Rocky Mountain Journal of Mathematics

Survey Article: Bellman function method and sharp inequalities for martingales

Adam Osȩkowski

Full-text: Open access

Abstract

The Bellman function method is an efficient device which enables relating certain types of estimates arising in probability and harmonic analysis to the existence of the associated special function satisfying appropriate majorization and concavity. This technique has gained considerable interest in recent years and led to many interesting results concerning the boundedness of wide classes of singular integrals, Fourier multipliers, maximal functions and other related objects. The objective of this survey is to describe the Bellman function approach to certain classical results for martingale transforms. We present the detailed study of the weak-type and moment estimates, and develop some arguments which allow us to simplify and extend the statements, originally proven by Burkholder and others.

Article information

Source
Rocky Mountain J. Math., Volume 43, Number 6 (2013), 1759-1823.

Dates
First available in Project Euclid: 25 February 2014

Permanent link to this document
https://projecteuclid.org/euclid.rmjm/1393336657

Digital Object Identifier
doi:10.1216/RMJ-2013-43-6-1759

Mathematical Reviews number (MathSciNet)
MR3178444

Zentralblatt MATH identifier
1286.60039

Subjects
Primary: 60G42: Martingales with discrete parameter
Secondary: 42A05: Trigonometric polynomials, inequalities, extremal problems 60G46: Martingales and classical analysis 49K20: Problems involving partial differential equations

Keywords
Martingale Bellman function best constants

Citation

Osȩkowski, Adam. Survey Article: Bellman function method and sharp inequalities for martingales. Rocky Mountain J. Math. 43 (2013), no. 6, 1759--1823. doi:10.1216/RMJ-2013-43-6-1759. https://projecteuclid.org/euclid.rmjm/1393336657


Export citation

References

  • R. Bañuelos, A. Bielaszewski and K. Bogdan, Fourier multipliers for non-symmetric Lévy processes, Banach Center Publ. 95 (2011), 9-25.
  • R. Bañuelos and G. Wang, Sharp inequalities for martingales with applications to the Beurling-Ahlfors and Riesz transformations, Duke Math. J. 80 (1995), 575-600.
  • A. Borichev, P. Janakiraman and A. Volberg, Subordination by conformal martingales in $L^p$ and zeros of Laguerre polynomials, Duke Math. J. 162 (2013), 889-924.
  • –––, On Burkholder function for orthogonal martingales and zeros of Legendre polynomials, Amer. J. Math. 135 (2012), 207-236.
  • D.L. Burkholder, Martingale transforms, Ann. Math. Stat. 37 (1966), 1494-1504.
  • –––, A sharp inequality for martingale transforms, Ann. Prob. 7 (1979), 858-863.
  • –––, A geometrical characterization of Banach spaces in which martingale difference sequences are unconditional, Ann. Prob. 9 (1981), 997-1011.
  • –––, Boundary value problems and sharp inequalities for martingale transforms, Ann. Prob. 12 (1984), 647-702.
  • –––, Martingales and Fourier analysis in Banach spaces, Prob. Anal., Lect. Notes Math. 1206, Springer, Berlin, 1986.
  • D.L. Burkholder, Explorations in martingale theory and its applications, Lect. Notes Math. 1464, Springer, Berlin, 1991.
  • –––, Strong differential subordination and stochastic integration, Ann. Prob. 22 (1994), 995-1025.
  • –––, Sharp norm comparison of martingale maximal functions and stochastic integrals, Proc. Symp. Appl. Math. 52, American Mathematical Society, Providence, 1997.
  • –––, The best constant in the Davis' inequality for the expectation of the martingale square function, Trans. Amer. Math. Soc. 354 (2002), 91-105.
  • K.P. Choi, Some sharp inequalities for martingale transforms, Trans. Amer. Math. Soc. 307 (1988), 279-300.
  • –––, A sharp inequality for martingale transforms and the unconditional basis constant of a monotone basis in $L^p(0,1)$, Trans. Amer. Math. Soc. 330 (1993), 509-529.
  • D.C. Cox, The best constant in Burkholder's weak-$L^1$ inequality for the martingale square function, Proc. Amer. Math. Soc. 85 (1982), 427-433.
  • –––, Some sharp martingale inequalities related to Doob's inequality, in Inequalities in statistics and probability, IMS Lect. Notes-Monograph Ser. 5 (1984), 78–83.
  • L.E. Dor and E. Odell, Monotone bases in $L_p$, Pac. J. Math. 60 (1975), 51-61.
  • I. Doust, Contractive projections on Banach spaces, Proc. Centre for Math. Anal., Austr. Natl. Univ. 20 (1988), 50-58.
  • O. Dragičević and A. Volberg, Bellman function, Littlewood-Paley estimates, and asymptotics of the Ahlfors-Beurling operator in $L^p(\c)$, $p > 1$, Indiana Math. J. 54 (2005), 971-995.
  • –––, Bellman function and dimensionless estimates of classical and Ornstein-Uhlenbeck Riesz transforms, J. Oper. Theor. 56 (2006), 167-198.
  • –––, Linear dimension-free estimates in the embedding theorem for Schrödinger operators, J. Lond. Math. Soc. 85 (2012), 191-222.
  • P. Ivanishvili, N.N. Osipov, D.M. Stolyarov, V. Vasyunin and P. Zatitskiy, On Bellman function for extremal problems in BMO, C.R. Math. Acad. Sci. 350 (2012), 561-564.
  • J.H.B. Kemperman, The general moment problem, a geometric approach, Ann. Math. Stat. 39 (1968), 93-122.
  • J. Marcinkiewicz, Quelques théorèmes sur les séries orthogonales, Ann. Soc. Polon. Math. 16 (1937), 84-96.
  • A.D. Melas, Sharp general local estimates for dyadic-like maximal operators and related Bellman functions, Adv. Math. 220 (2009), 367-426.
  • A.D. Melas and E.N. Nikolidakis, Dyadic-like maximal operators on integrable functions and Bellman functions related to Kolmogorov's inequality, Trans. Amer. Math. Soc. 362 (2010), 1571-1597.
  • F. Nazarov, A. Reznikov, V. Vasyunin and A. Volberg, $A_1$ conjecture: Weak norm estimates of weighted singular operators and Bellman function, http:/\!/sashavolberg.files.wordpress.com/2010/11/a11\uline 7loghilb11\uline 21\uline 2010.pdf.
  • F.L. Nazarov and S.R. Treil, The hunt for a Bellman function: Applications to estimates for singular integral operators and to other classical problems of harmonic analysis, St. Petersburg Math. J. 8 (1997), 721-824.
  • F.L. Nazarov, S.R. Treil and A. Volberg, The Bellman functions and two-weight inequalities for Haar multipliers, J. Amer. Math. Soc. 12 (1999), 909-928.
  • –––, Bellman function in stochastic optimal control and harmonic analysis $($how our Bellman function got its name$)$, Oper. Theory: Adv. Appl. 129 (2001), 393-424.
  • F. Nazarov and A. Volberg, Heating of the Ahlfors-Beurling operator and estimates of its norm, St. Petersburg Math. J. 15 (2004), 563-573.
  • A. Osękowski, Sharp martingale and semimartingale inequalities, Mono. Matem. 72 (2012), Birkhäuser. Basel.
  • –––, Sharp logarithmic inequalities for Riesz transforms, J. Funct. Anal. 263 (2012), 89-108.
  • A. Osękowski, Sharp inequalities for the dyadic square function in the BMO setting, Acta Math. Hungar. 139 (2013), 85-105.
  • –––, Some sharp estimates for the Haar system and other bases in $L^1(0,1)$, Math. Scand., to appear.
  • R.E.A.C. Paley, A remarkable series of orthogonal functions, Proc. Lond. Math. Soc. 34 (1932), 241-264.
  • A. Pełczyński and H.P. Rosenthal, Localization techniques in $L^p$ spaces, Stud. Math. 52 (1975), 263-289.
  • G. Peskir and A. Shiryaev, Optimal stopping and free-boundary problem, Lect. Math., ETH Zurich.
  • J. Schauder, Eine Eigenschaft des Haarschen Orthogonalsystems, Math. Z. 28 (1928), 317-320.
  • L. Slavin, A. Stokolos and V. Vasyunin, Monge-Ampère equations and Bellman functions: The dyadic maximal operator, C.R. Acad. Sci. 346 (2008), 585-588.
  • L. Slavin and V. Vasyunin, Sharp results in the integral-form John-Nirenberg inequality, Trans. Amer. Math. Soc. 363 (2011), 4135-4169.
  • –––, Sharp $L^ p$ estimates on BMO, Indiana Math. J. 61 (2012), 1051-1110.
  • Y. Suh, A sharp weak type $(p,p)$ inequality $(p>2)$ for martingale transforms and other subordinate martingales, Trans. Amer. Math. Soc. 357 (2005), 1545-1564.
  • V. Vasyunin, The exact constant in the inverse Hölder inequality for Muckenhoupt weights, Alg. Anal. 15 (2003), 73-117 (in Russian); St. Petersburg Math. J. 15 (2004), 49–79 (in English).
  • –––, Mutual estimates for $L^p$-norms and the Bellman function, Zap. Nauchn. Sem. S.-Petersburg. Otdel. Mat. Inst. Steklov. (POMI) 355 (2008), Issl. Lineinym Oper. Teor. Funkt. 36, 81-138, 237–238 (in Russian); J. Math. Sci. (N.Y.) 156 (2009), 766–798 (in English).
  • V. Vasyunin and A. Volberg, Bellman functions technique in harmonic analysis, sashavolberg.wordpress.com.
  • –––, The Bellman function for certain two weight inequality: The case study, St. Petersburg Math. J. 18 (2007), 201-222.
  • –––, Monge-Ampère equation and Bellman optimization of Carleson embedding theorems, Amer. Math. Soc. Transl. 226 (2009), 195-238.
  • –––, Burkholder's function via Monge-Ampère equation, Illinois J. Math. 54 (2010), 1393-1428.
  • G. Wang, Sharp square-function inequalities for conditionally symmetic Martingales, Trans. Amer. Math. Soc. 328 (1991), 393-419.
  • –––, Sharp inequalities for the conditional square function of a martingale, Ann. Probab. 19 (1991), 1679-1688.
  • –––, Differential subordination and strong differential subordination for continuous time martingales and related sharp inequalities, Ann. Probab. 23 (1995), 522-551. \noindentstyle