Explorations in Martingale Theory and Its Applications

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


These lectures center on new and other recent work giving sharp inequalities for martingales and stochastic integrals with some applications to harmonic analysis and the geometry of Banach spaces. But to introduce some of the ideas and notation, we shall begin with an example from earlier work. At the end of the introduction is a summary of the remaining chapters.


Banach Space Strict Inequality Good Constant Predictable Process Nondecreasing Sequence 
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.


  1. M. Abramowitz and I. A. Stegun, editors, "Handbook of mathematical functions," Dover, New York, 1970.Google Scholar
  2. D. J. Aldous, Unconditional bases and martingales in L P (F), Math. Proc. Cambridge Phil. Soc. 85(1979), 117-123.zbMATHMathSciNetCrossRefGoogle Scholar
  3. T. Ando, Contractive projections in L p spaces, Pacific J. Math. 17(1966), 391-405.zbMATHMathSciNetGoogle Scholar
  4. A. Baernstehin, Some sharp inequalities for conjugate functions, Indiana Univ. Math. J. 27 (1978), 833-852.MathSciNetCrossRefGoogle Scholar
  5. R. Bañuelos, A sharp good-X inequality with an application to Riesz trans f orras, Mich. Math. J. 35 (1988), 117-125.zbMATHCrossRefGoogle Scholar
  6. R. Bass, A probabilistic approach to the boundedness of singular integral operators, Séminaire de Probabilités XXIV 1988/89, Lecture Notes in Mathematics 1426(1990), 15-40.MathSciNetCrossRefGoogle Scholar
  7. A. Benedek, A. P. Calderon, and R. Panzone, Convolution operators on Banach space valued functions, Proc. Nat. Acad. Sei. 48(1962), 356-365.zbMATHMathSciNetCrossRefGoogle Scholar
  8. E. Berkson, T. A. Gillespie, and P. S. Muhly, Théorie spectrale dans les espaces UMD, C. R. Acad. Sei. Paris 302(1986), 155-158.zbMATHMathSciNetGoogle Scholar
  9. E. Berkson, T. A. Gillespie, and P. S. Muhly, Generalized analyticity in UMD spaces, Arkiv för Math. 27 (1989), 1-14.zbMATHMathSciNetCrossRefGoogle Scholar
  10. K. Bichteler, Stochastic integration and L p -iheory of sernirnartingales, Ann. Prob. 9 (1981), 49-89.zbMATHMathSciNetCrossRefGoogle Scholar
  11. O. Blasco, Hardy spaces of vector-valued functions: Duality, Trans. Amer. Math. Soc. 308(1988a), 495-507.zbMATHMathSciNetCrossRefGoogle Scholar
  12. O. Blasco, Boundary values of functions in vector-valued Hardy spaces and geometry on Banach spaces, J, Punct. Anal 78(1988b), 346-364.zbMATHMathSciNetCrossRefGoogle Scholar
  13. G. Blower, A multiplier characterization of analytic UMD spaces, Studia Math. 96 (1990), 117-124.zbMATHMathSciNetGoogle Scholar
  14. B. Bollobas, Martingale inequalities, Math. Proc. Cambridge Phil. Soc. 87(1980), 377-382.zbMATHMathSciNetCrossRefGoogle Scholar
  15. B. Bollobas, editor, "Littlewood’s Miscellany,’ Cambridge University Press, Cambridge, 1986.Google Scholar
  16. J. Bourgain, Some remarks on Banach spaces in which martingale difference sequences are unconditional, Ark. Mat. 21(1983), 163-168.zbMATHMathSciNetCrossRefGoogle Scholar
  17. J. Bourgain, Extension of a result of Benedek, Calderón, and Panzone, Ark. Mat. 22(1984), 91-95.zbMATHMathSciNetCrossRefGoogle Scholar
  18. J. Bourgain, Vector valued singular integrals and the H l -BMO duality, in "Probability Theory and Harmonic Analysis," edited by J. A. Chao and W. A. Woyczynski, Marcel Dekker, New York, 1986, pp. 1-19.Google Scholar
  19. A. V. Bukhvalov, Hardy spaces of vector-valued functions, J. Sov. Math. 16 (1981), 1051-1059.zbMATHCrossRefGoogle Scholar
  20. A. V. Bukhvalov, Continuity of operators in spaces of vector functions, with applications to the theory of bases, J. Sov. Math. 44 (1989), 749-762.zbMATHMathSciNetCrossRefGoogle Scholar
  21. D. L. Burkholder, Martingale transforms, Ann. Math. Statist. 37(1966), 1494-1504.zbMATHMathSciNetCrossRefGoogle Scholar
  22. D. L. Burkholder, Distribution function inequalities for martingales, Ann. Prob. 1(1973), 19-42.zbMATHMathSciNetCrossRefGoogle Scholar
  23. D. L. Burkholder, Exit times of Brownian motion, harmonic majorization, and Hardy spaces, Advances in Math. 26(1977), 182-205.zbMATHMathSciNetCrossRefGoogle Scholar
  24. D. L. Burkholder, A sharp inequality for martingale transforms, Ann. Prob. 7(1979), 858-863.zbMATHMathSciNetCrossRefGoogle Scholar
  25. D. L. Burkholder, A geometrical characterization of Banach spaces in which martingale difference sequences are unconditional, Ann. Prob. 9 (1981a), 997-1011.zbMATHMathSciNetCrossRefGoogle Scholar
  26. D. L. Burkholder, Martingale transforms and the geometry of Banach spaces, Proceedings of the Third International Conference on Probability in Banach Spaces, Tufts University, 1980, Lecture Notes in Mathematics 860(1981b), 35-50.Google Scholar
  27. D. L. Burkholder, A nonlinear partial differential equation and the unconditional constant of the Haar system in Lp, Bull Amer. Math. Soc. 7(1982), 591-595.zbMATHMathSciNetCrossRefGoogle Scholar
  28. D. L. Burkholder, A geometric condition that impies the existence of certain singular integrals of Banach-space-valued functions, in "Conference on Harmonic Analysis in Honor of Antoni Zyg-mund (Chicago, 1981)," edited by William Beckner, Alberto P. Calderón, Robert Fefferman, and Peter W. Jones. Wadsworth, Belmont, Califcrnia, 1983, pp. 270-286.Google Scholar
  29. D. L. Burkholder, Boundary value problems and sharp inequalities for martingale transforms, Ann. Prob. 12(1984), 647-702.zbMATHMathSciNetCrossRefGoogle Scholar
  30. D. L. Burkholder, An elementary proof of an inequality of R. E. A. C. Paley, Bull. London Math. Soc, 17(1985), 474-478.zbMATHMathSciNetCrossRefGoogle Scholar
  31. D. L. Burkholder, An extension of a classical martingale inequality, in "Probability Theory and Harmonic Analysis," edited by J. A. Chao and W. A. Woyczynski. Marcel Dekker, New York, 1986a, pp. 21-30.Google Scholar
  32. D. L. Burkholder, Mamingales and Fourier analysis in Banach spaces, C.LM.E. Lectures, Varenna (Como), Italy, 1985, Lecture Notes in Mathematics 1206(1986b), 61-108.MathSciNetGoogle Scholar
  33. D. L. Burkholder,A sharp and strict Lp-inequality for stochastic integrals, Ann. Prob. 15(1987), 268-273.zbMATHMathSciNetCrossRefGoogle Scholar
  34. D. L. Burkholder, A proof of Pelczyński’s conjecture for the Haar system, Studia Math. 91(1988a), 79-83.zbMATHMathSciNetGoogle Scholar
  35. D. L. Burkholder, Sharp inequalities for martingales and stochastic integrals, Colloque Paul Levy, Palaiseau, 1987, Astérisque 157-158(1988b), 75-94.Google Scholar
  36. D. L. Burkholder, Differential subordination of harmonic functions and martingales, Harmonic Analysis and Partial Differential Equations (El Escorial, 1987), Lecture Notes in Mathematics 1384(1989a), 1-23.MathSciNetCrossRefGoogle Scholar
  37. 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, 1989b, pp. 159-178.Google Scholar
  38. D. L. Burkholder and R. F. Gundy, Extrapolation and interpolation of quasi-linear operators on martingales, Acta. Math, 124(1970), 249-304.zbMATHMathSciNetCrossRefGoogle Scholar
  39. A. P. Calderón and A. Zygmund, On the existence of certain singular integrals, Acta Math. 88(1952), 85-139.zbMATHMathSciNetCrossRefGoogle Scholar
  40. A. P. Calderón and A. Zygmund, On singular integrals, Amer, J. Math. 78(1956), 289-309,zbMATHMathSciNetCrossRefGoogle Scholar
  41. S. D. Chatterji, Les martingales et leurs applications analytiques, Lecture Notes in Mathematics 307(1973), 27-164.MathSciNetCrossRefGoogle Scholar
  42. K. P. Choi, Some sharp inequalities for martingale transforms. Trans. Amer. Math. Soc. 307(1988), 279-300.zbMATHMathSciNetCrossRefGoogle Scholar
  43. F. Cobos, Some spaces in which martingale difference sequences are unconditional, Bull. Polish Acad, of Sei. Math. 34(1986), 695-703.zbMATHMathSciNetGoogle Scholar
  44. F. Cobos, Duality, UMD-properiy and Lorentz-Marcinkiewicz operator spaces, in "16 Colóquio Bra-sileiro de Matemática," Rio de Janeiro, 1988, pp. 97-106.Google Scholar
  45. F. Cobos and D. L. Fernandez, Hardy-Sobolev spaces and Besov spaces with a function parameter, Lecture Notes in Mathematics 1302(1988), 158-170.MathSciNetCrossRefGoogle Scholar
  46. T. Coulhon and D. Lamberton, Régularité L p pour les équations d’évolution, in "Séminaire d’Analyse Fonctionnelle, 1984/1985," PubL Math. Univ. Paris VII 26 (1986), 155-165.MathSciNetGoogle Scholar
  47. D. C. Cox, The best constant in Burkholder’s weak-L l inequality for the martingale square function, Proc. Amer. Math. Soc. 85(1982), 427-433.zbMATHMathSciNetGoogle Scholar
  48. D. C. Cox and R. P. Kertz, Common strict character of some sharp infinite-sequence martingale inequalities, Stochastic Process. Appl. 20(1985), 169-179.zbMATHMathSciNetCrossRefGoogle Scholar
  49. B. Davis, A comparison test f or martingale inequalities, Ann. Math. Statist. 40(1969), 505-508.zbMATHMathSciNetCrossRefGoogle Scholar
  50. B. Davis, On the weak type (1,1) inequality for conjugate functions, Proc. Amer. Math. Soc. 44(1974), 307-311.zbMATHMathSciNetGoogle Scholar
  51. B. Davis, On the Lpnorms of stochastic integrals and other martingales, Duke Math. J. 43(1976), 697-704.zbMATHMathSciNetCrossRefGoogle Scholar
  52. M. Defant, On the vector-valued Hubert transform, Math. Nachr. 141(1989), 251-265.zbMATHMathSciNetCrossRefGoogle Scholar
  53. C. Dellacherie and P. A. Meyer, "Probabilités et potentielirtneorie des martingales," Hermann, Paris, 1980.Google Scholar
  54. J. Diestel and J. J. Uhl, "Vector Measures," Math. Surveys 15, American Mathematical Society, Providence, Rhode Island, 1977.Google Scholar
  55. C. Doléans, Variation quadratique des martingales continues à droite, Ann. Math. Statist. 40(1969), 284-289.zbMATHMathSciNetCrossRefGoogle Scholar
  56. J. L. Doob, "Stochastic Processes," Wiley, New York, 1953.zbMATHGoogle Scholar
  57. J. L. Doob, Remarks on the boundary limits of harmonic functions, J. SIAM Numer. Anal. 3(1966), 229-235.zbMATHMathSciNetCrossRefGoogle Scholar
  58. J. L. Doob, "Classical Potential Theory and Its Probabilistic Counterpart," Springer, New York, 1984.zbMATHGoogle Scholar
  59. L. E. Dor and E. Odell, Monotone bases in L p, Pacific J. Math. 60 (1975), 51-61.zbMATHMathSciNetGoogle Scholar
  60. G. Dore and A. Venni, On the closedness of the sum of two closed operators, Math. Z. 196(1987), 189-201.zbMATHMathSciNetCrossRefGoogle Scholar
  61. I. Doust, Contractive projections on Banach spaces, Proc. Centre for Math. Anal., Australian National University 20(1988), 50-58.MathSciNetGoogle Scholar
  62. I. Doust,Well-bounded and scalar-type spectral operators on Lpspaces, J. London Math. Soc. 39(1989), 525-534.zbMATHMathSciNetCrossRefGoogle Scholar
  63. L. E. Dubins, Rises and upcrossings of nonnegative martingales, Illinois J, Math. 6(1962), 226-241.zbMATHMathSciNetGoogle Scholar
  64. P. Enflo, Banach spaces which can be given an equivalent uniformly convex norm, Israel J. Math. 13(1972), 281-288.MathSciNetGoogle Scholar
  65. M. Essén, A superharmonic proof of the M. Riesz conjugate function theorem, Ark. Math. 22(1984), 241-249.zbMATHCrossRefGoogle Scholar
  66. D. L. Fernandez, Vector-valued singular integral operators on L p -spaces with mixed norms and applications, Pacific J. Math. 129(1987), 257-275.zbMATHMathSciNetGoogle Scholar
  67. D. L. Fernandez, On Fourier multipliers of Banach-lattice valued functions, Rev. Roumaine Math, Pures Appl. 34(1989), 635-642.zbMATHMathSciNetGoogle Scholar
  68. D. L. Fernandez and J. B. Garcia, Interpolation of Orlicz-valued function spaces and U.M.D. property, 26° Semi’ario Brasileiro de Análise (Rio de Janeiro, 1987), Trabalhos Apresentados, 269-281.Google Scholar
  69. T. Figiel, On equivalence of some bases to the Haar system in spaces of vector-valued functions, Bull. Polon. Acad. Sei. 36(1988), 119-131.zbMATHMathSciNetGoogle Scholar
  70. T. Figiel, Singular integral operators: a martingale approach, to appear in the Proceedings of the Conference on the Geometry of Banach Spaces (Strobl, Austria, 1989).Google Scholar
  71. T. W. Gamelin, "Uniform Algebras and Jensen Measures," Cambridge University Press, London, 1978.zbMATHGoogle Scholar
  72. D. J. H. Garling, Brownian motion and UMD-spaces, Conference on Probability and Banach Spaces, Zaragoza, 1985, Lecture Notes in Mathematics 1221(1986), 36-49.Google Scholar
  73. Y. Giga and H. Sohr, Abstract L p estimates for the Cauchy problem with applications to the Navier-Stokes equations in exterior domains, preprint.Google Scholar
  74. D. Gilat, The best bound in the Llog L inequality of Hardy and Littlewood and its martingale counterpart, Proc. Amer. Math. Soc. 97(1986), 429-436.zbMATHMathSciNetGoogle Scholar
  75. S. Guerre, On the closedness of the sum of closed operators on a UMD space, in "Banach Space Theory," American Mathematical Society, Providence, Rhode Island, 1989, pp. 239-251.Google Scholar
  76. S. Guerre, Complex powers of operators and UMD spaces, manuscript.Google Scholar
  77. R. F. Gundy, "Some Topics in Probability and Analysis," Regional Conference Series in Mathematics 70, American Mathematical Society, Providence, Rhode Island, 1989.Google Scholar
  78. U. Haagerup, The best constants in the Khintchine inequality, Studia Math. 70(1982), 231-283.zbMATHMathSciNetGoogle Scholar
  79. U. Haagerup and G. Pisier, Factorization of analytic functions with values in non-commutative L 1 -spaces and applications, Can. J. Math. 41(1989), 882-906.zbMATHMathSciNetCrossRefGoogle Scholar
  80. G. H. Hardy, J. E. Littlewood, and G. Pólya, "Inequalities," Cambridge University Press, Cambridge, 1934.Google Scholar
  81. W. Hensgen, On complementation of vector-valued Hardy spaces, Proc. Amer. Math. Soc. 104(1988), 1153-1162.zbMATHMathSciNetGoogle Scholar
  82. W. Hensgen, On the dual space of H p (X), 1 < p <∞, J. Funct. Anal. 92(1990), 348-371.zbMATHMathSciNetCrossRefGoogle Scholar
  83. P. Hitczenko, Comparison of moments for tangent sequences of random variables, Probab. Th. Rel. Fields 78(1988), 223-230.zbMATHMathSciNetCrossRefGoogle Scholar
  84. P. Hitczenko, On tangent sequences of UMD-space valued random vectors, manuscript.Google Scholar
  85. P. Hitczenko,Upper bounds for the Lv-norms of martingales, Probab. Th. Rel. Fields 86(1990), 225-238.zbMATHMathSciNetCrossRefGoogle Scholar
  86. P. Hitczenko, Best constants in martingale version of RosenihaVs inequality, Ann. Probab. 18(1990), 1656-1668.zbMATHMathSciNetCrossRefGoogle Scholar
  87. R. C. James, Some self dual properties of norrned linear spaces, Ann. Math. Studies 69(1972a), 159-175.Google Scholar
  88. R. C. James, Super-reflexive spaces with bases, Pacific J. Math. 41(1972b), 409-419.Google Scholar
  89. R. C. James, Super-reflexive Banach spaces, Can. J. Math. 24(1972c), 896-904.zbMATHCrossRefGoogle Scholar
  90. W. B. Johnson and G. Schechtman, Martingale inequalities in rearrangement invariant function spaces, Israel J. Math. 64(1988), 267-275.MathSciNetCrossRefGoogle Scholar
  91. N. J. Kalton, Differentials of complex interpolation processes for Kothe function spaces, a paper delivered at the Conference on Function Spaces (Auburn University, 1989).Google Scholar
  92. G. Klincsek, A square function inequality, Ann. Prob. 5(1977), 823-825.zbMATHMathSciNetCrossRefGoogle Scholar
  93. A. N. Kolmogorov, Sur les fonctions harmoniques conjuguées et les séries de Fourier, Fund. Math. 7(1925), 24-29.Google Scholar
  94. H. König, Vector-valued multiplier theorems, in "Séminaire d’analyse fonctionnelle, 1985-1987," Publications mathématique de l’université Paris VII, 1988, pp. 131-140.Google Scholar
  95. H. Kunita, Stochastic integrals based on martingales taking values in Hubert space, Nagoya Math. J. 38(1970), 41-52.zbMATHMathSciNetGoogle Scholar
  96. S. Kwapieii, Isomorphic characterizations of inner product spaces by orthogonal series with vector valued coefficients, Studia Math. 44(1972), 583-595.MathSciNetGoogle Scholar
  97. S. Kwapien and W. A. Woyczynski, Tangent sequences of random variables: Basic inequalities and their applications, in "Almost Everywhere Convergence," edited by Gerald A. Edgar and Louis Sucheston. Academic Press, New York, 1989, pp. 237-265.Google Scholar
  98. J. Lindenstrauss and A. Pekzyński, Contributions to the theory of the classical Banach spaces, J. Funct. Anal. 8(1971), 225-249.zbMATHCrossRefGoogle Scholar
  99. J. Lindenstrauss and L. Tzafriri, "Classical Banach Spaces I: Sequence Spaces," Springer, New York, 1977.Google Scholar
  100. J. Lindenstrauss and L. Tzafriri, "Classical Banach Spaces II: Function Spaces," Springer, New York, 1979.Google Scholar
  101. A. Mandelbaum, L. A. Shepp, and R. Vanderbei, Optimal switching between a pair of Brownian motions, Ann. Prob. 18(1990), 1010-1033.zbMATHMathSciNetCrossRefGoogle Scholar
  102. J. Mareinkiewicz, Quelques théorèmes sur les séries orthogonales, Ann. Soc, Polon. Math. 16(1937), 84-96.Google Scholar
  103. B. Maurey, Système de Haar, in "Séminaire Maurey-Schwartz, 1974-1975," Ecole Polytechnique, Paris, 1975.Google Scholar
  104. T. R. McConnell, On Fourier multiplier transformations of Banach-valued functions, Trans. Amer. Math. Soc. 285(1984), 739-757.zbMATHMathSciNetCrossRefGoogle Scholar
  105. T. R. McConnell, A Skorohod-like representation in infinite dimensions, Probability in Banach Spaces V, Lecture Notes in Mathematics 1153(1985), 359-368.MathSciNetCrossRefGoogle Scholar
  106. T. R. McConnell, Decoupling and stochastic integration in UMD Banach spaces, Prob. Math. Stat. 10(1989), 283-295.zbMATHMathSciNetGoogle Scholar
  107. H. P. McKean, Geometry of differential space, Ann. Prob, 1(1973), 197-206.zbMATHMathSciNetCrossRefGoogle Scholar
  108. L Monroe, Martingale operator norms and local times, manuscript.Google Scholar
  109. A. A. Novikov, On stopping times for the Wiener process, (Russian, English summary), Teor. Vero-jatnost. i Primenen 16(1971), 458-465.Google Scholar
  110. A. M. Olevskiï, Fourier series and Lebesgue functions, (Russian), Uspehi Mat. Nauk 22(1967), 237-239.MathSciNetGoogle Scholar
  111. A. M. Olevskiï, "Fourier Series with Respect to General Orthogonal Systems," Springer, New York, 1975.zbMATHGoogle Scholar
  112. R. E. A. C. Paley, A remarkable series of orthogonal functionsƒ., Proc. London Math. Soc. 34(1932), 241-264.zbMATHCrossRefGoogle Scholar
  113. A. Pelezynski, Structural theory of Banach spaces and its interplay with analysis and probability, in "Proceedings of the International Congress of Mathematicians (Warsaw, 1983)," PWN, Warsaw, 1984, pp.237-269.Google Scholar
  114. A. Pelezynski, Norms of classical operators in function spaces, Colloque Laurent Schwartz, Astérisque 131(1985), 137-162.Google Scholar
  115. A. Pelczyiiski and H. Rosenthal, Localization techniques in L p spaces, Studia Math. 52(1975), 263-289.Google Scholar
  116. S. K. Pichorides, On the best values of the constants in the theorems of M. Riesz, Zygmund and Kolmogorov, Studia Math. 44(1972) 165-179.zbMATHMathSciNetGoogle Scholar
  117. G. Pisier, Un exemple concernant la super-réflexivité, in "Séminaire M aurey-Schwartz, 1974-75," École Polytechnique, Paris, 1975a.Google Scholar
  118. G. Pisier, Martingales with values in uniformly convex spaces, Israel J, Math. 20(1975b), 326-350.zbMATHMathSciNetCrossRefGoogle Scholar
  119. A. O. Pittenger, Note on a square function inequality, Ann. Prob. 7(1979), 907-908,zbMATHMathSciNetCrossRefGoogle Scholar
  120. M. Riesz, Sur les fonctions conjuguées, Math. Z. 27(1927), 218-244.zbMATHMathSciNetCrossRefGoogle Scholar
  121. J. L. Rubio de Francia, Martingale and integral transforms of Banach space valued functions, Conference on Probability and Banach Spaces, Zaragoza, 1985, Lecture Notes in Mathematics 1221(1986), 195-222.Google Scholar
  122. J. L. Rubio de Francia and J. L. Torrea, Some Banach techniques in vector valued Fourier analysis, Colloq. Math. 54(1987), 271-284.MathSciNetGoogle Scholar
  123. J. L. Rubio de Francia, F. J. Ruiz, and J. L. Torrea, Colder an-Zygmund theory for operator-valued kernels, Advances in Math. 62(1986), 7-48.zbMATHMathSciNetCrossRefGoogle Scholar
  124. J. Schwartz, A remark on inequalities of Colder on-Zygmund type for vector-valued functions, Comm. Pure Appl. Math. 14(1961), 785-799.zbMATHMathSciNetCrossRefGoogle Scholar
  125. L. A. Shepp, A first passage problem for the Wiener process, Ann. Math. Statist. 38(1967), 1912-1914.zbMATHMathSciNetCrossRefGoogle Scholar
  126. E. M. Stein, "Singular Integrals and Differentiability Properties of Functions," Princeton University Press, Princeton, 1970.zbMATHGoogle Scholar
  127. E. M. Stein and G. Weiss, On the theory of harmonic functions of several variables: L The theory of H p -spaces, Acta Math. 103(1960), 25-62.zbMATHMathSciNetCrossRefGoogle Scholar
  128. S. J. Szarek, On the best constants in the Khinchin inequality, Studia Math. 58 (1976), 197-208.zbMATHMathSciNetGoogle Scholar
  129. B. Tomaszewski, Sharp weak-type inequalities for analytic functions on the unit disc, Bull. London Math. Soc. 18(1986), 355-358.zbMATHMathSciNetCrossRefGoogle Scholar
  130. L. Tzafriri, Remarks on contractive projections in L p -spaces, Israel J. Math. 7(1969), 9-15.zbMATHMathSciNetCrossRefGoogle Scholar
  131. G. Wang, "Some Sharp Inequalities for Conditionally Symmetric Martingales," doctoral thesis, University of Illinois, Urbana, Illinois, 1989.Google Scholar
  132. G. Wang, Sharp square-function inequalities for conditionally symmetric martingales, Trans. Amer. Math. Soc. (to appear).Google Scholar
  133. G. Wang, Sharp maximal inequalities for conditionally symmetric martingales and Brownian motion, Proc. Amer. Math. Soc. (to appear).Google Scholar
  134. G. Wang, Sharp inequalities for the conditional square function of a martingale, Ann. Prob, (to appear).Google Scholar
  135. T. M. Wolmewicz, The Hubert transform in weighted spaces of integrable vector-valued functions, Colloq. Math. 53(1987), 103-108.MathSciNetGoogle Scholar
  136. M. Yor, Sur les intégrales stochastique à valeurs dans un Banach, C. R, Acad. Sei. Paris 277(1973), 467-469.zbMATHMathSciNetGoogle Scholar
  137. F. Zimmermann, On vector-valued Fourier multiplier theorems, Studia Math. 93(1989), 201-222.zbMATHMathSciNetGoogle Scholar
  138. J. Zinn, Comparison of martingale differences, Lecture Notes in Mathematics 1153(1985), 453-457.MathSciNetCrossRefGoogle Scholar
  139. A. Zygmund, "Trigonometric Series I, II," Cambridge University Press, Cambridge, 1959.Google Scholar

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© 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

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