Bellman function for extremal problems in BMO
HTML articles powered by AMS MathViewer
- by Paata Ivanisvili, Nikolay N. Osipov, Dmitriy M. Stolyarov, Vasily I. Vasyunin and Pavel B. Zatitskiy PDF
- Trans. Amer. Math. Soc. 368 (2016), 3415-3468 Request permission
Abstract:
We develop a general method for obtaining sharp integral estimates on BMO. Each such estimate gives rise to a Bellman function, and we show that for a large class of integral functionals, this function is a solution of a homogeneous Monge–Ampère boundary-value problem on a parabolic plane domain. Furthermore, we elaborate an essentially geometric algorithm for solving this boundary-value problem. This algorithm produces the exact Bellman function of the problem along with the optimizers in the inequalities being proved. The method presented subsumes several previous Bellman-function results for BMO, including the sharp John–Nirenberg inequality and sharp estimates of $L^p$-norms of BMO functions.References
- D. L. Burkholder, Boundary value problems and sharp inequalities for martingale transforms, Ann. Probab. 12 (1984), no. 3, 647–702. MR 744226
- Martin Dindoš and Treven Wall, The sharp $A_p$ constant for weights in a reverse-Hölder class, Rev. Mat. Iberoam. 25 (2009), no. 2, 559–594. MR 2569547, DOI 10.4171/RMI/576
- 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
- P. Ivanishvili, N. N. Osipov, D. M. Stolyarov, V. I. Vasyunin, P. B. Zatitskiy, Bellman function for extremal problems in BMO, preprint, 91pp., http://arxiv.org/abs/1205.7018; Russian version: http://www.pdmi.ras.ru/preprint/2011/rus-2011.html and http://www.pdmi.ras.ru/ preprint/2012/rus-2012.html.
- Ivo Klemes, A mean oscillation inequality, Proc. Amer. Math. Soc. 93 (1985), no. 3, 497–500. MR 774010, DOI 10.1090/S0002-9939-1985-0774010-0
- Paul Koosis, Introduction to $H_p$ spaces, 2nd ed., Cambridge Tracts in Mathematics, vol. 115, Cambridge University Press, Cambridge, 1998. With two appendices by V. P. Havin [Viktor Petrovich Khavin]. MR 1669574
- 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
- A. A. Logunov, L. Slavin, D. M. Stolyarov, V. Vasyunin, and P. B. Zatitskiy, Weak integral conditions for BMO, Proc. Amer. Math. Soc. 143 (2015), no. 7, 2913–2926. MR 3336616, DOI 10.1090/S0002-9939-2015-12424-0
- 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
- F. Nazarov, S. Treil, and A. Volberg, Bellman function in stochastic control and harmonic analysis, Systems, approximation, singular integral operators, and related topics (Bordeaux, 2000) Oper. Theory Adv. Appl., vol. 129, Birkhäuser, Basel, 2001, pp. 393–423. MR 1882704
- Adam Osękowski, Sharp martingale and semimartingale inequalities, Instytut Matematyczny Polskiej Akademii Nauk. Monografie Matematyczne (New Series) [Mathematics Institute of the Polish Academy of Sciences. Mathematical Monographs (New Series)], vol. 72, Birkhäuser/Springer Basel AG, Basel, 2012. MR 2964297, DOI 10.1007/978-3-0348-0370-0
- A. V. Pogorelov, Differential geometry, P. Noordhoff N. V., Groningen, 1959. Translated from the first Russian ed. by L. F. Boron. MR 0114163
- Alexander Reznikov, Sharp weak type estimates for weights in the class $A_{p_1,p_2}$, Rev. Mat. Iberoam. 29 (2013), no. 2, 433–478. MR 3047424, DOI 10.4171/RMI/726
- Leonid Slavin, Bellman function and BMO, ProQuest LLC, Ann Arbor, MI, 2004. Thesis (Ph.D.)–Michigan State University. MR 2706427
- Leonid Slavin, Alexander Stokolos, and Vasily Vasyunin, Monge-Ampère equations and Bellman functions: the dyadic maximal operator, C. R. Math. Acad. Sci. Paris 346 (2008), no. 9-10, 585–588 (English, with English and French summaries). MR 2412802, DOI 10.1016/j.crma.2008.03.003
- 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
- Elias M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series, vol. 43, Princeton University Press, Princeton, NJ, 1993. With the assistance of Timothy S. Murphy; Monographs in Harmonic Analysis, III. MR 1232192
- V. Vasyunin, The sharp constant in the John–Nirenberg inequality, preprint POMI no. 20, 2003; http://www.pdmi.ras.ru/preprint/2003/index.html
- V. I. Vasyunin, The exact constant in the inverse Hölder inequality for Muckenhoupt weights, Algebra i Analiz 15 (2003), no. 1, 73–117 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 15 (2004), no. 1, 49–79. MR 1979718, DOI 10.1090/S1061-0022-03-00802-1
- V. I. Vasyunin, Mutual estimates for $L^p$-norms and the Bellman function, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 355 (2008), no. Issledovaniya po Lineĭnym Operatoram i Teorii Funktsiĭ. 36, 81–138, 237–238 (Russian, with English and Russian summaries); English transl., J. Math. Sci. (N.Y.) 156 (2009), no. 5, 766–798. MR 2744535, DOI 10.1007/s10958-009-9288-3
- V. Vasyunin, Sharp constants in the classical weak form of the John–Nirenberg inequality, preprint POMI, no. 10, 2011, 1–9; http://www.pdmi.ras.ru/preprint/2011/eng-2011.html
- V. Vasyunin and A. Vol′berg, The Bellman function for the simplest two-weight inequality: an investigation of a particular case, Algebra i Analiz 18 (2006), no. 2, 24–56 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 18 (2007), no. 2, 201–222. MR 2244935, DOI 10.1090/S1061-0022-07-00953-3
- Vasily Vasyunin and Alexander Volberg, Monge-Ampère equation and Bellman optimization of Carleson embedding theorems, Linear and complex analysis, Amer. Math. Soc. Transl. Ser. 2, vol. 226, Amer. Math. Soc., Providence, RI, 2009, pp. 195–238. MR 2500520, DOI 10.1090/trans2/226/16
Additional Information
- Paata Ivanisvili
- Affiliation: Department of Mathematics, Michigan State University, East Lansing, Michigan 48824
- MR Author ID: 921909
- Email: ivanishvili.paata@gmail.com
- Nikolay N. Osipov
- Affiliation: St. Petersburg Department of Steklov Mathematical Institute, Russian Academy of Sciences, Fontanka 27, St. Petersburg, Russia
- Email: nicknick@pdmi.ras.ru
- Dmitriy M. Stolyarov
- Affiliation: Chebyshev Laboratory, SPbU, 14th Line 29B, Vasilyevsky Island, St. Petersburg, Russia – and – St. Petersburg Department of Steklov Mathematical Institute, Russian Academy of Sciences, Fontanka 27, St. Petersburg, Russia
- MR Author ID: 895114
- Email: dms@pdmi.ras.ru
- Vasily I. Vasyunin
- Affiliation: St. Petersburg Department of Steklov Mathematical Institute, Russian Academy of Sciences, Fontanka 27, St. Petersburg, Russia
- Email: vasyunin@pdmi.ras.ru
- Pavel B. Zatitskiy
- Affiliation: Chebyshev Laboratory, SPbU, 14th Line 29B, Vasilyevsky Island, St. Petersburg, Russia – and – St. Petersburg Department of Steklov Mathematical Institute, Russian Academy of Sciences, Fontanka 27, St. Petersburg, Russia
- MR Author ID: 895184
- Email: paxa239@yandex.ru
- Received by editor(s): December 25, 2013
- Received by editor(s) in revised form: March 17, 2014
- Published electronically: September 15, 2015
- Additional Notes: The first author was supported by Chebyshev Laboratory of SPbU (RF Government grant No. 11.G34.31.0026)
The second author was supported by Chebyshev Laboratory of SPbU (RF Government grant No. 11.G34.31.0026), by RFBR (grant No. 11-01-00526), and by a Rokhlin grant.
The third author was supported by Chebyshev Laboratory of SPbU (RF Government grant No. 11.G34.31.0026) and by RFBR (grant No. 11-01-00526).
The fourth author was supported by RFBR (grant No. 11-01-00584).
The fifth author was supported by Chebyshev Laboratory of SPbU (RF Government grant No. 11.G34.31.0026) and by a Rokhlin grant. - © Copyright 2015 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 368 (2016), 3415-3468
- MSC (2010): Primary 42A05, 42B35, 49K20
- DOI: https://doi.org/10.1090/tran/6460
- MathSciNet review: 3451882