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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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On Fourier multiplier transformations of Banach-valued functions
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by Terry R. McConnell PDF
Trans. Amer. Math. Soc. 285 (1984), 739-757 Request permission

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
  • 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 133653, DOI 10.1073/pnas.48.3.356
    • J. Bourgain, A generalization of a theorem of Benedek, Calderón, and Panzone, manuscript.
  • J. Bourgain, Some remarks on Banach spaces in which martingale difference sequences are unconditional, Ark. Mat. 21 (1983), no. 2, 163–168. MR 727340, DOI 10.1007/BF02384306
  • D. L. Burkholder, Martingale transforms, Ann. Math. Statist. 37 (1966), 1494–1504. MR 208647, DOI 10.1214/aoms/1177699141
  • 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
  • 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
  • J. L. Doob, Conditional Brownian motion and the boundary limits of harmonic functions, Bull. Soc. Math. France 85 (1957), 431–458. MR 109961
  • R. E. Edwards and G. I. Gaudry, Littlewood-Paley and multiplier theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 90, Springer-Verlag, Berlin-New York, 1977. MR 0618663
  • 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
  • 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
    • L. Hörmander, Estimates for translation invariant operators in ${L^p}$ spaces, Acta Math. 104 (1960) 93-139.
  • Kiyosi Itô and Henry P. McKean Jr., Diffusion processes and their sample paths, Die Grundlehren der mathematischen Wissenschaften, Band 125, Springer-Verlag, Berlin-New York, 1974. Second printing, corrected. MR 0345224
  • S. Kwapień, Isomorphic characterizations of inner product spaces by orthogonal series with vector valued coefficients, Studia Math. 44 (1972), 583–595. MR 341039, DOI 10.4064/sm-44-6-583-595
    • 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. —, Theorems on Fourier series and power series. II, Proc. London Math. Soc. 42 (1936), 52-89. —, Theorems on Fourier series and power series. III, Proc. London Math. Soc. 43 (1937), 105-126.
  • S. G. Mihlin, On the multipliers of Fourier integrals, Dokl. Akad. Nauk SSSR (N.S.) 109 (1956), 701–703 (Russian). MR 0080799
  • J. Neveu, Discrete-parameter martingales, Revised edition, North-Holland Mathematical Library, Vol. 10, North-Holland Publishing Co., Amsterdam-Oxford; American Elsevier Publishing Co., Inc., New York, 1975. Translated from the French by T. P. Speed. MR 0402915
  • Gian-Carlo Rota, An “Alternierende Verfahren” for general positive operators, Bull. Amer. Math. Soc. 68 (1962), 95–102. MR 133847, DOI 10.1090/S0002-9904-1962-10737-X
  • R. Salem and A. Zygmund, On lacunary trigonometric series, Proc. Nat. Acad. Sci. U.S.A. 33 (1947), 333–338. MR 22263, DOI 10.1073/pnas.33.11.333
  • M. J. Sharpe, Some transformations of diffusions by time reversal, Ann. Probab. 8 (1980), no. 6, 1157–1162. MR 602388
  • Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095
  • 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
  • Nicolas Th. Varopoulos, Aspects of probabilistic Littlewood-Paley theory, J. Functional Analysis 38 (1980), no. 1, 25–60. MR 583240, DOI 10.1016/0022-1236(80)90055-5
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Additional Information
  • © Copyright 1984 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 285 (1984), 739-757
  • MSC: Primary 42B15; Secondary 46E40
  • DOI: https://doi.org/10.1090/S0002-9947-1984-0752501-X
  • MathSciNet review: 752501