Sharp inequalities for BMO functions

Abstract

The purpose of the paper is to study sharp weak-type bounds for functions of bounded mean oscillation. Let 0 < p < ∞ be a fixed number and let I be an interval contained in ℝ. The author shows that for any φ: I → ℝ and any subset E ⊂ I of positive measure, $\begin{array}{*{20}c} {\frac{{\left| I \right|^{ - \tfrac{1}{p}} }}{{\left| E \right|^{1 - \tfrac{1}{p}} }}\int_E {\left| {\phi - \frac{1}{{\left| I \right|}}\int_I {\phi dy} } \right|dx \leqslant \left\| \phi \right\|_{BMO(I)} , 0 < p \leqslant 2,} } \\ {\frac{{\left| I \right|^{ - \tfrac{1}{p}} }}{{\left| E \right|^{1 - \tfrac{1}{p}} }}\int_E {\left| {\phi - \frac{1}{{\left| I \right|}}\int_I {\phi dy} } \right|dx \leqslant \frac{p}{{2^{\tfrac{2}{p}} }}e^{\tfrac{2}{p} - 1} \left\| \phi \right\|_{BMO(I)} , p \geqslant 2.} } \\ \end{array}$ For each p, the constant on the right-hand side is the best possible. The proof rests on the explicit evaluation of the associated Bellman function. The result is a complement of the earlier works of Slavin, Vasyunin and Volberg concerning weak-type, L ^p and exponential bounds for the BMO class.

Keywords

BMO / Bellman function / Weak-type inequality / Best constants

References

[1] Bellman R. Dynamic programming, 2010, Princeton: Princeton University Press

[2] Burkholder D L. Boundary value problems and sharp inequalities for martingale transforms. Ann. Probab., 1984, 12: 647-702

[3] Burkholder D L. Explorations in martingale theory and its applications, École d’Ete de Probabilités de Saint-Flour XIX—1989, 1991, Berlin: Springer-Verlag 1-66

[4] Fefferman C. Characterizations of bounded mean oscillation. Bull. Amer. Math. Soc., 1971, 77: 587-588

[5] Grafakos L. Classical and Modern Fourier Analysis, 2004, New Jersey: Pearson Education, Inc.

[6] Ivanishvili P, Osipov N, Stolyarov D On Bellman function for extremal problems in BMO. C. R. Math. Acad. Sci. Paris, 2012, 350: 561-564

[7] Ivanishvili, P., Osipov, N., Stolyarov, D., et al., Bellman functions for the extremal problems on BMO (in Russian), http://www.pdmi.ras.ru/preprint/2011/rus-2011.html.

[8] John F, Nirenberg L. On functions of bounded mean oscillation. Comm. Pure and Appl. Math., 1961, 14: 415-426

[9] Korenovskii A. The connection between mean oscillations and exact exponents of summability of functions. Math. USSR-Sb., 1992, 71(2): 561-567

[10] Melas A D. The Bellman functions of dyadic-like maximal operators and related inequalities. Adv. Math., 2005, 192: 410-340

[11] Nazarov F, Treil S. The hunt for Bellman function: Applications to estimates of singular integral operators and to other classical problems in harmonic analysis. Algebra i Analis, 1997, 8: 42-162

[12] Nazarov F, Treil S, Volberg A. The Bellman functions and two-weight inequalities for Haar multipliers. J. Amer. Math. Soc., 1999, 12: 909-928

[13] Osekowski A. Sharp martingale and semimartingale inequalities, 2012, Basel AG, Basel: Birkhäuser, Springer-Verlag

[14] Slavin L. Bellman function and BMO, 2004

[15] Slavin L, Vasyunin V. Sharp results in the integral-form John-Nirenberg inequality. Trans. Amer. Math. Soc., 2011, 363: 4135-4169

[16] Slavin L, Vasyunin V. Sharp L ^p-estimates on BMO. Indiana Univ. Math. J., 2012, 61: 1051-1110

[17] Slavin L, Volberg A. Bellman function and the H ¹ — BMO duality, Harmonic analysis, partial differential equations, and related topics, 2007, Providence, RI: Amer. Math. Soc. 113-126

[18] Vasyunin, V., The sharp constant in John-Nirenberg inequality, http://www.pdmi.ras.ru/preprint/2003/index.html.

[19] Vasyunin V, Volberg A. Monge-Ampére equation and Bellman optimization of Carleson embedding theorems, linear and complex analysis, 2009, Providence, RI: Amer. Math. Soc. 195-238

[20] Vasyunin V, Volberg A. Sharp constants in the classical weak form of the John-Nirenberg inequality. Proc. Lond. Math. Soc., 2014, 108: 1417-1434