Advertisement

Donald Burkholder’s Work in Martingales and Analysis

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

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.

Keywords

Brownian Motion Harmonic Function Maximal Function Quasiconformal Mapping Singular Integral Operator 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    N. Arcozzi, Riesz transforms on compact Lie groups, spheres and Gauss space, Ark. Mat. 36 (1998), 201-231.zbMATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    N. Arcozzi and X. Li, Riesz transforms on spheres, Math. Res. Lett. 4 (1997), 401-12.zbMATHMathSciNetGoogle Scholar
  3. 3.
    K. Astala, Area Distortion of Quasiconformal Mappings, Acta Math. 173(1994), 37-60.zbMATHMathSciNetCrossRefGoogle Scholar
  4. 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. 5.
    D.G. Austin, A sample function property of martingales, Ann. Math. Statist. 37, (1966), 1396-1397.zbMATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    J.M. Ball, Convexity conditions and existence theorems in nonlinear elasticity, Arch. Rational Mech. Anal. 63 (1977), 337-403.zbMATHCrossRefGoogle Scholar
  7. 7.
    R. Bañuelos and K. Bogdan, Symmetric stable processes in cones, Potential Anal, 21(2004), 263-288.zbMATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    R. Bañuelos and K. Bogdan, Levy processes and FMrilr multipliers, J. Funct. Anal. 250(2007), 197-213.zbMATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    R. Bañuelos and P. Janakiraman, L p -bounds for the Beurling-Ahlfors transform, Trans. Amer. Math. Soc. 360(2008), 3603-3612.zbMATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    R. Bañuelos and A.J. Lindeman, A Martingale study of the Beurling-Ahlfors transform inIRn, Journal of Functional Analysis. 145(1997), 224-265.zbMATHCrossRefGoogle Scholar
  11. 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.zbMATHGoogle Scholar
  12. 12.
    R. Bañuelos and C. Moore, Probabilistic behavior of harmonic functions, Birkäuser, 1999.Google Scholar
  13. 13.
    R. Bañuelos and R. Smits, Brownian motion in cones, Probab. Theory Related Fields, 108(1997), 299-319.zbMATHMathSciNetCrossRefGoogle Scholar
  14. 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.zbMATHMathSciNetCrossRefGoogle Scholar
  15. 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. 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. 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. 18.
    M.T. Barlow and M. Yor, (Semi) martingale inequalities and local times, Z. Wahrscheinlichkeitstheorie Verw Geb. 55(1981), 237-354.zbMATHMathSciNetCrossRefGoogle Scholar
  19. 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.zbMATHMathSciNetCrossRefGoogle Scholar
  20. 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. 21.
    R. Bass, Probabilistic Techniques in Analysis, Springer-Verlag, 1995.Google Scholar
  22. 22.
    J. Bourgain, Some remarks on Banach spaces in which martingale difference sequences are unconditional, Ark. Mat., 21(1983), 163-168.zbMATHMathSciNetCrossRefGoogle Scholar
  23. 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.MathSciNetGoogle Scholar
  24. 24.
    J. Brossard, Densité de l’intégrale d’aire dansIRv+1 + et limites non tangentiales, Invent. Math. 93 (1988), 297-308.zbMATHMathSciNetCrossRefGoogle Scholar
  25. 25.
    J. Brossard and L. Chevalier, Problème de Fatou ponctuel et dérivabilitié de mesures, Acta Math. 164(1990), 237-263.zbMATHMathSciNetCrossRefGoogle Scholar
  26. 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.zbMATHMathSciNetGoogle Scholar
  27. 27.
    J. Brossard and L. Chevalier, Un réciproque optimale du théorème de Fatou ponctuel, Adv. in Math. 115(1995), 300-318.zbMATHMathSciNetCrossRefGoogle Scholar
  28. 28.
    D.L. Burkholder, Maximal inequalities as necessary conditions for almost everywhere convergence, Z. Wahrschein-lichkeitstheorie und Verw. Gebiete 3 (1964), 75-88.zbMATHMathSciNetCrossRefGoogle Scholar
  29. 29.
    D.L. Burkholder, Martingale transforms, Ann. Math. Statist. 37 (1966), 1494-1504.zbMATHMathSciNetCrossRefGoogle Scholar
  30. 30.
    D.L. Burkholder, Inequalities for operators on martingales, Proc. International Congress of Mathematicians (Nice, France, 1970) 2 (1971), 551-557.MathSciNetGoogle Scholar
  31. 31.
    D.L. Burkholder, Distribution function inequalities for martingales, Ann. Probab. 1 (1973), 19-42.zbMATHMathSciNetCrossRefGoogle Scholar
  32. 32.
    D.L. Burkholder, One-sided maximal functions and H p, J. Funct. Anal. 18 (1975), 429-454.zbMATHMathSciNetCrossRefGoogle Scholar
  33. 33.
    D.L. Burkholder, Exit times of Brownian motion, harmonic majorization, and Hardy spaces, Advances in Math. 26 (1977), 182-205.zbMATHMathSciNetCrossRefGoogle Scholar
  34. 34.
    D.L. Burkholder, Boundary value estimation of the range of an analytic function, Michigan Math. J. 25 (1978), 197-211.zbMATHMathSciNetCrossRefGoogle Scholar
  35. 35.
    D.L. Burkholder, A sharp inequality for martingale transforms, Ann. Prob 7 (1979), 858-863.zbMATHMathSciNetCrossRefGoogle Scholar
  36. 36.
    D.L. Burkholder, Martingale theory and harmonic analysis in Euclidean spaces, Profof Symposia in Pure Math. 35 (1979), 283-301.MathSciNetGoogle Scholar
  37. 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, 97-118Google Scholar
  38. 38.
    D.L. Burkholder, A geometrical characterization of Banach spaces in which martingale difference sequences are un- conditional, Ann. Probab. 9 (1981), 997-1011.zbMATHMathSciNetCrossRefGoogle Scholar
  39. 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.zbMATHMathSciNetCrossRefGoogle Scholar
  40. 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. 41.
    D.L. Burkholder, Boundary value problems and sharp inequalities for martingale transforms, Ann. Probab. 12 (1984), 647-702.zbMATHMathSciNetCrossRefGoogle Scholar
  42. 42.
    D.L. Burkholder, An elementary proof of an inequality ofR. E. A. C. Paley, Bull. London Math. Soc. 17 (1985), 474-478.zbMATHMathSciNetCrossRefGoogle Scholar
  43. 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. 44.
    D.L. Burkholder, A sharp and strict L p -inequality for stochastic integrals, Ann. Probab. 15 (1987), 268-273.zbMATHMathSciNetCrossRefGoogle Scholar
  45. 45.
    D.L. Burkholder, A proof of Pelczy’nski’s conjecture for the Haar system, Studia Math. 91 (1988), 79-83.zbMATHMathSciNetGoogle Scholar
  46. 46.
    D.L. Burkholder, Sharp inequalities for martingales and stochastic integrals, Colloque Paul Levy (Palaiseau,1987), Ast’erisque 157-158 (1988), 75-94.MathSciNetGoogle Scholar
  47. 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. 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. 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.MathSciNetCrossRefGoogle Scholar
  50. 50.
    D.L. Burkholder, Strong differential subordination and stochastic integration, Ann. Probab. 22 (1994), 995-1025.zbMATHMathSciNetCrossRefGoogle Scholar
  51. 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. 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. 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.CrossRefGoogle Scholar
  54. 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.zbMATHMathSciNetCrossRefGoogle Scholar
  55. 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.MathSciNetGoogle Scholar
  56. 56.
    D.L. Burkholder and R. F. Gundy, Extrapolation and interpolation of quasi-linear operators on martingales, Acta Math. 124(1970), 249-304.zbMATHMathSciNetCrossRefGoogle Scholar
  57. 57.
    D.L. Burkholder and R. F. Gundy, Distribution function inequalities for the area integral, Studia Math. 44(1972), 527-544.zbMATHMathSciNetGoogle Scholar
  58. 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.zbMATHMathSciNetGoogle Scholar
  59. 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.zbMATHMathSciNetGoogle Scholar
  60. 60.
    A. P. Calderón, On the behavior of harmonic functions on the boundary, Trans. Amer. Math. Soc. 68 (1950), 47-54.zbMATHMathSciNetCrossRefGoogle Scholar
  61. 61.
    A. P. Calderón, On a theorem of Marcinkiewicz and Zygmund, Proc. Amer. Math. Soc. 1 (1950), 533-535.zbMATHMathSciNetCrossRefGoogle Scholar
  62. 62.
    A.P. Calderón and A. Torchinsky, Parabolic maximal functions associated with a distribution, II, Advances in Math. 24 (1977), 101-171.zbMATHMathSciNetCrossRefGoogle Scholar
  63. 63.
    R.R. Coifman, Distribution function inequalities for singular integrals, Proc. Nat. Acad. Sei. U.S.A. 69 (1972), 2838-2839.zbMATHMathSciNetCrossRefGoogle Scholar
  64. 64.
    R.R. Coifman and C. Fefferman, Weighted norm inequalities for maximal functions and singular integrals. Studia Math., 51(1974), 241-250.zbMATHMathSciNetGoogle Scholar
  65. 65.
    J. Cheeger, Spectral geometry of singular Riemannian spaces, J. Diff. Geo. 18 (1983), 575-657.zbMATHMathSciNetGoogle Scholar
  66. 66.
    C. Choi, A submartingale inequality, Proc. Amer. Math. Soc. 124(1996), 2549-2553.zbMATHMathSciNetCrossRefGoogle Scholar
  67. 67.
    C. Choi, A weak-type inequality for differentially subordinate harmonic functions, Tran. Amer. Math. Soc. 350(1998), 2687-2696.zbMATHCrossRefGoogle Scholar
  68. 68.
    K.P. Choi, Some sharp inequalities for martingale transforms, Trans. Amer. Math. Soc. 307(1988), 279-300.zbMATHMathSciNetCrossRefGoogle Scholar
  69. 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.zbMATHCrossRefGoogle Scholar
  70. 70.
    R. Courant, K Friedrichs, and H. Lewy, Über die partiellen Differenzengleichungen der mathematischen Physik, Math- ematische Annalen, 100(1928), 32-74.zbMATHMathSciNetCrossRefGoogle Scholar
  71. 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.zbMATHMathSciNetGoogle Scholar
  72. 72.
    B. Dacoronga, Direct Methods in the Calculus of Variations, Springer 1989.Google Scholar
  73. 73.
    R. DeBlassie, Exit times from cones inIRn of Brownian motion, Probab. Theory Related Fields 74(1987), 1-29.MathSciNetCrossRefGoogle Scholar
  74. 74.
    R. DeBlassie, Remark on: "Exit times from cones inŒT of Brownian motion", Probab. Theory Related Fields 79 (1988), 95-97.MathSciNetCrossRefGoogle Scholar
  75. 75.
    B. Davis, On the integrability of the martingale square function, Israel J. Math. 8 (1970), 187-190.zbMATHMathSciNetCrossRefGoogle Scholar
  76. 76.
    B. Davis, On the Barlow-Yor inequalities for local time, Séminaire de Probabilités, XXI, 218-220, Lecture Notes in Math. 1247, Springer, Berlin, 1987CrossRefGoogle Scholar
  77. 77.
    B. Davis, On stopping times for n dimensional Brownian motion, Ann. Prob. 6 (1978), 651-659.zbMATHCrossRefGoogle Scholar
  78. 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.MathSciNetGoogle Scholar
  79. 79.
    B. Dahlberg, D. Jerison and K. Kenig, Area integral estimates for elliptic operators with nonsmooth coefficients, Arkiv Mat. 22(1984), 97-108.zbMATHMathSciNetCrossRefGoogle Scholar
  80. 80.
    S. Donaldson and D. Sullivan, Quasiconformal 4-manifolds, Acta Math. 163(1989), 181-252.zbMATHMathSciNetCrossRefGoogle Scholar
  81. 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.zbMATHMathSciNetGoogle Scholar
  82. 82.
    O. Dragicevic and A. Volberg, Bellman functions and dimensionless estimates of Littlewood-Paley type, J. Operator Theory 56(2006), 167-198.zbMATHMathSciNetGoogle Scholar
  83. 83.
    O. Dragicevic and A. Volberg, Sharp estimate of the Ahlfors-Beurling operator via averaging martingale transforms, Michigan Math. J. 51(2003), 415-435.zbMATHMathSciNetCrossRefGoogle Scholar
  84. 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.zbMATHMathSciNetCrossRefGoogle Scholar
  85. 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.zbMATHMathSciNetGoogle Scholar
  86. 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.zbMATHMathSciNetGoogle Scholar
  87. 87.
    R. Durrett, Brownian Motion and Martingales in Analysis, Wadsworth, Belmont, CA, 1984.zbMATHGoogle Scholar
  88. 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. 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.zbMATHMathSciNetCrossRefGoogle Scholar
  90. 90.
    C. Fefferman and E.M. Stein, H p spaces in several variables, Acta Math. 129(1972) 137-193.zbMATHMathSciNetCrossRefGoogle Scholar
  91. 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.zbMATHMathSciNetCrossRefGoogle Scholar
  92. 92.
    J. Garnett, Bounded Analytic Functions, Academic Press, New York, 1980.Google Scholar
  93. 93.
    A. Garsia, The Burgess Davis inequalities via Fefferman’s inequalities, Arkiv. für Mathematik 11 (1973) 229-237.zbMATHMathSciNetCrossRefGoogle Scholar
  94. 94.
    A. Garsia, On a convex function inequality for martingales, Ann. Prob. 1 (1973), 171-174.zbMATHMathSciNetCrossRefGoogle Scholar
  95. 95.
    A. Garsia, Martingale Inequalities, Seminar Notes on Recent Progress, W. A. Benjamin Mathematics Lecture Note Series, 1973.Google Scholar
  96. 96.
    E. Geiss, S. Mongomery-Smith, E. Saksman, On singular integral and martingale transforms, Trans. Amer. Math. Soc. 362(2010), 553-575.zbMATHMathSciNetCrossRefGoogle Scholar
  97. 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. 98.
    L. Grafakos, Classical and Modern Fourier Analysis, Pearson Education Inc., 2004.Google Scholar
  99. 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. 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 zbMATHMathSciNetGoogle Scholar
  101. 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.MathSciNetGoogle Scholar
  102. 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.zbMATHMathSciNetCrossRefGoogle Scholar
  103. 103.
    W. Hammack, Sharp maximal inequalities for stochastic integrals in which the integrator is a submartingale, Proc. Amer. Math. Soc. 124(1996), 931-938.zbMATHMathSciNetCrossRefGoogle Scholar
  104. 104.
    G. H. Hardy and J. E. Littlewood, A maximal function with function theoretic applications, Acta Math. 54 (1930), 81-116.MathSciNetCrossRefGoogle Scholar
  105. 105.
    T.P. Hytönen: On the norm of the Beurling-Ahlfors operator in several dimensions, (Preprint). Google Scholar
  106. 106.
    T. Iwaniec, Extremal inequalities in Sobolev spaces and quasiconformal mappings, Z. Anal. Anwendungen 1 (1982), 1-16.zbMATHMathSciNetGoogle Scholar
  107. 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. 108.
    T. Iwaniec, Nonlinear Cauchy-Riemann operators in R n, Trans. Amer. Math. Soc. 354(2002), 1961-1995.zbMATHMathSciNetCrossRefGoogle Scholar
  109. 109.
    T. Iwaniec, and GJ. Martin, Quasiregular mappings in even dimensions, Acta Math. 170(1993), 29-81.zbMATHMathSciNetCrossRefGoogle Scholar
  110. 110.
    T. Iwaniec and G. Martin, Riesz transforms and related singular integrals, J. Reine Angew. Math. 473(1996), 25-57.zbMATHMathSciNetGoogle Scholar
  111. 111.
    T. Iwaniec, and G. J. Martin, Geometric Function Theory and Nonlinear Analysis, Oxford University Press, 2001.Google Scholar
  112. 112.
    P. Janakiraman, Best weak-type (p,p) constants, 1 ≤ p2, for orthogonal harmonic functions and martingales, Illinois J. Math. 48 (2004), 909-921.zbMATHMathSciNetGoogle Scholar
  113. 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. 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. 115.
    O. Lehto, Remarks on the integrability of the derivatives of quasiconformal mappings, Ann. Acad. Sei. Fenn. Series AI Math. 371(1965), 8 pp.MathSciNetGoogle Scholar
  116. 116.
    J. Marcinkiewicz and A. Zygmund, A theorem of Lusin, Duke Math. J. 4 (1938), 473-485.MathSciNetCrossRefGoogle Scholar
  117. 117.
    T. McConnell, On Fourier multiplier transformations of Banach-valuedfunctions, Trans. Amer. Math. Soc. 285(1984), 739-757.zbMATHMathSciNetCrossRefGoogle Scholar
  118. 118.
    A. D. Mêlas, The Bellman functions of dyadic-like maximal operators and related inequalities, Advances in Mathe- matics, 192(2005), 310-340.zbMATHCrossRefGoogle Scholar
  119. 119.
    A. D. Melas, Dyadic-like maximal operators on LlogL functions, Journal of Functional Analysis 257(2009), 1631-1654.zbMATHMathSciNetCrossRefGoogle Scholar
  120. 120.
    P. W. Millar, Martingale integrals, Trans. Amer. Math. Soc, 133(1966), 145-166.MathSciNetCrossRefGoogle Scholar
  121. 121.
    C.B. Morrey, Quasi-convexity and the lower semicontinuity of multiple integrals, Pacific J. Math., 2 (1952), 25-53.zbMATHMathSciNetGoogle Scholar
  122. 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. 123.
    T. Murai and A. Uchiyama, Good-X inequalities for the area integral and the nontangential maximal function, Studia Math. 83 (1986), 251-262.MathSciNetGoogle Scholar
  124. 124.
    A. Miyachi and K. Yabuta, On good-X inequalities, Bull. Fac. Sei., Ibaraki Univ., Math. 16 (1984), 1-11.MathSciNetCrossRefGoogle Scholar
  125. 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.MathSciNetGoogle Scholar
  126. 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.MathSciNetCrossRefGoogle Scholar
  127. 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.MathSciNetGoogle Scholar
  128. 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.MathSciNetCrossRefGoogle Scholar
  129. 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.MathSciNetGoogle Scholar
  130. 130.
    A. A. Novikov, On moment inequalities for stochastic integrals(Russian; English summary), Teor. Verejatnost i Primenen 16 (1971), 548-551.Google Scholar
  131. 131.
    A. Osekowski, Sharp inequality for bounded submartingales and their differential subordinates, Electron. Commun. Probab. 13 (2008), 660-675.zbMATHMathSciNetGoogle Scholar
  132. 132.
    A. Osekowski, Sharp maximal inequality for stochastic integrals. Proc. Amer. Math. Soc. 136(2008), 2951-2958.zbMATHMathSciNetCrossRefGoogle Scholar
  133. 133.
    A. Osekowski, Sharp weak-type inequalities for differentially subordinated martingales, Bernoulli 15 (2009), 871-897.zbMATHMathSciNetCrossRefGoogle Scholar
  134. 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.zbMATHMathSciNetGoogle Scholar
  135. 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.zbMATHMathSciNetCrossRefGoogle Scholar
  136. 136.
    A. Osekowski, Weak type inequality for the square function of a nonnegative submartingale, Bull. Pol. Acad. Sei. Math. 57(2009), 81-89.MathSciNetCrossRefGoogle Scholar
  137. 137.
    A. Osekowski, Sharp maximal inequality for martingales and stochastic integrals. Electron, Commun. Probab. 14 (2009), 17-30.zbMATHMathSciNetGoogle Scholar
  138. 138.
    R. E. A. C. Paley, A remarkable series of orthogonalfunctions I, Proc. London Math. Soc. 34 (1932) 241-264.zbMATHCrossRefGoogle Scholar
  139. 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.zbMATHMathSciNetCrossRefGoogle Scholar
  140. 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.MathSciNetGoogle Scholar
  141. 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.zbMATHMathSciNetGoogle Scholar
  142. 142.
    I. Privalov, sur les formions conjuguées, Bull. Soc. Math. France (1916), 100-103.Google Scholar
  143. 143.
    D. Revuz and M. Yor, Continuous Martingales and Brownian Motion, Springer-Verlag, 293, 1980.Google Scholar
  144. 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.MathSciNetGoogle Scholar
  145. 145.
    L. Slavin and V Vasyunin, Bellman function for the sharp classical and dyadic John-Nirenberg inequality, (Preprint). Google Scholar
  146. 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. 147.
    D. C. Spencer, A function theoretic identity, Amer. J. Math. 65 (1943), 147-160.zbMATHMathSciNetCrossRefGoogle Scholar
  148. 148.
    E.M. Stein, On the theory of harmonic functions of several variables. II, Acta Math. 106(1961), 137-174.zbMATHMathSciNetCrossRefGoogle Scholar
  149. 149.
    E. M. Stein, Singular integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, 1970.zbMATHGoogle Scholar
  150. 150.
    E.M. Šteáin, The development of square functions in the work of A. Zygmund, Bull. Amer. Math. Soc. 7 (1982), 359-376.MathSciNetCrossRefGoogle Scholar
  151. 151.
    E.M. Šteáin, Some results in Harmonic Analysis inIRn for n→ ∞. Bull. Amer. Math. Soc. 9 (1983), 71-73.MathSciNetCrossRefGoogle Scholar
  152. 152.
    E.M. Šteáin, Problems in harmonic analysis related to curvature and oscillatory integrals, Proceedings of the Interna- tional Congress of Mathematicians, Berkeley, CA., 1986Google Scholar
  153. 153.
    E.M. Šteáin, Harmonic Analysis, Princeton Mathematical Series, 43, 1993.Google Scholar
  154. 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.zbMATHMathSciNetCrossRefGoogle Scholar
  155. 155.
    V. Sverâk, Examples of rank-one convex functions, Proc. Roy. Soc. Edinburgh 114A(1990), 237-242.Google Scholar
  156. 156.
    V. Sverâk, Rank-one convexity does not imply quasiconvexity, Proc. Roy. Soc. Edinburgh 120A(1992), 185-189.Google Scholar
  157. 157.
    V. Sverâk, New examples of quasiconvex functions, Arch. Rational Mech. Anal. 119(1992), 293-300.zbMATHMathSciNetCrossRefGoogle Scholar
  158. 158.
    A. Torchinsky, Real Variable Methods in Harmonic Analysis, Academic Press, Inc. Orlando, FL, 1986.zbMATHGoogle Scholar
  159. 159.
    N. Th. Varopoulos, Aspects of probabilistic Littlewood-Paley theory, J. Funct. Anal. 38 (1980), no. 1, 25-60.zbMATHMathSciNetCrossRefGoogle Scholar
  160. 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. 161.
    V. Vasyunin and A. Volberg, Bellman functions technique in harmonic analysis, (sashavolberg.wordpress.com).Google Scholar
  162. 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. 163.
    G. Wang, Sharp inequalities for the conditional square function of a martingale, Ann. Probab. 19 (1991), 1679-1688.zbMATHMathSciNetCrossRefGoogle Scholar
  164. 164.
    G. Wang, Sharp maximal inequalities for conditionally symmetric martingales and Brownian motion, Proc. Amer. Math. Soc. 112(1991), no. 2, 579-586zbMATHMathSciNetCrossRefGoogle Scholar
  165. 165.
    G. Wang, Sharp square-function inequalities for conditionally symmetric martingales, Trans. Amer. Math. Soc. 328(1991), 393-419.zbMATHMathSciNetCrossRefGoogle Scholar
  166. 166.
    G. Wang, Differential subordination and strong differential subordination for continuous-time martingales and related sharp inequalities, Ann. Probab. 23(1995), 522-551.zbMATHMathSciNetCrossRefGoogle Scholar
  167. 167.
    A. Zygmund, Trigonometrical Series, 2nd ed. Cambridge Univ. Press, Cambridge 1959.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