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Bellman Function for Extremal Problems in BMO II: Evolution
About this Title
Paata Ivanisvili, Kent State University, Kent, Ohio 44243, Dmitriy M. Stolyarov, Chebyshev Laboratory, St. Petersburg State University, 14th Line, 29b, St. Petersburg, 199178, Russia – and – St. Petersburg Department of Steklov Mathematical Institute, Russian Academy of Sciences, 27 Fontanka, St. Petersburg, 191023, Russia, Vasily I. Vasyunin, St. Petersburg Department of Steklov Mathematical Institute, Russian Academy of Sciences, 27 Fontanka, St. Petersburg, 191023, Russia – and – Chebyshev Laboratory, St. Petersburg State University, 14th Line, 29b, St. Petersburg, 199178, Russia and Pavel B. Zatitskiy, Chebyshev Laboratory, St. Petersburg State University, 14th Line, 29b, St. Petersburg, 199178, Russia – and – St. Petersburg Department of Steklov Mathematical Institute, Russian Academy of Sciences, 27 Fontanka, St. Petersburg, 191023, Russia
Publication: Memoirs of the American Mathematical Society
Publication Year:
2018; Volume 255, Number 1220
ISBNs: 978-1-4704-2954-6 (print); 978-1-4704-4817-2 (online)
DOI: https://doi.org/10.1090/memo/1220
Published electronically: August 2, 2018
Keywords: Bellman function,
bounded mean oscialltion
MSC: Primary 42B35, 26D07, 52A10, 35E10
Table of Contents
Chapters
- 1. Introduction
- 2. Setting and sketch of proof
- 3. Patterns for Bellman candidates
- 4. Evolution of Bellman candidates
- 5. Optimizers
- 6. Related questions and further development
Abstract
In a previous study, the authors built the Bellman function for integral functionals on the $\mathrm {BMO}$ space. The present paper provides a development of the subject. We abandon the majority of unwanted restrictions on the function that generates the functional. It is the new evolutional approach that allows us to treat the problem in its natural setting. What is more, these new considerations lighten dynamical aspects of the Bellman function, in particular, evolution of its picture- O. Beznosova and A. Reznikov, Sharp estimates involving $A_\infty$ and $L\log L$ constants, and their applications to PDE, Algebra i Analiz 26 (2014), no. 1, 40–67; English transl., St. Petersburg Math. J. 26 (2015), no. 1, 27–47. MR 3234812, DOI 10.1090/S1061-0022-2014-01329-5
- D. L. Burkholder, Boundary value problems and sharp inequalities for martingale transforms, Ann. Probab. 12 (1984), no. 3, 647–702. MR 744226
- Olof Hanner, On the uniform convexity of $L^p$ and $l^p$, Ark. Mat. 3 (1956), 239–244. MR 77087, DOI 10.1007/BF02589410
- 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, London Mathematical Society Lecture Note Series, vol. 40, Cambridge University Press, Cambridge-New York, 1980. With an appendix on Wolff’s proof of the corona theorem. MR 565451
- 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
- P. B. Zatitskiĭ, P. Ivanisvili, and D. M. Stolyarov, Bellman vs Beurling: sharp estimates of uniform convexity for $L^p$ spaces, Algebra i Analiz 27 (2015), no. 2, 218–231 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 27 (2016), no. 2, 333–343. MR 3444467, DOI 10.1090/S1061-0022-2016-01390-9
- P. Ivanishvili, D. M. Stolyarov, V. I. Vasyunin, P. B. Zatitskiy, Bellman function for extremal problems on $\mathrm {BMO}$ II: evolution, http://arxiv.org/abs/1510.01010.
- 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
- Paata Ivanisvili, Nikolay N. Osipov, Dmitriy M. Stolyarov, Vasily I. Vasyunin, and Pavel B. Zatitskiy, Bellman function for extremal problems in BMO, Trans. Amer. Math. Soc. 368 (2016), no. 5, 3415–3468. MR 3451882, DOI 10.1090/S0002-9947-2015-06460-5
- P. Ivanishvili, N. N. Osipov, D. M. Stolyarov, V. I. Vasyunin, P. B. Zatitskiy, Bellman function for extremal problems on $\mathrm {BMO}$, PDMI preprints 19/2011 (in Russian).
- 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
- Paata Ivanisvili, Nikolay N. Osipov, Dmitriy M. Stolyarov, Vasily I. Vasyunin, and Pavel B. Zatitskiy, Sharp estimates of integral functionals on classes of functions with small mean oscillation, C. R. Math. Acad. Sci. Paris 353 (2015), no. 12, 1081–1085 (English, with English and French summaries). MR 3427912, DOI 10.1016/j.crma.2015.07.016
- Sergey Kislyakov and Natan Kruglyak, Extremal problems in interpolation theory, Whitney-Besicovitch coverings, and singular integrals, Instytut Matematyczny Polskiej Akademii Nauk. Monografie Matematyczne (New Series) [Mathematics Institute of the Polish Academy of Sciences. Mathematical Monographs (New Series)], vol. 74, Birkhäuser/Springer Basel AG, Basel, 2013. MR 2975808
- 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
- Antonios D. Melas, The Bellman functions of dyadic-like maximal operators and related inequalities, Adv. Math. 192 (2005), no. 2, 310–340. MR 2128702, DOI 10.1016/j.aim.2004.04.013
- 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
- Adam Osȩkowski, Survey article: Bellman function method and sharp inequalities for martingales, Rocky Mountain J. Math. 43 (2013), no. 6, 1759–1823. MR 3178444, DOI 10.1216/RMJ-2013-43-6-1759
- Adam Osȩkowski, Sharp inequalities for BMO functions, Chin. Ann. Math. Ser. B 36 (2015), no. 2, 225–236. MR 3305705, DOI 10.1007/s11401-015-0887-7
- Adam Osȩkowski, Sharp estimates for Lipschitz class, J. Geom. Anal. 26 (2016), no. 2, 1346–1369. MR 3472838, DOI 10.1007/s12220-015-9593-7
- 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
- A. Reznikov, V. Vasyunin, A. Volberg, An observation: cut-off of the weight $w$ does not increase the $A_{p_{1}, p_{2}}$-‘‘norm’’ of $w$, http://arxiv.org/abs/1008.3635.
- Leonid Slavin, Bellman function and BMO, ProQuest LLC, Ann Arbor, MI, 2004. Thesis (Ph.D.)–Michigan State University. MR 2706427
- Leonid Slavin, Best constants for a family of Carleson sequences, Adv. Math. 289 (2016), 685–724. MR 3439697, DOI 10.1016/j.aim.2015.11.004
- L. Slavin, The John–Nirenberg constant for $\mathrm {BMO}^p$, $1 \leq p \leq 2$, submitted, http://arxiv.org/abs/1506.04969.
- 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
- Leonid Slavin and Vasily Vasyunin, Inequalities for BMO on $\alpha$-trees, Int. Math. Res. Not. IMRN 13 (2016), 4078–4102. MR 3544629, DOI 10.1093/imrn/rnv258
- L. Slavin and V. Vasyunin, The John–Nirenberg constant of $BMO^p$, $p>2$, Alg. i Anal. 28:2 (2016), 72–96 (in Russian), to be translated in St. Petersburg Math. J..
- 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
- D. M. Stolyarov and P. B. Zatitskiy, Theory of locally concave functions and its applications to sharp estimates of integral functionals, Adv. Math. 291 (2016), 228–273. MR 3459018, DOI 10.1016/j.aim.2015.11.048
- Dmitriy M. Stolyarov, Vasily I. Vasyunin, and Pavel B. Zatitskiy, Monotonic rearrangements of functions with small mean oscillation, Studia Math. 231 (2015), no. 3, 257–267. MR 3471053, DOI 10.4064/sm8326-2-2016
- V. Vasyunin, The sharp constant in the John–Nirenberg inequality, preprint POMI no. 20, 2003.
- 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. Vasyunin, Sharp constants in the classical weak form of the John–Nirenberg inequality, PDMI preprint, no. 10/2011 (in Russian).
- V. I. Vasyunin, An example of the construction of a Bellman function for extremal problems in BMO space, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 424 (2014), no. Issledovaniya po Lineĭnym Operatorami Teorii Funktsiĭ. 42, 33–125 (Russian, with English summary); English transl., J. Math. Sci. (N.Y.) 209 (2015), no. 5, 683–742. MR 3481444, DOI 10.1007/s10958-015-2521-3
- 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
- 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
- A. Volberg, Bellman approach to some problems in harmonic analysis, Séminaire Équations aux dérivées partielles (2001-2002), 1–14.
- A. Volberg, Bellman function technique in Harmonic Analysis, Lectures of INRIA Summer School in Antibes, June 2011, http://arxiv.org/abs/1106.3899.