Sharp estimates of integral functionals on classes of functions with small mean oscillation

Paata Ivanisvili; Nikolay N. Osipov; Dmitriy M. Stolyarov; Vasily I. Vasyunin; Pavel B. Zatitskiy

doi:10.1016/j.crma.2015.07.016

Mathematical analysis/Harmonic analysis

Sharp estimates of integral functionals on classes of functions with small mean oscillation

Presented by: Gilles Pisier

Paata Ivanisvili ¹; Nikolay N. Osipov ^2,³; Dmitriy M. Stolyarov ^2,⁴; Vasily I. Vasyunin ²; Pavel B. Zatitskiy ^2,⁴

¹ Department of Mathematics, Michigan State University, East Lansing, MI 48823, USA
² St. Petersburg Department of Steklov Mathematical Institute RAS, Fontanka 27, St. Petersburg, Russia
³ Norwegian University of Science and Technology (NTNU), IME Faculty, Dep. of Math. Sci., Alfred Getz' vei 1, Trondheim, Norway
⁴ Chebyshev Laboratory, St. Petersburg State University, 14th Line, 29b, St. Petersburg, 199178 Russia

Comptes Rendus. Mathématique, Volume 353 (2015) no. 12, pp. 1081-1085.

Abstract (Original language)
Résumé

We unify several Bellman function problems treated in [1,2,4–6,9–12,14–16,18–25]. For that purpose, we define a class of functions that have, in a sense, small mean oscillation (this class depends on two convex sets in $R^{2}$ ). We show how the unit ball in the BMO space, or a Muckenhoupt class, or a Gehring class can be described in such a fashion. Finally, we consider a Bellman function problem on these classes, discuss its solution and related questions.

Nous unifions plusieurs problèmes concernant la fonction de Bellman traités dans [1,2,4–6,9–12,14–16,18–25]. Dans ce but, nous introduisons une classe de fonctions dont l'oscillation moyenne est petite dans un certain sens (cette classe depend de deux sous-ensembles convexes de $R^{2}$ ). Nous démontrons que la boule unité de l'espace BMO, ou de la classe de Muckenhoupt, ou de la classe de Gehring, peut être décrite de cette façon. Finalement, nous considérons un problème de fonction de Bellman sur chacune de ces classes et discutons sa résolution ainsi que des questions voisines.

Received: 2014-12-21
Accepted: 2015-07-24
Published online: 2015-10-29

DOI: 10.1016/j.crma.2015.07.016

Author's affiliations:

Paata Ivanisvili ¹; Nikolay N. Osipov ^{2, 3}; Dmitriy M. Stolyarov ^{2, 4}; Vasily I. Vasyunin ²; Pavel B. Zatitskiy ^{2, 4}

¹ Department of Mathematics, Michigan State University, East Lansing, MI 48823, USA
² St. Petersburg Department of Steklov Mathematical Institute RAS, Fontanka 27, St. Petersburg, Russia
³ Norwegian University of Science and Technology (NTNU), IME Faculty, Dep. of Math. Sci., Alfred Getz' vei 1, Trondheim, Norway
⁴ Chebyshev Laboratory, St. Petersburg State University, 14th Line, 29b, St. Petersburg, 199178 Russia

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     author = {Paata Ivanisvili and Nikolay N. Osipov and Dmitriy M. Stolyarov and Vasily I. Vasyunin and Pavel B. Zatitskiy},
     title = {Sharp estimates of integral functionals on classes of functions with small mean oscillation},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1081--1085},
     publisher = {Elsevier},
     volume = {353},
     number = {12},
     year = {2015},
     doi = {10.1016/j.crma.2015.07.016},
     language = {en},
}

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Paata Ivanisvili; Nikolay N. Osipov; Dmitriy M. Stolyarov; Vasily I. Vasyunin; Pavel B. Zatitskiy. Sharp estimates of integral functionals on classes of functions with small mean oscillation. Comptes Rendus. Mathématique, Volume 353 (2015) no. 12, pp. 1081-1085. doi : 10.1016/j.crma.2015.07.016. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2015.07.016/

References
Cited by

[1] O. Beznosova; A. Reznikov Sharp estimates involving $A_{\infty}$ and $L \log L$ constants, and their applications to PDE, Algebra Anal., Volume 26 (2014) no. 1, pp. 40-67

[2] M. Dindoš; T. Wall The sharp $A_{p}$ constant for weights in a reverse-Hölder class, Rev. Mat. Iberoam., Volume 25 (2009) no. 2, pp. 559-594

[3] P. Ivanishvili; D.M. Stolyarov; P.B. Zatitskiy Bellman VS Beurling: sharp estimates of uniform convexity for $L^{p}$ spaces, Algebra Anal., Volume 27 (2015) no. 2, pp. 218-231 (in Russian); to be translated in St. Petersburg Math. J. | arXiv

[4] P. Ivanishvili; D.M. Stolyarov; N.N. Osipov; V.I. Vasyunin; P.B. Zatitskiy Bellman function for extremal problems in BMO, Trans. Amer. Math. Soc. (2015) http://www.ams.org/journals/tran/0000-000-00/S0002-9947-2015-06460-5/home.html (in press) | arXiv | DOI

[5] P. Ivanishvili; N.N. Osipov; D.M. Stolyarov; V.I. Vasyunin; P.B. Zatitskiy On Bellman function for extremal problems in BMO, C. R. Acad. Sci. Paris, Ser. I, Volume 350 (2012) no. 11, pp. 561-564

[6] P. Ivanishvili, D.M. Stolyarov, V.I. Vasyunin, P.B. Zatitskiy, Bellman function for extremal problems on BMO II: evolution, 2015, . | arXiv

[7] A.A. Korenovskii The exact continuation of a reverse Hölder inequality and Muckenhoupt's conditions, Mat. Zametki, Volume 52 (1992) no. 6, pp. 32-44 (in Russian); translated in Math. Notes, 52, 6, 1992, pp. 1192-1201

[8] A.A. Korenovskii; A.M. Stokolos Some applications of equimeasurable rearrangements, Recent Advances in Harmonic Analysis and Applications, Proc. in Math. & Stat., vol. 25, Springer, 2013, pp. 181-196

[9] A.A. Logunov; L. Slavin; D.M. Stolyarov; V. Vasyunin; P.B. Zatitskiy Weak integral conditions for BMO, Proc. Amer. Math. Soc., Volume 143 (2015), pp. 2913-2926 | arXiv

[10] A. Osȩkowski Sharp inequalities for BMO functions, Chin. Ann. Math., Ser. B, Volume 36 (March 2015) no. 2, pp. 225-236

[11] A. Reznikov Sharp weak type estimates for weights in the class $A_{p_{1}, p_{2}}$ , Rev. Mat. Iberoam., Volume 29 (2013) no. 2, pp. 433-478

[12] L. Slavin Bellman function and BMO, Michigan State University, 2004 (Ph.D. thesis)

[13] L. Slavin, The John–Nirenberg constant of ${BMO}^{p}$ , $1 ⩽ p ⩽ 2$ , 2015, . | arXiv

[14] L. Slavin; A. Stokolos; V. Vasyunin Monge–Ampère equations and Bellman functions: the dyadic maximal operator, C. R. Acad. Sci. Paris, Ser. I, Volume 346 (2008) no. 1, pp. 585-588

[15] L. Slavin; V. Vasyunin Sharp results in the integral form John–Nirenberg inequality, Trans. Amer. Math. Soc., Volume 363 (2011) no. 8, pp. 4135-4169

[16] L. Slavin; V. Vasyunin Sharp $L^{p}$ estimates on BMO, Indiana Univ. Math. J., Volume 61 (2012) no. 3, pp. 1051-1110

[17] D.M. Stolyarov, P.B. Zatitskiy, Theory of locally concave functions and its applications to sharp estimates of integral functionals, 2015, . | arXiv

[18] V. Vasyunin The sharp constant in the John–Nirenberg inequality, POMI Prepr., Volume 20 (2003)

[19] V.I. Vasyunin The sharp constant in the reverse Hölder inequality for Muckenhoupt weights, Algebra Anal., Volume 15 (2003) no. 1, pp. 73-117 (in Russian); translated in St. Petersburg Math. J., 15, 1, 2004, pp. 49-79

[20] V.I. Vasyunin Mutual estimates of $L^{p}$ -norms and the Bellman function, Zap. Nauč. Semin. POMI, Volume 355 (2008) no. 5, pp. 81-138 (in Russian); translated in J. Math. Sci., 156, 2009, pp. 766-798

[21] V. Vasyunin Sharp constants in the classical weak form of the John–Nirenberg inequality, POMI Prepr., Volume 10 (2011), pp. 1-9

[22] V.I. Vasyunin An example of constructing Bellman function for extremal problems in BMO, Zap. Nauč. Semin. POMI, Volume 424 (2014), pp. 33-125 (in Russian)

[23] V. Vasyunin; A. Volberg The Bellman function for a certain two-weight inequality: a case study, Algebra Anal., Volume 18 (2006) no. 2, pp. 24-56 (in Russian); translated in St. Petersburg Math. J., 18, 2, 2007, pp. 201-222

[24] V. Vasyunin; A. Volberg Monge–Ampère equation and Bellman optimization of Carleson embedding theorems, Amer. Math. Soc. Transl., Ser. 2, Volume 226 (2009), pp. 195-238

[25] V. Vasyunin; A. Volberg Sharp constants in the classical weak form of the John–Nirenberg inequality, Proc. Lond. Math. Soc., Volume 108 (2014) no. 6, pp. 1417-1434

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