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Explorations in martingale theory and its applications

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References

  • M. Abramowitz and I. A. Stegun, editors, “Handbook of mathematical functions,” Dover, New York, 1970.

    Google Scholar 

  • D. J. Aldous, Unconditional bases and martingales in L p(F), Math. Proc. Cambridge Phil. Soc. 85 (1979), 117–123.

    Article  MathSciNet  MATH  Google Scholar 

  • T. Ando, Contractive projections in L p spaces, Pacific J. Math. 17 (1966), 391–405.

    Article  MathSciNet  MATH  Google Scholar 

  • A. Baernstein, Some sharp inequalities for conjugate functions, Indiana Univ. Math. J. 27 (1978), 833–852.

    Article  MathSciNet  MATH  Google Scholar 

  • R. Bañuelos, A sharp good-γ inequality with an application to Riesz transforms, Mich. Math. J. 35 (1988), 117–125.

    Article  MathSciNet  MATH  Google Scholar 

  • 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.

    Article  MathSciNet  Google Scholar 

  • A. Benedek, A. P. Calderón, and R. Panzone, Convolution operators on Banach space valued functions, Proc. Nat. Acad. Sci. 48 (1962), 356–365.

    Article  MathSciNet  MATH  Google Scholar 

  • E. Berkson, T. A. Gillespie, and P. S. Muhly, Théorie spectrale dans les espaces UMD, C. R. Acad. Sci. Paris 302 (1986), 155–158.

    MathSciNet  MATH  Google Scholar 

  • ___, Generalized analyticity in UMD spaces, Arkiv för Math. 27 (1989), 1–14.

    Article  MathSciNet  MATH  Google Scholar 

  • K. Bichteler, Stochastic integration and L p-theory of semimartingales, Ann. Prob. 9 (1981), 49–89.

    Article  MathSciNet  MATH  Google Scholar 

  • O. Blasco, Hardy spaces of vector-valued functions: Duality, Trans. Amer. Math. Soc. 308 (1988a), 495–507.

    Article  MathSciNet  MATH  Google Scholar 

  • ___, Boundary values of functions in vector-valued Hardy spaces and geometry on Banach spaces, J. Funct. Anal. 78 (1988b), 346–364.

    Article  MathSciNet  MATH  Google Scholar 

  • G. Blower, A multiplier characterization of analytic UMD spaces, Studia Math. 96 (1990), 117–124.

    MathSciNet  MATH  Google Scholar 

  • B. Bollobás, Martingale inequalities, Math. Proc. Cambridge Phil. Soc. 87 (1980), 377–382.

    Article  MathSciNet  MATH  Google Scholar 

  • ___, editor, “Littlewood's Miscellany,” Cambridge University Press, Cambridge, 1986.

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  • ___, Extension of a result of Benedek, Calderón, and Panzone, Ark. Mat. 22 (1984), 91–95.

    Article  MathSciNet  MATH  Google Scholar 

  • ___, Vector valued singular integrals and the H 1-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 

  • A. V. Bukhvalov, Hardy spaces of vector-valued functions, J. Sov. Math. 16 (1981), 1051–1059.

    Article  MATH  Google Scholar 

  • _____, Continuity of operators in spaces of vector functions, with applications to the theory of bases, J. Sov. Math. 44 (1989), 749–762.

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  • _____, Distribution function inequalities for martingales, Ann. Prob. 1 (1973), 19–42.

    Article  MathSciNet  MATH  Google Scholar 

  • _____, Exit times of Brownian motion, harmonic majorization, and Hardy spaces, Advances in Math. 26 (1977), 182–205.

    Article  MathSciNet  MATH  Google Scholar 

  • _____, A sharp inequality for martingale transforms, Ann. Prob. 7 (1979), 858–863.

    Article  MathSciNet  MATH  Google Scholar 

  • _____, A geometrical characterization of Banach spaces in which martingale difference sequences are unconditional, Ann. Prob. 9 (1981a), 997–1011.

    Article  MathSciNet  MATH  Google Scholar 

  • _____, 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.

    Article  MathSciNet  MATH  Google Scholar 

  • _____, 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  MathSciNet  MATH  Google Scholar 

  • _____, A geometric condition that imples 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 

  • _____, Boundary value problems and sharp inequalities for martingale transforms, Ann. Prob. 12 (1984), 647–702.

    Article  MathSciNet  MATH  Google Scholar 

  • _____, An elementary proof of an inequality of R. E. A. C. Paley, Bull. London Math. Soc. 17 (1985), 474–478.

    Article  MathSciNet  MATH  Google Scholar 

  • _____, 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 

  • , Martingales and Fourier analysis in Banach spaces, C.I.M.E. Lectures, Varenna (Como), Italy, 1985, Lecture Notes in Mathematics 1206 (1986b), 61–108.

    Article  MathSciNet  MATH  Google Scholar 

  • _____, A sharp and strict L p-inequality for stochastic integrals, Ann. Prob. 15 (1987), 268–273.

    Article  MathSciNet  MATH  Google Scholar 

  • _____, A proof of Pelczyński's conjecture for the Haar system, Studia Math. 91 (1988a), 79–83.

    MathSciNet  MATH  Google Scholar 

  • _____, Sharp inequalities for martingales and stochastic integrals, Colloque Paul Lévy, Palaiseau, 1987, Astérisque 157–158 (1988b), 75–94.

    MathSciNet  MATH  Google Scholar 

  • _____, Differential subordination of harmonic functions and martingales, Harmonic Analysis and Partial Differential Equations (El Escorial, 1987), Lecture Notes in Mathematics 1384 (1989a), 1–23.

    Article  MathSciNet  MATH  Google Scholar 

  • _____, 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 

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

    Article  MathSciNet  MATH  Google Scholar 

  • A. P. Calderón and A. Zygmund, On the existence of certain singular integrals, Acta Math. 88 (1952), 85–139.

    Article  MathSciNet  MATH  Google Scholar 

  • _____, On singular integrals, Amer. J. Math. 78 (1956), 289–309.

    Article  MathSciNet  MATH  Google Scholar 

  • S. D. Chatterji, Les martingales et leurs applications analytiques, Lecture Notes in Mathematics 307 (1973), 27–164.

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  • F. Cobos, Some spaces in which martingale difference sequences are unconditional, Bull. Polish Acad. of Sci. Math. 34 (1986), 695–703.

    MathSciNet  MATH  Google Scholar 

  • ___, Duality, UMD-property and Lorentz-Marcinkiewicz operator spaces, in “16 Colóquio Brasileiro de Matemática,” Rio de Janeiro, 1988, pp. 97–106.

    Google Scholar 

  • F. Cobos and D. L. Fernandez, Hardy-Sobolev spaces and Besov spaces with a function parameter, Lecture Notes in Mathematics 1302 (1988), 158–170.

    Article  MathSciNet  MATH  Google Scholar 

  • 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.

    Google Scholar 

  • 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.

    MathSciNet  MATH  Google Scholar 

  • D. C. Cox and R. P. Kertz, Common strict character of some sharp infinite-sequence martingale inequalities, Stochastic Process. Appl. 20 (1985), 169–179.

    Article  MathSciNet  MATH  Google Scholar 

  • B. Davis, A comparison test for martingale inequalities, Ann. Math. Statist. 40 (1969), 505–508.

    Article  MathSciNet  MATH  Google Scholar 

  • ___, On the weak type (1,1) inequality for conjugate functions, Proc. Amer. Math. Soc. 44 (1974), 307–311.

    MathSciNet  MATH  Google Scholar 

  • ___, On the L p norms of stochastic integrals and other martingales, Duke Math. J. 43 (1976), 697–704.

    Article  MathSciNet  MATH  Google Scholar 

  • M. Defant, On the vector-valued Hilbert transform, Math. Nachr. 141 (1989), 251–265.

    Article  MathSciNet  MATH  Google Scholar 

  • C. Dellacherie and P. A. Meyer, “Probabilités et potentiel: théorie des martingales,” Hermann, Paris, 1980.

    MATH  Google Scholar 

  • J. Diestel and J. J. Uhl, “Vector Measures,” Math. Surveys 15, American Mathematical Society, Providence, Rhode Island, 1977.

    MATH  Google Scholar 

  • C. Doléans, Variation quadratique des martingales continues à droite, Ann. Math. Statist. 40 (1969), 284–289.

    Article  MathSciNet  MATH  Google Scholar 

  • J. L. Doob, “Stochastic Processes,” Wiley, New York, 1953.

    MATH  Google Scholar 

  • ____, Remarks on the boundary limits of harmonic functions, J. SIAM Numer. Anal. 3 (1966), 229–235.

    Article  MathSciNet  MATH  Google Scholar 

  • ____, “Classical Potential Theory and Its Probabilistic Counterpart,” Springer, New York, 1984.

    Book  MATH  Google Scholar 

  • L. E. Dor and E. Odell, Monotone bases in L p, Pacific J. Math. 60 (1975), 51–61.

    Article  MathSciNet  MATH  Google Scholar 

  • G. Dore and A. Venni, On the closedness of the sum of two closed operators, Math. Z. 196 (1987), 189–201.

    Article  MathSciNet  MATH  Google Scholar 

  • I. Doust, Contractive projections on Banach spaces, Proc. Centre for Math. Anal., Australian National University 20 (1988), 50–58.

    MathSciNet  MATH  Google Scholar 

  • ___, Well-bounded and scalar-type spectral operators on L p spaces, J. London Math. Soc. 39 (1989), 525–534.

    Article  MathSciNet  MATH  Google Scholar 

  • L. E. Dubins, Rises and upcrossings of nonnegative martingales, Illinois J. Math. 6 (1962), 226–241.

    MathSciNet  MATH  Google Scholar 

  • P. Enflo, Banach spaces which can be given an equivalent uniformly convex norm, Israel J. Math. 13 (1972), 281–288.

    Article  MathSciNet  Google Scholar 

  • M. Essén, A superharmonic proof of the M. Riesz conjugate function theorem, Ark. Math. 22 (1984), 241–249.

    Article  MathSciNet  MATH  Google Scholar 

  • D. L. Fernandez, Vector-valued singular integral operators on L p-spaces with mixed norms and applications, Pacific J. Math. 129 (1987), 257–275.

    Article  MathSciNet  MATH  Google Scholar 

  • _____, On Fourier multipliers of Banach-lattice valued functions, Rev. Roumaine Math. Pures Appl. 34 (1989), 635–642.

    MathSciNet  MATH  Google Scholar 

  • 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 

  • T. Figiel, On equivalence of some bases to the Haar system in spaces of vector-valued functions, Bull. Polon. Acad. Sci. 36 (1988), 119–131.

    MathSciNet  MATH  Google Scholar 

  • ___, 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 

  • T. W. Gamelin, “Uniform Algebras and Jensen Measures,” Cambridge University Press, London, 1978.

    MATH  Google Scholar 

  • 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.

    Article  MathSciNet  MATH  Google Scholar 

  • 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 

  • D. Gilat, The best bound in the L log L inequality of Hardy and Littlewood and its martingale counterpart, Proc. Amer. Math. Soc. 97 (1986), 429–436.

    MathSciNet  MATH  Google Scholar 

  • 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.

    Chapter  Google Scholar 

  • ___, Complex powers of operators and UMD spaces, manuscript.

    Google Scholar 

  • R. F. Gundy, “Some Topics in Probability and Analysis,” Regional Conference Series in Mathematics 70, American Mathematical Society, Providence, Rhode Island, 1989.

    Book  MATH  Google Scholar 

  • U. Haagerup, The best constants in the Khintchine inequality, Studia Math. 70 (1982), 231–283.

    MathSciNet  MATH  Google Scholar 

  • 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.

    Article  MathSciNet  MATH  Google Scholar 

  • G. H. Hardy, J. E. Littlewood, and G. Pólya, “Inequalities,” Cambridge University Press, Cambridge, 1934.

    MATH  Google Scholar 

  • W. Hensgen, On complementation of vector-valued Hardy spaces, Proc. Amer. Math. Soc. 104 (1988), 1153–1162.

    Article  MathSciNet  MATH  Google Scholar 

  • ____, On the dual space of H p(X), 1<p<∞, J. Funct. Anal. 92 (1990), 348–371.

    Article  MathSciNet  MATH  Google Scholar 

  • P. Hitczenko, Comparison of moments for tangent sequences of random variables, Probab. Th. Rel. Fields 78 (1988), 223–230.

    Article  MathSciNet  MATH  Google Scholar 

  • ____, On tangent sequences of UMD-space valued random vectors, manuscript.

    Google Scholar 

  • ____, Upper bounds for the L p-norms of martingales, Probab. Th. Rel. Fields 86 (1990), 225–238.

    Article  MathSciNet  MATH  Google Scholar 

  • ____, Best constants in martingale version of Rosenthal's inequality, Ann. Probab. 18 (1990), 1656–1668.

    Article  MathSciNet  MATH  Google Scholar 

  • R. C. James, Some self dual properties of normed linear spaces, Ann. Math. Studies 69 (1972a), 159–175.

    MathSciNet  Google Scholar 

  • ____, Super-reflexive spaces with bases, Pacific J. Math. 41 (1972b), 409–419.

    Article  MathSciNet  MATH  Google Scholar 

  • ____, Super-reflexive Banach spaces, Can. J. Math. 24 (1972c), 896–904.

    Article  MathSciNet  MATH  Google Scholar 

  • W. B. Johnson and G. Schechtman, Martingale inequalities in rearrangement invariant function spaces, Israel J. Math. 64 (1988), 267–275.

    Article  MathSciNet  MATH  Google Scholar 

  • N. J. Kalton, Differentials of complex interpolation processes for Köthe function spaces, a paper delivered at the Conference on Function Spaces (Auburn University, 1989).

    Google Scholar 

  • G. Klincsek, A square function inequality, Ann. Prob. 5 (1977), 823–825.

    Article  MathSciNet  MATH  Google Scholar 

  • A. N. Kolmogorov, Sur les fonctions harmoniques conjuguées et les séries de Fourier, Fund. Math. 7 (1925), 24–29.

    MATH  Google Scholar 

  • 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 

  • H. Kunita, Stochastic integrals based on martingales taking values in Hilbert space, Nagoya Math. J. 38 (1970), 41–52.

    MathSciNet  MATH  Google Scholar 

  • S. Kwapień, Isomorphic characterizations of inner product spaces by orthogonal series with vector valued coefficients, Studia Math. 44 (1972), 583–595.

    MathSciNet  MATH  Google Scholar 

  • S. Kwapień 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 

  • J. Lindenstrauss and A. Pelczyński, Contributions to the theory of the classical Banach spaces, J. Funct. Anal. 8 (1971), 225–249.

    Article  MathSciNet  MATH  Google Scholar 

  • J. Lindenstrauss and L. Tzafriri, “Classical Banach Spaces I: Sequence Spaces,” Springer, New York, 1977.

    Book  MATH  Google Scholar 

  • ______, “Classical Banach Spaces II: Function Spaces,” Springer, New York, 1979.

    Book  MATH  Google Scholar 

  • A. Mandelbaum, L. A. Shepp, and R. Vanderbei, Optimal switching between a pair of Brownian motions, Ann. Prob. 18 (1990), 1010–1033.

    Article  MathSciNet  MATH  Google Scholar 

  • J. Marcinkiewicz, Quelques théorèmes sur les séries orthogonales, Ann. Soc. Polon. Math. 16 (1937), 84–96.

    MATH  Google Scholar 

  • B. Maurey, Système de Haar, in “Séminaire Maurey-Schwartz, 1974–1975,” École Polytechnique, Paris, 1975.

    Google Scholar 

  • T. R. McConnell, On Fourier multiplier transformations of Banach-valued functions, Trans. Amer. Math. Soc. 285 (1984), 739–757.

    Article  MathSciNet  MATH  Google Scholar 

  • _____, A Skorohod-like representation in infinite dimensions, Probability in Banach Spaces V, Lecture Notes in Mathematics 1153 (1985), 359–368.

    Article  MathSciNet  MATH  Google Scholar 

  • _____, Decoupling and stochastic integration in UMD Banach spaces, Prob. Math. Stat. 10 (1989), 283–295.

    MathSciNet  MATH  Google Scholar 

  • H. P. McKean, Geometry of differential space, Ann. Prob. 1 (1973), 197–206.

    Article  MathSciNet  MATH  Google Scholar 

  • I. Monroe, Martingale operator norms and local times, manuscript.

    Google Scholar 

  • A. A. Novikov, On stopping times for the Wiener process, (Russian, English summary), Teor. Verojatnost. i Primenen 16 (1971), 458–465.

    MathSciNet  Google Scholar 

  • A. M. Olevskii, Fourier series and Lebesgue functions, (Russian), Uspehi Mat. Nauk 22 (1967), 237–239.

    MathSciNet  Google Scholar 

  • _____A. M. Olevskii, “Fourier Series with Respect to General Orthogonal Systems,” Springer, New York, 1975.

    Book  Google Scholar 

  • R. E. A. C. Paley, A remarkable series of orthogonal functions I., Proc. London Math. Soc. 34 (1932), 241–264.

    Article  MathSciNet  MATH  Google Scholar 

  • A. Pełczyński, 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.

    MATH  Google Scholar 

  • ___, Norms of classical operators in function spaces, Colloque Laurent Schwartz, Astérisque 131 (1985), 137–162.

    MathSciNet  MATH  Google Scholar 

  • A. Pełczyński and H. Rosenthal, Localization techniques in L p spaces, Studia Math. 52 (1975), 263–289.

    MATH  Google Scholar 

  • 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.

    MathSciNet  MATH  Google Scholar 

  • G. Pisier, Un exemple concernant la super-réflexivité, in “Séminaire Maurey-Schwartz, 1974–75,” École Polytechnique, Paris, 1975a.

    Google Scholar 

  • ____, Martingales with values in uniformly convex spaces, Israel J. Math. 20 (1975b), 326–350.

    Article  MathSciNet  MATH  Google Scholar 

  • A. O. Pittenger, Note on a square function inequality, Ann. Prob. 7 (1979), 907–908.

    Article  MathSciNet  MATH  Google Scholar 

  • M. Riesz, Sur les fonctions conjuguées, Math. Z. 27 (1927), 218–244.

    Article  MathSciNet  MATH  Google Scholar 

  • 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.

    Article  MathSciNet  Google Scholar 

  • J. L. Rubio de Francia and J. L. Torrea, Some Banach techniques in vector valued Fourier analysis, Colloq. Math. 54 (1987), 271–284.

    MathSciNet  MATH  Google Scholar 

  • J. L. Rubio de Francia, F. J. Ruiz, and J. L. Torrea, Calderón-Zygmund theory for operator-valued kernels, Advances in Math. 62 (1986), 7–48.

    Article  MathSciNet  MATH  Google Scholar 

  • J. Schwartz, A remark on inequalities of Calderón-Zygmund type for vector-valued functions, Comm. Pure Appl. Math. 14 (1961), 785–799.

    Article  MathSciNet  MATH  Google Scholar 

  • L. A. Shepp, A first passage problem for the Wiener process, Ann. Math. Statist. 38 (1967), 1912–1914.

    Article  MathSciNet  MATH  Google Scholar 

  • E. M. Stein, “Singular Integrals and Differentiability Properties of Functions,” Princeton University Press, Princeton, 1970.

    MATH  Google Scholar 

  • E. M. Stein and G. Weiss, On the theory of harmonic functions of several variables: I. The theory of H p-spaces, Acta Math. 103 (1960), 25–62.

    Article  MathSciNet  MATH  Google Scholar 

  • S. J. Szarek, On the best constants in the Khinchin inequality, Studia Math. 58 (1976), 197–208.

    MathSciNet  MATH  Google Scholar 

  • B. Tomaszewski, Sharp weak-type inequalities for analytic functions on the unit disc, Bull. London Math. Soc. 18 (1986), 355–358.

    Article  MathSciNet  MATH  Google Scholar 

  • L. Tzafriri, Remarks on contractive projections in L p-spaces, Israel J. Math. 7 (1969), 9–15.

    Article  MathSciNet  MATH  Google Scholar 

  • G. Wang, “Some Sharp Inequalities for Conditionally Symmetric Martingales,” doctoral thesis, University of Illinois, Urbana, Illinois, 1989.

    Google Scholar 

  • ____, Sharp square-function inequalities for conditionally symmetric martingales, Trans. Amer. Math. Soc. (to appear).

    Google Scholar 

  • ____, Sharp maximal inequalities for conditionally symmetric martingales and Brownian motion, Proc. Amer. Math. Soc. (to appear).

    Google Scholar 

  • ____, Sharp inequalities for the conditional square function of a martingale, Ann. Prob. (to appear).

    Google Scholar 

  • T. M. Wolniewicz, The Hilbert transform in weighted spaces of integrable vector-valued functions, Colloq. Math. 53 (1987), 103–108.

    MathSciNet  MATH  Google Scholar 

  • M. Yor, Sur les intégrales stochastique à valeurs dans un Banach, C. R. Acad. Sci. Paris 277 (1973), 467–469.

    MATH  Google Scholar 

  • F. Zimmermann, On vector-valued Fourier multiplier theorems, Studia Math. 93 (1989), 201–222.

    MathSciNet  MATH  Google Scholar 

  • J. Zinn, Comparison of martingale differences, Lecture Notes in Mathematics 1153 (1985), 453–457.

    Article  MathSciNet  Google Scholar 

  • A. Zygmund, “Trigonometric Series I, II,” Cambridge University Press, Cambridge, 1959.

    MATH  Google Scholar 

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Burkholder, D.L. (1991). Explorations in martingale theory and its applications. In: Hennequin, PL. (eds) Ecole d'Eté de Probabilités de Saint-Flour XIX — 1989. Lecture Notes in Mathematics, vol 1464. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0085167

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