Don Burkholder’s work on Banach spaces

  • Burgess Davis
  • Renming Song
Part of the Selected Works in Probability and Statistics book series (SWPS)


Martingale theory and especially Burkholder’s work fascinated me right from the start. Mar- tingales were extremely popular when I started as a PhD student in Paris in late 1972. Of course they still are, but they were a particularly "hot topic" at the Paris VI probability seminar under the leadership of Jaques Neveu (and the monitoring of P.A. Meyer from Strasbourg). The papers [16] and [17] (as well as Garsia’s book) had appeared not long before and their impact was still visible. I remember vividly Neveu’s lectures on Burkholder’s inequalities and the iJ1-BMO duality. There was also intense interest for martingales within the Harmonic Analysis group in Orsay where Gundy made frequent visits and worked with Varopoulos. There, martingale came to the center stage through the links revealed with Harmonic functions, area integrals, the Fefferman Stein theory of H p -spaces, and so on. Meanwhile, the potential theory community had to face the shocking news that its subject was now also a part of probability theory.


Banach Space Singular Integral Operator Fourier Multiplier Haar System Bellman Function 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    E. Berkson, T. Gillespie and P. Muhly, Abstract spectral decompositions guaranteed by the Hilbert transform. Proc. London Math. Soc. 53 (1986), no. 3, 489-517.MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    P. Biane and R. Speicher, Stochastic calculus with respect to free Brownian motion and analysis on Wigner space, Probab. Theory Related Fields 112 (1998), 373-409.MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    J. Bourgain, Some remarks on Banach spaces in which martingale difference sequences are unconditional, Ark. Mat. 21 (1983) 163-168.MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    J. Bourgain, Extension of a result of Benedek, Calderón and Panzone, Ark. Mat. 22 (1984) 91 95.MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    J. Bourgain, Vector valued singular integrals and the H1-BMO duality, Probability Theory and Harmonic Analysis, Chao-Woyczynski (ed.), Dekker, New York, 1986, pp. 1-19.Google Scholar
  6. 6.
    D.L. Burkholder, Maximal inequalities as necessary conditions for almost everywhere convergence, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 3 (1964) 75 88.MathSciNetGoogle Scholar
  7. 7.
    D.L. Burkholder, Martingale transforms. Ann. Math. Statist. 37 (1966) 1494-1504.MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    D.L. Burkholder, A sharp inequality for martingale transforms. Ann. Probab. 7 (1979) 858-863.MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    D.L. Burkholder, A geometrical characterization of Banach spaces in which martingale differ- ence sequences are unconditional, Ann. Probab. 9 (1981), 997-1011.MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    D.L. Burkholder, A nonlinear partial differential equation and the unconditional constant of the Haar system in D>. Bull. Amer. Math. Soc. 7 (1982) 591 595.MathSciNetGoogle Scholar
  11. 11.
    D.L. Burkholder, A geometric condition that implies the existence of certain singular inte- grals of Banach space-valued functions, Conference on Harmonic Analysis in honor of Antoni Zygmund, Vol. I, II (Chicago, 111., 1981), pp. 270-286, Wadsworth Math. Ser., Wadsworth, Belmont, CA, 1983.Google Scholar
  12. 12.
    D.L. Burkholder, Martingales and Fourier analysis in Banach spaces, Probability and Analysis (Varenna, 1985), 61-108, Lecture Notes in Math., 1206, Springer, Berlin, 1986.Google Scholar
  13. 13.
    D.L. Burkholder, A proof of Pelczyhski’s conjecture for the Haar syste JÉ Studia Math. 91 (1988) 79-83.MATHMathSciNetGoogle Scholar
  14. 14.
    D.L. Burkholder, Explorations in martingale theory and its applications, École d’Été de Prob-abilités de Saint-Flour XIX 1989, 1 66, Lecture Notes in Math., 1464, Springer, Berlin, 1991.Google Scholar
  15. 15.
    D.L. Burkholder, Martingales and singular integrals in Banach spaces, Handbook of the Ge- ometry of Banach Spaces, Vol. I, North-Holland, Amsterdam, 2001, pp. 233-269.Google Scholar
  16. 16.
    D.L. Burkholder and R.F. Gundy, Extrapolation and interpolation of quasi-linear operators on martingales, Acta Math. 124 (1970) 249-304.MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    D.L. Burkholder, R.F. Gundy, and M.L. Silverstein, A maximal function characterization of the class H p, Trans. Amer. Math. Soc. 157 (1971) 137-153.MATHMathSciNetGoogle Scholar
  18. 18.
    B. Davis. On the integrability of the martingale square function. Israel J. Math. 8 (1970) 187-190.MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    P. Enflo, Banach spaces which can be given an equivalent uniformly convex norm. Israel J. Math. 13 (1972) 281-288.CrossRefMathSciNetGoogle Scholar
  20. 20.
    D.J.H. Garling, Brownian motion and UMD-spaces. Probability and Banach spaces (Zaragoza, 1985), 36 49, Lecture Notes in Math., 1221, Springer, Berlin, 1986.Google Scholar
  21. 21.
    T. Hytönen and L. Weis, Singular convolution integrals with operator-valued kernel. Math. Z. 255 (2007) 393-425.MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    M. Junge, Doob’s inequality for non-commutative martingales. J. Reine Angew. Math. 549 (2002) 149-190.MATHMathSciNetGoogle Scholar
  23. 23.
    M. Junge, C. Le Merdy and Q. Xu, H∞ functional calculus and square functions on noncom- mutative Lp-spaces. Astérisque No. 305 (2006) vi+138 pp.Google Scholar
  24. 24.
    M. Junge and Q. Xu, On the best constants in some non-commutative martingale inequalities. Bull. London Math. Soc. 37 (2005) 243-253.MATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    J.P. Kahane, Some Random Series of Runctions, Second edition, Cambridge Studies in Ad- vanced Mathematics, 5, Cambridge University Press, Cambridge, 1985.Google Scholar
  26. 26.
    N. Kalton, S. V. Konyagin, L. Vesely. Delta-semidefinite and delta-convex quadratic forms in Banach spaces. Positivity 12 (2008) 221-240.MATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    P. Kunstmann, L. Weis, Maximal L p - regularity for Parabolic Equations, Fourier Multiplier Teorems and H∞- functional Calculus, in: Functional Analytic methods for Evolution Equa- tions, Springer Lecture notes 1855, (2004) p. 65-312.Google Scholar
  28. 28.
    T. McConnell, On Fourier multiplier transformations of Banach-valued functions, Trans. Amer. Math. Soc. 285 (1984) 739-757.MATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    B. Maurey, Systèmes de Haar, Séminaire Maurey-Schwartz, 74-75, Ecole Polytechnique, Paris.Google Scholar
  30. 30.
    B. Maurey, Type, cotype and K-convexity. Handbook of the geometry of Banach spaces, Vol. 2, 1299-1332, North-Holland, Amsterdam, 2003.CrossRefGoogle Scholar
  31. 31.
    B. Maurey and G. Pisier, Séries de variables aléatoires vectorielles indépendantes et propriétés géométriques des espaces de Banach, Studia Math. 58 (1976) 45-90.MATHMathSciNetGoogle Scholar
  32. 32.
    M. Musat, On the operator space UMD property and non-commutative martingale inequalities. PhD Thesis, University of Illinois at Urbana-Champaign, 2002.Google Scholar
  33. 33.
    F. Nazarov, S. Treil, The hunt for a Bellman function: applicationl^É^estimates for singular integral operators and to other classical problems of  harmonic analysis. St. Petersburg Math. J. 8 (1997) 721-824MathSciNetGoogle Scholar
  34. 34.
    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), 393 423, Oper. Theory Adv. Appl., 129, Birkhuser, Basel, 2001.Google Scholar
  35. 35.
    S. Petermichl, CR. Acad. Sei. Paris Sér. I MaSL 330 (2000) 455-460.MATHMathSciNetGoogle Scholar
  36. 36.
    G. Pisier, Martingales with values in uniformly convex spaces, Israel J. Math. 20 (1975) 326- 350.MATHCrossRefMathSciNetGoogle Scholar
  37. 37.
    G. Pisier, Non-commutative vector valued Lp-spaces and completely p-summing maps. Astérisque No. 247 (1998).Google Scholar
  38. 38.
    G. Pisier and Q. Xu, Non-commutative martingale inequalities, Comm. Math. Phys. 189 (1997)667-698.MATHCrossRefMathSciNetGoogle Scholar
  39. 39.
    N. Randrianantoanina, Non-commutative martingale transforms. J. Funct. Anal. 194 (2002) 181-212.MATHMathSciNetGoogle Scholar
  40. 40.
    J. Parcet and N. Randrianantoanina, Non-commutative Gundy decompositions. Proc. London Math. Soc.Google Scholar
  41. 41.
    E.M. Stein, Topics in harmonic analysis related to the Littlewood-Paley theory. Annals of Mathematics Studies, No. 63 Princeton University Press, Princeton 1970.Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Department of Mathematics and Department of StatisticsPurdue UniversityWest LafayetteUSA
  2. 2.Department of MathematicsUniversity of IllinoisUrbanaUSA

Personalised recommendations