Skip to main content
SpringerLink
Log in

Donald Burkholder’s Work in Martingales and Analysis

  • Chapter
  • First Online:
Selected Works of Donald L. Burkholder

Part of the book series: Selected Works in Probability and Statistics ((SWPS))

  • 1755 Accesses

Abstract

The two mathematicians who have most advanced martingale theory in the last seventy years are Joseph Doob and Donald Burkholder. Martingales as a remarkably flexible tool are used throughout probability and its applications to other areas of mathematics. They are central to modern stochastic analysis. And martingales, which can be defined in terms of fair games, lie at the core of mathematical finance. Burkholder’s research has profoundly advanced not only martingale theory but also, via martingale connections, harmonic and functional analysis.

Date: April 14, 2010.

R. Baňuelos is supported in part by NSF Grant # 0603701-DMS.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. N. Arcozzi, Riesz transforms on compact Lie groups, spheres and Gauss space, Ark. Mat. 36 (1998), 201-231.

    Article  MATH  MathSciNet  Google Scholar 

  2. N. Arcozzi and X. Li, Riesz transforms on spheres, Math. Res. Lett. 4 (1997), 401-12.

    MATH  MathSciNet  Google Scholar 

  3. K. Astala, Area Distortion of Quasiconformal Mappings, Acta Math. 173(1994), 37-60.

    Article  MATH  MathSciNet  Google Scholar 

  4. K. Astala, T. Iwaniec and G. Martin, Elliptic Partial Differential Equations and Quasiconformal Mappings in the Plane, Princeton University Press, 2009.

    Google Scholar 

  5. D.G. Austin, A sample function property of martingales, Ann. Math. Statist. 37, (1966), 1396-1397.

    Article  MATH  MathSciNet  Google Scholar 

  6. J.M. Ball, Convexity conditions and existence theorems in nonlinear elasticity, Arch. Rational Mech. Anal. 63 (1977), 337-403.

    Article  MATH  Google Scholar 

  7. R. Bañuelos and K. Bogdan, Symmetric stable processes in cones, Potential Anal, 21(2004), 263-288.

    Article  MATH  MathSciNet  Google Scholar 

  8. R. Bañuelos and K. Bogdan, Levy processes and FMrilr multipliers, J. Funct. Anal. 250(2007), 197-213.

    Article  MATH  MathSciNet  Google Scholar 

  9. R. Bañuelos and P. Janakiraman, L p -bounds for the Beurling-Ahlfors transform, Trans. Amer. Math. Soc. 360(2008), 3603-3612.

    Article  MATH  MathSciNet  Google Scholar 

  10. R. Bañuelos and A.J. Lindeman, A Martingale study of the Beurling-Ahlfors transform inIRn, Journal of Functional Analysis. 145(1997), 224-265.

    Article  MATH  Google Scholar 

  11. R. Bañuelos and P. Méndez-Hernândez, Space-time Brownian motion and the Beurling-Ahlfors transform, Indiana Uni- versity Math J. 52(2003), 981-990.

    MATH  Google Scholar 

  12. R. Bañuelos and C. Moore, Probabilistic behavior of harmonic functions, Birkäuser, 1999.

    Google Scholar 

  13. R. Bañuelos and R. Smits, Brownian motion in cones, Probab. Theory Related Fields, 108(1997), 299-319.

    Article  MATH  MathSciNet  Google Scholar 

  14. R. Bañuelos and G. Wang, Sharp inequalities for martingales with applications to the Beurling-Ahlfors and Riesz transforms, Duke Math. J. 80(1995), 575-600.

    Article  MATH  MathSciNet  Google Scholar 

  15. R. Bañuelos and G. Wang, Sharp inequalities for martingales under orthogonality and differential subordination, Illinois Journal of Mathematics 40(1996), 687-691.

    Google Scholar 

  16. R. Bañuelos and G. Wang, Davis’s inequality for orthogonal martingales under differential subordination, Michigan Math. J., 47(2000) 120-124.

    Google Scholar 

  17. A. Baernstein II and S. J. Montgomery-Smith, Some conjectures about integral means of df and dfin Complex Analysis and Differential Equations (Uppsala, Sweden, 1999), ed. Ch. Kiselman, Acta. Univ. Upsaliensis Univ. C Organ. Hist. 64, Uppsala Univ. Press, Uppsala, Sweden, (1999), 92-109.

    Google Scholar 

  18. M.T. Barlow and M. Yor, (Semi) martingale inequalities and local times, Z. Wahrscheinlichkeitstheorie Verw Geb. 55(1981), 237-354.

    Article  MATH  MathSciNet  Google Scholar 

  19. M.T. Barlow and M. Yor, Semi-martingale inequalities via the Garsia-Rodemich-Rumsey lemma and applications to local times, J. Func.Anal. 49 (1982), 198-229.

    Article  MATH  MathSciNet  Google Scholar 

  20. R. Bass, L p inequalities for functional of Brownian motion, Séminaire des Probabilités, XXI, (1987), 206-217, Lecture Notes in Math. 1247, Springer, New York.

    Google Scholar 

  21. R. Bass, Probabilistic Techniques in Analysis, Springer-Verlag, 1995.

    Google Scholar 

  22. J. Bourgain, Some remarks on Banach spaces in which martingale difference sequences are unconditional, Ark. Mat., 21(1983), 163-168.

    Article  MATH  MathSciNet  Google Scholar 

  23. J. Brossard, Comportemnent nontangentiel et Brownien de fonctions harmonique dans un demi-space: Demostration probabiliste dûn théorème de Calderón et Stein, Sem. Ill, Springer LNM 649(1976), 378-397.

    MathSciNet  Google Scholar 

  24. J. Brossard, Densité de l’intégrale d’aire dansIRv+1 + et limites non tangentiales, Invent. Math. 93 (1988), 297-308.

    Article  MATH  MathSciNet  Google Scholar 

  25. J. Brossard and L. Chevalier, Problème de Fatou ponctuel et dérivabilitié de mesures, Acta Math. 164(1990), 237-263.

    Article  MATH  MathSciNet  Google Scholar 

  26. J. Brossard and L. Chevalier, Limites non tangentiales, limites browniennes en probabilité et limites semi-fines, J. Reine Angew. Math. 421(1991), 141-157.

    MATH  MathSciNet  Google Scholar 

  27. J. Brossard and L. Chevalier, Un réciproque optimale du théorème de Fatou ponctuel, Adv. in Math. 115(1995), 300-318.

    Article  MATH  MathSciNet  Google Scholar 

  28. D.L. Burkholder, Maximal inequalities as necessary conditions for almost everywhere convergence, Z. Wahrschein-lichkeitstheorie und Verw. Gebiete 3 (1964), 75-88.

    Article  MATH  MathSciNet  Google Scholar 

  29. D.L. Burkholder, Martingale transforms, Ann. Math. Statist. 37 (1966), 1494-1504.

    Article  MATH  MathSciNet  Google Scholar 

  30. D.L. Burkholder, Inequalities for operators on martingales, Proc. International Congress of Mathematicians (Nice, France, 1970) 2 (1971), 551-557.

    MathSciNet  Google Scholar 

  31. D.L. Burkholder, Distribution function inequalities for martingales, Ann. Probab. 1 (1973), 19-42.

    Article  MATH  MathSciNet  Google Scholar 

  32. D.L. Burkholder, One-sided maximal functions and H p, J. Funct. Anal. 18 (1975), 429-454.

    Article  MATH  MathSciNet  Google Scholar 

  33. D.L. Burkholder, Exit times of Brownian motion, harmonic majorization, and Hardy spaces, Advances in Math. 26 (1977), 182-205.

    Article  MATH  MathSciNet  Google Scholar 

  34. D.L. Burkholder, Boundary value estimation of the range of an analytic function, Michigan Math. J. 25 (1978), 197-211.

    Article  MATH  MathSciNet  Google Scholar 

  35. D.L. Burkholder, A sharp inequality for martingale transforms, Ann. Prob 7 (1979), 858-863.

    Article  MATH  MathSciNet  Google Scholar 

  36. D.L. Burkholder, Martingale theory and harmonic analysis in Euclidean spaces, Profof Symposia in Pure Math. 35 (1979), 283-301.

    MathSciNet  Google Scholar 

  37. D.L. Burkholder, Brownian motion and the Hardy spaces H p, in "Aspects of Contemporary Complex Analysis," edited by D. A. Brannan and J. G. Clunie, Academic Press, London, 1980, 97-118

    Google Scholar 

  38. D.L. Burkholder, A geometrical characterization of Banach spaces in which martingale difference sequences are un- conditional, Ann. Probab. 9 (1981), 997-1011.

    Article  MATH  MathSciNet  Google Scholar 

  39. D.L. Burkholder, A nonlinear partial differential equation and the unconditional constant of the Haar system in L p, Bull. Amer. Math. Soc. 7 (1982), 591-595.

    Article  MATH  MathSciNet  Google Scholar 

  40. D.L. Burkholder, A geometric condition that implies the existence of certain singular integrals of Banach-space-valued functions, in "Conference on Harmonic Analysis in Honor of Antoni Zygmund," (Chicago, 1981), edited by William Beckner, Alberto P. Calderón, Robert Fefferman, and Peter W. Jones. Wadsworth, Belmont, California, 1983, pp. 270-286.

    Google Scholar 

  41. D.L. Burkholder, Boundary value problems and sharp inequalities for martingale transforms, Ann. Probab. 12 (1984), 647-702.

    Article  MATH  MathSciNet  Google Scholar 

  42. D.L. Burkholder, An elementary proof of an inequality ofR. E. A. C. Paley, Bull. London Math. Soc. 17 (1985), 474-478.

    Article  MATH  MathSciNet  Google Scholar 

  43. D.L. Burkholder, Martingales and Fourier analysis in Banach spaces, C.I.M.E. Lectures (Varenna (Como), Italy, 1985), Lecture Notes in Mathematics 1206 (1986), 61-108.

    Google Scholar 

  44. D.L. Burkholder, A sharp and strict L p -inequality for stochastic integrals, Ann. Probab. 15 (1987), 268-273.

    Article  MATH  MathSciNet  Google Scholar 

  45. D.L. Burkholder, A proof of Pelczy’nski’s conjecture for the Haar system, Studia Math. 91 (1988), 79-83.

    MATH  MathSciNet  Google Scholar 

  46. D.L. Burkholder, Sharp inequalities for martingales and stochastic integrals, Colloque Paul Levy (Palaiseau,1987), Ast’erisque 157-158 (1988), 75-94.

    MathSciNet  Google Scholar 

  47. D.L. Burkholder, Differential subordination of harmonic functions and martingales, Harmonic Analysis and Partial Differential Equations, (El Escorial, 1987), Lecture Notes in Mathematics 1384 (1989), 1-23.

    Google Scholar 

  48. D.L. Burkholder, On the number of escapes of a martingale and its geometrical significance, in "Almost Everywhere Convergence," edited by Gerald A. Edgar and Louis Sucheston. Academic Press, New York, 1989, 159-178.

    Google Scholar 

  49. D.L. Burkholder, Explorations in martingale theory and its applications, ’Ecole d’Et’e de Probabilités de Saint-Flour XIX-1989, Lecture Notes in Mathematics 1464 (1991), 1-66.

    Article  MathSciNet  Google Scholar 

  50. D.L. Burkholder, Strong differential subordination and stochastic integration, Ann. Probab. 22 (1994), 995-1025.

    Article  MATH  MathSciNet  Google Scholar 

  51. D.L. Burkholder, Sharp norm comparison of martingale maximal functions and stochastic integrals, Proceedings of the Norbert Wiener Centenary Congress (East Lansing, MI, 1994), 343-358, Proc. Sympos. Appl. Math. 52, Amer. Math. Soc, Providence, RI (1997).

    Google Scholar 

  52. D.L. Burkholder, Some extremal problems in martingale theory and harmonic analysis, Harmonic Analysis and Partial Differential Equations (Chicago, 1996), 99-115, Chicago Lectures in Math. Univ. Chicago Press, Chicago, 1999.

    Google Scholar 

  53. D.L. Burkholder, Martingales and singular integrals, in Banach spaces, Handbook on the Geometry of Banach spaces, Volume 1, edited by William B. Johnson and Joram Lindenstrauss, Elsevier, (2001) 233-269.

    Chapter  Google Scholar 

  54. D.L. Burkholder, The best constant in the Davis inequality for the expectation of the martingale square function, Trans. Amer. Math. Soc. 354(2002), 91-105.

    Article  MATH  MathSciNet  Google Scholar 

  55. D.L. Burkholder, B. J. Davis and R. F. Gundy, Integral inequalities for convex functions of operators on martingales, Proc. Sixth Berkeley Symp. Math. Statist, and Prob. 2 (1972), 223-240.

    MathSciNet  Google Scholar 

  56. D.L. Burkholder and R. F. Gundy, Extrapolation and interpolation of quasi-linear operators on martingales, Acta Math. 124(1970), 249-304.

    Article  MATH  MathSciNet  Google Scholar 

  57. D.L. Burkholder and R. F. Gundy, Distribution function inequalities for the area integral, Studia Math. 44(1972), 527-544.

    MATH  MathSciNet  Google Scholar 

  58. D.L. Burkholder and R.F. Gundy, Boundary behaviour of harmonic functions in a half-space and Brownian motion, Ann. Inst. Fourier (Grenoble) 23(1973), 195-212.

    MATH  MathSciNet  Google Scholar 

  59. D.L. Burkholder, R. F. Gundy and M. L. Silverstein, A maximal f unction characterization of the class H p, Trans. Amer. Math. Soc. 157(1971), 137-153.

    MATH  MathSciNet  Google Scholar 

  60. A. P. Calderón, On the behavior of harmonic functions on the boundary, Trans. Amer. Math. Soc. 68 (1950), 47-54.

    Article  MATH  MathSciNet  Google Scholar 

  61. A. P. Calderón, On a theorem of Marcinkiewicz and Zygmund, Proc. Amer. Math. Soc. 1 (1950), 533-535.

    Article  MATH  MathSciNet  Google Scholar 

  62. A.P. Calderón and A. Torchinsky, Parabolic maximal functions associated with a distribution, II, Advances in Math. 24 (1977), 101-171.

    Article  MATH  MathSciNet  Google Scholar 

  63. R.R. Coifman, Distribution function inequalities for singular integrals, Proc. Nat. Acad. Sei. U.S.A. 69 (1972), 2838-2839.

    Article  MATH  MathSciNet  Google Scholar 

  64. R.R. Coifman and C. Fefferman, Weighted norm inequalities for maximal functions and singular integrals. Studia Math., 51(1974), 241-250.

    MATH  MathSciNet  Google Scholar 

  65. J. Cheeger, Spectral geometry of singular Riemannian spaces, J. Diff. Geo. 18 (1983), 575-657.

    MATH  MathSciNet  Google Scholar 

  66. C. Choi, A submartingale inequality, Proc. Amer. Math. Soc. 124(1996), 2549-2553.

    Article  MATH  MathSciNet  Google Scholar 

  67. C. Choi, A weak-type inequality for differentially subordinate harmonic functions, Tran. Amer. Math. Soc. 350(1998), 2687-2696.

    Article  MATH  Google Scholar 

  68. K.P. Choi, Some sharp inequalities for martingale transforms, Trans. Amer. Math. Soc. 307(1988), 279-300.

    Article  MATH  MathSciNet  Google Scholar 

  69. K.P. Choi, A sharp inequality for martingale transforms and the unconditional basis constant of a monotone basis inLp(0,1), Trans. Amer. Math. Soc. 330(1992), 509-529.

    Article  MATH  Google Scholar 

  70. R. Courant, K Friedrichs, and H. Lewy, Über die partiellen Differenzengleichungen der mathematischen Physik, Math- ematische Annalen, 100(1928), 32-74.

    Article  MATH  MathSciNet  Google Scholar 

  71. 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.

    MATH  MathSciNet  Google Scholar 

  72. B. Dacoronga, Direct Methods in the Calculus of Variations, Springer 1989.

    Google Scholar 

  73. R. DeBlassie, Exit times from cones inIRn of Brownian motion, Probab. Theory Related Fields 74(1987), 1-29.

    Article  MathSciNet  Google Scholar 

  74. R. DeBlassie, Remark on: "Exit times from cones inŒT of Brownian motion", Probab. Theory Related Fields 79 (1988), 95-97.

    Article  MathSciNet  Google Scholar 

  75. B. Davis, On the integrability of the martingale square function, Israel J. Math. 8 (1970), 187-190.

    Article  MATH  MathSciNet  Google Scholar 

  76. B. Davis, On the Barlow-Yor inequalities for local time, Séminaire de Probabilités, XXI, 218-220, Lecture Notes in Math. 1247, Springer, Berlin, 1987

    Article  Google Scholar 

  77. B. Davis, On stopping times for n dimensional Brownian motion, Ann. Prob. 6 (1978), 651-659.

    Article  MATH  Google Scholar 

  78. B. Dahlberg, Weighted norm inequalities for the Lusin area integral and the nontangential maximal functions for func- tions harmqmc in a Lipschitz domain, Studia Math. 47(1980), 297-314.

    MathSciNet  Google Scholar 

  79. B. Dahlberg, D. Jerison and K. Kenig, Area integral estimates for elliptic operators with nonsmooth coefficients, Arkiv Mat. 22(1984), 97-108.

    Article  MATH  MathSciNet  Google Scholar 

  80. S. Donaldson and D. Sullivan, Quasiconformal 4-manifolds, Acta Math. 163(1989), 181-252.

    Article  MATH  MathSciNet  Google Scholar 

  81. O. Dragicevic, S. Petermichl and A. Volberg, A rotation method which gives linear L p estimates for powers of the Ahlfors-Beurling operator, J. Math. Pures Appl. 86 (2006), no. 6, 492-509.

    MATH  MathSciNet  Google Scholar 

  82. O. Dragicevic and A. Volberg, Bellman functions and dimensionless estimates of Littlewood-Paley type, J. Operator Theory 56(2006), 167-198.

    MATH  MathSciNet  Google Scholar 

  83. O. Dragicevic and A. Volberg, Sharp estimate of the Ahlfors-Beurling operator via averaging martingale transforms, Michigan Math. J. 51(2003), 415-435.

    Article  MATH  MathSciNet  Google Scholar 

  84. O. Dragicevic and A. Volberg, Bellman function, Littlew ood-Paley estimates and asymptotic s for the Ahlfors-Beurling operator in L P (C), Indiana Univ. Math. J. 54(2005), no. 4, 971-995.

    Article  MATH  MathSciNet  Google Scholar 

  85. O. Dragicevic and A. Volberg, Bellman function for the estimates of Littlewood-Paley type and asymptotic estimates in thep-1 problem, C. R. Math. Acad. Sei. Paris 340(2005), no. 10, 731-734.

    MATH  MathSciNet  Google Scholar 

  86. J. Duoandikoetxea and J.L. Rubio de Francia, Estimations indépendantes de la dimension pour les transformées de Riesz, C. R. Acad. Sei. Paris Sér. I Math. 300(1985),193-196.

    MATH  MathSciNet  Google Scholar 

  87. R. Durrett, Brownian Motion and Martingales in Analysis, Wadsworth, Belmont, CA, 1984.

    MATH  Google Scholar 

  88. M. Essén, K. Haliste, J.L. Lewis, and D.F. Shea, Classical analysis and Burkholder’s results on harmonic majorization and Hardy spaces, Complex analysis and applications (Varna, 1983), 61-14, Publ. House Bulgar. Acad. Sei., Sofia, 1985.

    Google Scholar 

  89. M. Essén, K. Haliste, J.L. Lewis, and D.F. Shea, Harmonic majorization and classical analysis, J. London Math. Soc. 32 (1985), 506-520.

    Article  MATH  MathSciNet  Google Scholar 

  90. C. Fefferman and E.M. Stein, H p spaces in several variables, Acta Math. 129(1972) 137-193.

    Article  MATH  MathSciNet  Google Scholar 

  91. R. Fefferman, R.F. Gundy, M. Silverstein and E.M. Stein, Inequalities for ratios of ’functional of harmonic functions, Proc. Nat. Acad. Sei. U.S.A. 79(1982), 7958-7960.

    Article  MATH  MathSciNet  Google Scholar 

  92. J. Garnett, Bounded Analytic Functions, Academic Press, New York, 1980.

    Google Scholar 

  93. A. Garsia, The Burgess Davis inequalities via Fefferman’s inequalities, Arkiv. für Mathematik 11 (1973) 229-237.

    Article  MATH  MathSciNet  Google Scholar 

  94. A. Garsia, On a convex function inequality for martingales, Ann. Prob. 1 (1973), 171-174.

    Article  MATH  MathSciNet  Google Scholar 

  95. A. Garsia, Martingale Inequalities, Seminar Notes on Recent Progress, W. A. Benjamin Mathematics Lecture Note Series, 1973.

    Google Scholar 

  96. E. Geiss, S. Mongomery-Smith, E. Saksman, On singular integral and martingale transforms, Trans. Amer. Math. Soc. 362(2010), 553-575.

    Article  MATH  MathSciNet  Google Scholar 

  97. F. W. Gehring and E. Reich, Area distortion under quasiconformal mappings, Ann. Acad. Sei. Fenn. Ser A 1388(1966), 1-15.

    Google Scholar 

  98. L. Grafakos, Classical and Modern Fourier Analysis, Pearson Education Inc., 2004.

    Google Scholar 

  99. R.F. Gundy, Some Topics in Probability and Analysis. CBMS Regional Conference Series in Mathematics, 70 American Mathematical Society, Providence, RI, 1989.

    Google Scholar 

  100. R.F. Gundy and R.L. Wheeden, Weighted integral inequalities for the non-tangential maximal function, Lusin area function, and Walsh-Paley series, Studia Math. 49 (1974), 107-124

    MATH  MathSciNet  Google Scholar 

  101. R.F. Gundy and N. Th. Varopoulos, Les transformations de Riesz et les intégrales stochastiques, C. R. Acad. Sei. Paris Sér. A-B 289(1979), A13-A16.

    MathSciNet  Google Scholar 

  102. W. Hammack, Sharp inequalities for the distribution of a stochastic integral in which the integrator is a bounded submartingale, Ann. Probab. 23(1995), 223-235.

    Article  MATH  MathSciNet  Google Scholar 

  103. W. Hammack, Sharp maximal inequalities for stochastic integrals in which the integrator is a submartingale, Proc. Amer. Math. Soc. 124(1996), 931-938.

    Article  MATH  MathSciNet  Google Scholar 

  104. G. H. Hardy and J. E. Littlewood, A maximal function with function theoretic applications, Acta Math. 54 (1930), 81-116.

    Article  MathSciNet  Google Scholar 

  105. T.P. Hytönen: On the norm of the Beurling-Ahlfors operator in several dimensions, (Preprint).

    Google Scholar 

  106. T. Iwaniec, Extremal inequalities in Sobolev spaces and quasiconformal mappings, Z. Anal. Anwendungen 1 (1982), 1-16.

    MATH  MathSciNet  Google Scholar 

  107. T. Iwaniec, L p -theory of quasiregular mappings, in Quasiconformal Space Mappings, Ed. Matti Vuorinen, Lecture Notes in Math. 1508, Springer, Berlin, 1992.

    Google Scholar 

  108. T. Iwaniec, Nonlinear Cauchy-Riemann operators in R n, Trans. Amer. Math. Soc. 354(2002), 1961-1995.

    Article  MATH  MathSciNet  Google Scholar 

  109. T. Iwaniec, and GJ. Martin, Quasiregular mappings in even dimensions, Acta Math. 170(1993), 29-81.

    Article  MATH  MathSciNet  Google Scholar 

  110. T. Iwaniec and G. Martin, Riesz transforms and related singular integrals, J. Reine Angew. Math. 473(1996), 25-57.

    MATH  MathSciNet  Google Scholar 

  111. T. Iwaniec, and G. J. Martin, Geometric Function Theory and Nonlinear Analysis, Oxford University Press, 2001.

    Google Scholar 

  112. P. Janakiraman, Best weak-type (p,p) constants, 1 ≤ p≤ 2, for orthogonal harmonic functions and martingales, Illinois J. Math. 48 (2004), 909-921.

    MATH  MathSciNet  Google Scholar 

  113. J. L. Journé, Colderón-Zygmund Operators, Pseudo-Differential Operators and the Cauchy Integral of Calderón, Lecture Notes in Math. 994, Springer-Verlag, New York, 1983.

    Google Scholar 

  114. C. Kenig, Harmonic Analysis Techniques for Second Order Elliptic Boundary Value Problems, CBMS Regional Conference Series in Mathematics 83American Mathematical Society, Providence, RI, 1994.

    Google Scholar 

  115. O. Lehto, Remarks on the integrability of the derivatives of quasiconformal mappings, Ann. Acad. Sei. Fenn. Series AI Math. 371(1965), 8 pp.

    MathSciNet  Google Scholar 

  116. J. Marcinkiewicz and A. Zygmund, A theorem of Lusin, Duke Math. J. 4 (1938), 473-485.

    Article  MathSciNet  Google Scholar 

  117. T. McConnell, On Fourier multiplier transformations of Banach-valuedfunctions, Trans. Amer. Math. Soc. 285(1984), 739-757.

    Article  MATH  MathSciNet  Google Scholar 

  118. A. D. Mêlas, The Bellman functions of dyadic-like maximal operators and related inequalities, Advances in Mathe- matics, 192(2005), 310-340.

    Article  MATH  Google Scholar 

  119. A. D. Melas, Dyadic-like maximal operators on LlogL functions, Journal of Functional Analysis 257(2009), 1631-1654.

    Article  MATH  MathSciNet  Google Scholar 

  120. P. W. Millar, Martingale integrals, Trans. Amer. Math. Soc, 133(1966), 145-166.

    Article  MathSciNet  Google Scholar 

  121. C.B. Morrey, Quasi-convexity and the lower semicontinuity of multiple integrals, Pacific J. Math., 2 (1952), 25-53.

    MATH  MathSciNet  Google Scholar 

  122. C.B. Morrey, Multiple integrals in the calculus of variations, Die Grundlehren der mathematischen Wissenschaften, Band 130 Springer-Verlag New York, Inc., New York 1966.

    Google Scholar 

  123. T. Murai and A. Uchiyama, Good-X inequalities for the area integral and the nontangential maximal function, Studia Math. 83 (1986), 251-262.

    MathSciNet  Google Scholar 

  124. A. Miyachi and K. Yabuta, On good-X inequalities, Bull. Fac. Sei., Ibaraki Univ., Math. 16 (1984), 1-11.

    Article  MathSciNet  Google Scholar 

  125. 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.

    MathSciNet  Google Scholar 

  126. F. Nazarov, S. Treil and A. Volberg, The Bellman functions and two-weight inequalities for Haar multipliers, J. Amer. Math. Soc. 4 (1999), 909-928.

    Article  MathSciNet  Google Scholar 

  127. F. Nazarov, S. Treil and A. Volberg, Bellman function in stochastic optimal control and harmonic analysis (how our Bellman function got its name), Oper. Theory: Adv. Appl. 129(2001), 393-424.

    MathSciNet  Google Scholar 

  128. F Nazarov and A. Volberg, Bellman function, two weight Hubert transform and imbedding for the model space K, Volume in the memory of Tom Wolff. J. d’Analyse Math. 87, (2003), 385-414.

    Article  MathSciNet  Google Scholar 

  129. F Nazarov and A. Volberg, Heat extension of the Beurling operator and estimates for its norm, St. Petersburg Math. J. 15, (2004), 563-573.

    MathSciNet  Google Scholar 

  130. A. A. Novikov, On moment inequalities for stochastic integrals(Russian; English summary), Teor. Verejatnost i Primenen 16 (1971), 548-551.

    Google Scholar 

  131. A. Osekowski, Sharp inequality for bounded submartingales and their differential subordinates, Electron. Commun. Probab. 13 (2008), 660-675.

    MATH  MathSciNet  Google Scholar 

  132. A. Osekowski, Sharp maximal inequality for stochastic integrals. Proc. Amer. Math. Soc. 136(2008), 2951-2958.

    Article  MATH  MathSciNet  Google Scholar 

  133. A. Osekowski, Sharp weak-type inequalities for differentially subordinated martingales, Bernoulli 15 (2009), 871-897.

    Article  MATH  MathSciNet  Google Scholar 

  134. A. Osekowski, Sharp norm inequalities for stochastic integrals in which the integrator is a nonnegative supermartingale, Probab. Math. Statist. 29(2009), no. 1, 29-42.

    MATH  MathSciNet  Google Scholar 

  135. A. Osekowski, On the best constant in the weak type inequality for the square function of a conditionally symmetric martingale, Statist. Probab. Lett. 79 (2009), 1536-1538.

    Article  MATH  MathSciNet  Google Scholar 

  136. A. Osekowski, Weak type inequality for the square function of a nonnegative submartingale, Bull. Pol. Acad. Sei. Math. 57(2009), 81-89.

    Article  MathSciNet  Google Scholar 

  137. A. Osekowski, Sharp maximal inequality for martingales and stochastic integrals. Electron, Commun. Probab. 14 (2009), 17-30.

    MATH  MathSciNet  Google Scholar 

  138. R. E. A. C. Paley, A remarkable series of orthogonalfunctions I, Proc. London Math. Soc. 34 (1932) 241-264.

    Article  MATH  Google Scholar 

  139. S. Petermichl and A. Volberg, Heating of the Beurling operator: weakly quasiregular maps on the plane are quasiregular, Duke Math. 112(2002), 281-305.

    Article  MATH  MathSciNet  Google Scholar 

  140. G. Pisier, Riesz transforms: simpler analytic proof of P.-A. Meyer’s inequality, Séminaire de Probabilités, XXII, 485-501, Lecture Notes in Math. 1321, Springer, Berlin, 1988.

    MathSciNet  Google Scholar 

  141. S.K. Pichorides, On the best values of the constants in the theorems of M. Riesz, Zygmund and Kolmogorov, Collection of articles honoring the completion by Antoni Zygmund of 50 years of scientific activity, II. Studia Math. 44(1972), 165-179.

    MATH  MathSciNet  Google Scholar 

  142. I. Privalov, sur les formions conjuguées, Bull. Soc. Math. France (1916), 100-103.

    Google Scholar 

  143. D. Revuz and M. Yor, Continuous Martingales and Brownian Motion, Springer-Verlag, 293, 1980.

    Google Scholar 

  144. L. Slavin, A. Stokolos and V Vasyunin, Monge-Ampére equations and Bellman functions: The dyadic maximal operator, C. R. Acad. Sei. Paris, Ser. 1346(2008), 585-588.

    MathSciNet  Google Scholar 

  145. L. Slavin and V Vasyunin, Bellman function for the sharp classical and dyadic John-Nirenberg inequality, (Preprint).

    Google Scholar 

  146. L. Slavin and A. Volberg, The explicit BFfor a dyadic Chang-Wilson-Wolff theorem. The s-function and the exponential integral, Contemp. Math. 444(2007).

    Google Scholar 

  147. D. C. Spencer, A function theoretic identity, Amer. J. Math. 65 (1943), 147-160.

    Article  MATH  MathSciNet  Google Scholar 

  148. E.M. Stein, On the theory of harmonic functions of several variables. II, Acta Math. 106(1961), 137-174.

    Article  MATH  MathSciNet  Google Scholar 

  149. E. M. Stein, Singular integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, 1970.

    MATH  Google Scholar 

  150. E.M. Šteáin, The development of square functions in the work of A. Zygmund, Bull. Amer. Math. Soc. 7 (1982), 359-376.

    Article  MathSciNet  Google Scholar 

  151. E.M. Šteáin, Some results in Harmonic Analysis inIRn for n→ ∞. Bull. Amer. Math. Soc. 9 (1983), 71-73.

    Article  MathSciNet  Google Scholar 

  152. E.M. Šteáin, Problems in harmonic analysis related to curvature and oscillatory integrals, Proceedings of the Interna- tional Congress of Mathematicians, Berkeley, CA., 1986

    Google Scholar 

  153. E.M. Šteáin, Harmonic Analysis, Princeton Mathematical Series, 43, 1993.

    Google Scholar 

  154. J. 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.

    Article  MATH  MathSciNet  Google Scholar 

  155. V. Sverâk, Examples of rank-one convex functions, Proc. Roy. Soc. Edinburgh 114A(1990), 237-242.

    Google Scholar 

  156. V. Sverâk, Rank-one convexity does not imply quasiconvexity, Proc. Roy. Soc. Edinburgh 120A(1992), 185-189.

    Google Scholar 

  157. V. Sverâk, New examples of quasiconvex functions, Arch. Rational Mech. Anal. 119(1992), 293-300.

    Article  MATH  MathSciNet  Google Scholar 

  158. A. Torchinsky, Real Variable Methods in Harmonic Analysis, Academic Press, Inc. Orlando, FL, 1986.

    MATH  Google Scholar 

  159. N. Th. Varopoulos, Aspects of probabilistic Littlewood-Paley theory, J. Funct. Anal. 38 (1980), no. 1, 25-60.

    Article  MATH  MathSciNet  Google Scholar 

  160. V. Vasyunin and A. Volberg, The Bellman functions for a certain two weight inequality: The case study, Algebra I Analiz 18 (2006).

    Google Scholar 

  161. V. Vasyunin and A. Volberg, Bellman functions technique in harmonic analysis, (sashavolberg.wordpress.com).

    Google Scholar 

  162. A. Volberg, Bellman approach to some problems in harmonic analysis, in Séminaire aux équations dérives partielles, 20, Ecole Polytechnique, Palaiseau, (2002), 1-14.

    Google Scholar 

  163. G. Wang, Sharp inequalities for the conditional square function of a martingale, Ann. Probab. 19 (1991), 1679-1688.

    Article  MATH  MathSciNet  Google Scholar 

  164. G. Wang, Sharp maximal inequalities for conditionally symmetric martingales and Brownian motion, Proc. Amer. Math. Soc. 112(1991), no. 2, 579-586

    Article  MATH  MathSciNet  Google Scholar 

  165. G. Wang, Sharp square-function inequalities for conditionally symmetric martingales, Trans. Amer. Math. Soc. 328(1991), 393-419.

    Article  MATH  MathSciNet  Google Scholar 

  166. G. Wang, Differential subordination and strong differential subordination for continuous-time martingales and related sharp inequalities, Ann. Probab. 23(1995), 522-551.

    Article  MATH  MathSciNet  Google Scholar 

  167. A. Zygmund, Trigonometrical Series, 2nd ed. Cambridge Univ. Press, Cambridge 1959.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Department of Statistics, Purdue University, West Lafayette, IN, 47907-2068, USA

    Burgess Davis

  2. Department of Mathematics, University of Illinois, Urbana, IL, 61801, USA

    Renming Song

Authors
  1. Burgess Davis
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Renming Song
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Burgess Davis .

Editor information

Editors and Affiliations

  1. , Department of Mathematics and, Purdue University, N. University Street 150, West Lafayette, 47907-2068, Indiana, USA

    Burgess Davis

  2. , Department of Mathematics, University of Illinois, Urbana, W. Green St. 1409, Urbana, 61801, Illinois, USA

    Renming Song

Rights and permissions

Reprints and permissions

Copyright information

© 2011 Springer Science+Business Media, LLC

About this chapter

Cite this chapter

Davis, B., Song, R. (2011). Donald Burkholder’s Work in Martingales and Analysis. In: Davis, B., Song, R. (eds) Selected Works of Donald L. Burkholder. Selected Works in Probability and Statistics. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-7245-3_1

Download citation

Publish with us

Policies and ethics

Search

Navigation