Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
D. J. Aldous, Unconditional bases and martingales in Lp(F), Math. Proc. Cambridge Phil. Soc. 85 (1979), 117–123.
A. Benedek, A. P. Calderón, and R. Panzone, Convolution operators on Banach space valued functions, Proc. Nat. Acad. Sci. 48 (1962), 356–365.
E. Berkson, T. A. Gillespie, and P. S. Muhly, Théorie spectrale dans les espaces UMD, C. R. Acad. Sc. Paris (1986).
O. Blasco, Espacios de Hardy de funciones con valores vectoriales, Publicaciones del Seminario Matematico, Universidad de Zaragoza, 1985.
S. Bochner and A. E. Taylor, Linear functionals on certain spaces of abstractly-valued functions, Ann. Math. 39 (1938), 913–944.
J. Bourgain, Some remarks on Banach spaces in which martingale difference sequences are unconditional, Ark. Mat. 21 (1983), 163–168.
J. Bourgain, Extension of a result of Benedek, Calderón, and Panzone, Ark. Mat. 22 (1984), 91–95.
J. Bourgain, On martigale transforms in finite dimensional lattices with an appendix on the K-convexity constant, Math. Nachr. 119 (1984), 41–53.
J. Bourgain, Vector valued singular integrals and the H1 — BMO duality, Probability Theory and Harmonic Analysis, J. A. Chao and W. A. Woyczynski, editors, Marcel Dekker, New York (1986), 1–19.
J. Bourgain and W. J. Davis, Martingale transforms and complex uniform convexity, Trans. Amer. Math. Soc.
D. L. Burkholder, Martingale transforms, Ann. Math. Statist. 37 (1966), 1494–1504.
D. L. Burkholder, Distribution function inequalities for martingales, Ann. Probab. 1 (1973), 19–42.
D. L. Burkholder, A geometrical characterization of Banach spaces in which martingale difference sequences are unconditional, Ann. Probab. 9 (1981), 997–1011.
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 (1981), 35–50.
D. L. Burkholder, A geometric condition that implies the existence of certain singular integrals of Banach-space-valued functions, Conference on Harmonic Analysis in Honor of Antoni Zygmund, University of Chicago, 1981, Wadsworth International Group, Belmont, California, 1 (1983), 270–286.
D. L. Burkholder, Boundary value problems and sharp inequalities for martingale transforms, Ann. Probab. 12 (1984), 647–702.
D. L. Burkholder, An elementary proof of an inequality of R. E. A. C. Paley, Bull. London Math. Soc. 17 (1985), 474–478.
D. L. Burkholder, An extension of a classical martingale inequality, Probability Theory and Harmonic Analysis, J. A. Chao and W. A. Woyczynski, editors, Marcel Dekker, New York (1986), 21–30.
D. L. Burkholder, A sharp and strict Lp-inequality for stochastic integrals, Ann. Probab. 14 (1986).
D. L. Burkholder and R. F. Gundy, Extrapolation and interpolation of quasilinear operators on martingales, Acta Math. 124 (1970), 249–304.
D. L. Burkholder, R. F. Gundy, and M. L. Silverstein, A maximal function characterization of the class Hp, Trans. Amer. Math. Soc. 157 (1971), 137–153.
A. P. Calderón and A. Zygmund, On the existence of certain singular integrals, Acta Math. 88 (1952), 85–139.
A. P. Calderón and A. Zygmund, On singular integrals, Amer. J. of Math. 78 (1956), 289–309.
M. L. Cartwright, Manuscripts of Hardy, Littlewood, Marcel Riesz and Titchmarsh, Bull. London Math. Soc. 14 (1982), 472–532.
C. Dellacherie and P.-A. Meyer, Probabilités et potentiel: Théorie des martingales, Hermann, Paris, 1980.
J. Diestel and J. J. Uhl, Vector Measures, Math. Surveys 15, American Mathematical Society, Providence, Rhode Island, 1977.
J. L. Doob, Stochastic Processes, Wiley, New York, 1953.
L. E. Dor and E. Odell, Monotone bases in Lp, Pacific J. Math. 60 (1975), 51–61.
P. Enflo, Banach spaces which can be given an equivalent uniformly convex norm, Israel J. Math. 13 (1972), 281–288.
C. Fefferman and E. M. Stein, Hp spaces of several variables, Acta Math. 129 (1972), 137–193.
L. Gårding, Marcel Riesz in memoriam, Acta Math. 127 (1970), i–xi.
D. J. H. Garling, Brownian motion and UMD-spaces, Conference on Probability and Banach Spaces, Zaragoza, 1985, Lecture Notes in Mathematics.
J. A. Gutiérrez, On the Boundedness of the Banach Space-Valued Hilbert Transform, Ph.D. Dissertation, University of Texas, Austin, 1982.
G. H. Hardy and J. E. Littlewood, A maximal theorem with function-theoretic applications, Acta Math. 54 (1930), 81–116.
S. Kwapień, Isomorphic characterizations of inner product spaces by orthogonal series with vector valued coefficients, Studia Math. 44 (1972), 583–595.
J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces I: Sequence Spaces, Springer, New York, 1977.
B. Maurey, Système de Haar, Séminaire Maurey-Schwartz (1974–75), École Polytechnique, Paris, 1975.
T. R. McConnell, On Fourier multiplier transformations of Banach-valued functions, Trans. Amer. Math. Soc. 285 (1984), 739–757.
T. R. McConnell, A Skorohod-like representation in infinite dimensions, Probability in Banach Spaces V, Tufts University, 1984, Lecture Notes in Mathematics, 1153 (1985), 359–368.
A. Pelczyński and H. P. Rosenthal, Localization techniques in Lp spaces, Studia Math. 52 (1975), 263–289.
G. Pisier, Un exemple concernant la super-réflexivité, Séminaire Maurey-Schwartz (1974–75), École Polytechnique, Paris, 1975.
M. Riesz, Les fonctions conjuguées et les séries de Fourier, C. R. Acad. Sci. Paris, 178 (1924), 1464–1467.
M. Riesz, Sur les fonctions conjuguées, Math. Z. 27 (1927), 218–244.
J. L. Rubio de Francia, Fourier series and Hilbert transforms with values in UMD Banach spaces, Studia Math. 81 (1985), 95–105.
J. L. Rubio de Francia, F. J. Ruiz, and J. L. Torrea, Calderón-Zygmund theory for operator-valued kernels, Advances in Math.
J. Schwartz, A remark on inequalities of Galderón-Zygmund type for vector-valued functions, Comm. Pure Appl. Math. 14 (1961), 785–799.
E. M. Stein, Singular Integrals and Differentiability Properties of Functions Princeton, New Jersey, Princeton University Press, 1970.
S. Vági, A remark on Plancherel's theorem for Banach space valued functions, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 23 (1969), 305–315.
B. Virot, Quelques inégalitiés concernant les transformées de Hilbert des fonctions à valeurs vectorielles, C. R. Acad. Sc. Paris, 293 (1981), 459–462.
A. Zygmund, Trigonometric Series I, II, New York, Cambridge University Press, 1959.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1986 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Burkholder, D.L. (1986). Martingales and Fourier analysis in Banach spaces. In: Letta, G., Pratelli, M. (eds) Probability and Analysis. Lecture Notes in Mathematics, vol 1206. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0076300
Download citation
DOI: https://doi.org/10.1007/BFb0076300
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-16787-7
Online ISBN: 978-3-540-40955-7
eBook Packages: Springer Book Archive