Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

On Fourier multiplier transformations of Banach-valued functions


Author: Terry R. McConnell
Journal: Trans. Amer. Math. Soc. 285 (1984), 739-757
MSC: Primary 42B15; Secondary 46E40
MathSciNet review: 752501
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We obtain analogues of the Mihlin multiplier theorem and Littlewood-Paley inequalities for functions with values in a suitable Banach space $ B$. The requirement on $ B$ is that it have the unconditionality property for martingale difference sequences.


References [Enhancements On Off] (What's this?)

  • [1] A. Benedek, A.-P. Calderón, and R. Panzone, Convolution operators on Banach space valued functions, Proc. Nat. Acad. Sci. U.S.A. 48 (1962), 356–365. MR 0133653
  • [2] J. Bourgain, A generalization of a theorem of Benedek, Calderón, and Panzone, manuscript.
  • [3] J. Bourgain, Some remarks on Banach spaces in which martingale difference sequences are unconditional, Ark. Mat. 21 (1983), no. 2, 163–168. MR 727340, 10.1007/BF02384306
  • [4] D. L. Burkholder, Martingale transforms, Ann. Math. Statist. 37 (1966), 1494–1504. MR 0208647
  • [5] D. L. Burkholder, A geometrical characterization of Banach spaces in which martingale difference sequences are unconditional, Ann. Probab. 9 (1981), no. 6, 997–1011. MR 632972
  • [6] 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, Vol. I, II (Chicago, Ill., 1981) Wadsworth Math. Ser., Wadsworth, Belmont, CA, 1983, pp. 270–286. MR 730072
  • [7] J. L. Doob, Conditional Brownian motion and the boundary limits of harmonic functions, Bull. Soc. Math. France 85 (1957), 431–458. MR 0109961
  • [8] R. E. Edwards and G. I. Gaudry, Littlewood-Paley and multiplier theory, Springer-Verlag, Berlin-New York, 1977. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 90. MR 0618663
  • [9] Richard F. Gundy and Martin L. Silverstein, On a probabilistic interpretation for the Riesz transforms, Functional analysis in Markov processes (Katata/Kyoto, 1981) Lecture Notes in Math., vol. 923, Springer, Berlin-New York, 1982, pp. 199–203. MR 661625
  • [10] Richard F. Gundy and Nicolas Th. Varopoulos, Les transformations de Riesz et les intégrales stochastiques, C. R. Acad. Sci. Paris Sér. A-B 289 (1979), no. 1, A13–A16 (French, with English summary). MR 545671
  • [11] L. Hörmander, Estimates for translation invariant operators in $ {L^p}$ spaces, Acta Math. 104 (1960) 93-139.
  • [12] Kiyosi Itô and Henry P. McKean Jr., Diffusion processes and their sample paths, Springer-Verlag, Berlin-New York, 1974. Second printing, corrected; Die Grundlehren der mathematischen Wissenschaften, Band 125. MR 0345224
  • [13] S. Kwapień, Isomorphic characterizations of inner product spaces by orthogonal series with vector valued coefficients, Studia Math. 44 (1972), 583–595. Collection of articles honoring the completion by Antoni Zygmund of 50 years of scientific activity, VI. MR 0341039
  • [14] J. E. Littlewood and R. E. A. C. Paley, Theorems on Fourier series and power series. I, J. London Math. Soc. 6 (1931), 230-233.
  • [15] -, Theorems on Fourier series and power series. II, Proc. London Math. Soc. 42 (1936), 52-89.
  • [16] -, Theorems on Fourier series and power series. III, Proc. London Math. Soc. 43 (1937), 105-126.
  • [17] S. G. Mihlin, On the multipliers of Fourier integrals, Dokl. Akad. Nauk SSSR (N.S.) 109 (1956), 701–703 (Russian). MR 0080799
  • [18] J. Neveu, Discrete-parameter martingales, Revised edition, North-Holland Publishing Co., Amsterdam-Oxford; American Elsevier Publishing Co., Inc., New York, 1975. Translated from the French by T. P. Speed; North-Holland Mathematical Library, Vol. 10. MR 0402915
  • [19] Gian-Carlo Rota, An “Alternierende Verfahren” for general positive operators, Bull. Amer. Math. Soc. 68 (1962), 95–102. MR 0133847, 10.1090/S0002-9904-1962-10737-X
  • [20] R. Salem and A. Zygmund, On lacunary trigonometric series, Proc. Nat. Acad. Sci. U. S. A. 33 (1947), 333–338. MR 0022263
  • [21] M. J. Sharpe, Some transformations of diffusions by time reversal, Ann. Probab. 8 (1980), no. 6, 1157–1162. MR 602388
  • [22] Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095
  • [23] Elias M. Stein, Topics in harmonic analysis related to the Littlewood-Paley theory., Annals of Mathematics Studies, No. 63, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1970. MR 0252961
  • [24] Nicolas Th. Varopoulos, Aspects of probabilistic Littlewood-Paley theory, J. Funct. Anal. 38 (1980), no. 1, 25–60. MR 583240, 10.1016/0022-1236(80)90055-5

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 42B15, 46E40

Retrieve articles in all journals with MSC: 42B15, 46E40


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1984-0752501-X
Keywords: Fourier multiplier, martingale transform, $ {L^p}$ inequalities, vector-valued function, unconditionality
Article copyright: © Copyright 1984 American Mathematical Society