articles powered by AMS MathViewer

by Adam Osękowski PDF

Proc. Amer. Math. Soc. 149 (2021), 333-343 Request permission

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 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 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 10.1007/s004400050218
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
Adriano M. Garsia, Martingale inequalities: Seminar notes on recent progress, Mathematics Lecture Note Series, W. A. Benjamin, Inc., Reading, Mass.-London-Amsterdam, 1973. MR 0448538
R. K. Getoor and M. J. Sharpe, Conformal martingales, Invent. Math. 16 (1972), 271–308. MR 305473, DOI 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 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 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), Lecture Notes in Math., Vol. 511, Springer, Berlin, 1976, pp. 401–413. 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 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
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 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 10.1112/plms/pdt063