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Vector valued inequalities of Marcinkiewicz-Zygmund and Grothendieck type for Toeplitz forms

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Harmonic Analysis

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 992))

Abstract

The generalized Bochner-Herglotz theorem for generalized Toeplitz kernels (GTKs) [10] contains as special cases the solutions of several classical moment problems that, in turn, contain the germs of Grothendieck's theory of bilinear forms. In this paper some Grothendickian properties of the GTKs are studied, through the consideration of matrix-valued Hilbertian forms. Generalizations for GTKs of the Bochner-Eberlein-Horn theorems and of the vector-valued Marcinkiewicz-Zygmund and Grothendieck inequalities are given. Some applications to vector-valued weighted norm inequalities for the Hilbert transform and to Toeplitz and Hankel operators are outlined.

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References

  1. V. M. Adamjan, D. Z. Arov, and M. G. Krein, Infinite Hankel matrices and problems of Carathéodory and Fejér, Func. Anal. Appl., 2 (1968), 1–19.

    Article  Google Scholar 

  2. R. Arocena, M. Cotlar, and C. Sadosky, Weighted inequalities in L2 and lifting properties, Math. Anal. & Appl., Part A, Adv. in Math. Suppl. Studies, 7A (1981), 95–128.

    MathSciNet  MATH  Google Scholar 

  3. R. Arocena and M. Cotlar, Generalized Toeplitz kernels and moment problems of Adamjan-Arov-Krein, Integral Eq. and Operator Theory, 5 (1982), 37–55.

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  5. G. Bennet, Schur multipliers, Duke Math. J., 44 (1977), 603–639.

    Article  MathSciNet  MATH  Google Scholar 

  6. R. Blei, Uniformity property for Λ(2) sets and Grothendieck's inequality, Symp. Math., 22 (1977), 321–337.

    MathSciNet  Google Scholar 

  7. S. Bochner, Vorlesungen über Fouriersche Integrale, Akad. Verlagsgesellschaft, Leipszig, 1932. English transl., Ann. of Math. Studies, 42, Princeton University Press, Princeton, 1959.

    MATH  Google Scholar 

  8. S. Bochner, A theorem on Fourier-Stieltjes integrals, Bull. Amer. Math. Soc., 40 (1934), 271–276.

    Article  MathSciNet  MATH  Google Scholar 

  9. A. Córdoba and R. Fefferman, A weighted norm inequality for singular integrals, Studia Math., 57 (1976), 97–101.

    MathSciNet  MATH  Google Scholar 

  10. M. Cotlar and C. Sadosky, On the Helson-Szegö theorem and a related class of modified Toeplitz kernels, Proc. Symp. Pure Math. AMS, 35: I (1979), 383–407.

    Article  MathSciNet  MATH  Google Scholar 

  11. M. Cotlar and C. Sadosky, On some LP versions of the Helson-Szegö theorem, in Harmonic Analysis Conference in honor of Prof. A. Zygmund, Wadsworth Intl. Math. Series (1982), 306–317.

    Google Scholar 

  12. M. Cotlar and C. Sadosky, Majorized Toeplitz forms and weighted inequalities with general norms, in Harmonic Analysis (Ed.: F. Ricci & G. Weiss), Lecture Notes in Math. #908, Springer-Verlag (1982), 139–168.

    Google Scholar 

  13. M. Cotlar and C. Sadosky, Transformée de Hilbert, theorème de Bochner et le problème des moments, II, C.R. Acad. Sci. Paris, A, 285 (1977), 661–665.

    MathSciNet  MATH  Google Scholar 

  14. N. Dincoleanu, Vector Measures, Pergamon Press, New York, 1967.

    Book  Google Scholar 

  15. J. E. Gilbert, Harmonic analysis and the Grothendieck fundamental theorem, Symp. Math., 22 (1977), 393–420.

    MathSciNet  Google Scholar 

  16. J. E. Gilbert, Nikisin-Stein theory and factorization with fundamental theorem, Symp. Math., 22 (1977), 393–420.

    Google Scholar 

  17. J. E. Gilbert, Nikisin-Stein theory and factorization with applications, Proc. Symp. Pure Math. AMS, 35:II(1979), 233–267.

    Article  MathSciNet  MATH  Google Scholar 

  18. A. Grothendieck, Resumé de la théorie métrique des produits tensoriels topologiques, Bol. da Soc. Mat. São Paul, 8 (1956), 1–79.

    MATH  Google Scholar 

  19. R. A. Horn, Quadratic forms in Harmonic analysis and Bochner-Eberlein theorem. Proc. Amer. Math. Soc., 52 (1975), 263–270.

    Article  MathSciNet  MATH  Google Scholar 

  20. J. Marcinkiewicz and A. Zygmund, Quelques inégalités pour les operations linéaires, Fund. Math., 32 (1939), 115–121.

    MATH  Google Scholar 

  21. P. Masani, Propagators and dilatations, in Probability Theory in Vector Spaces (Ed.: A. Weron), Lecture Notes in Math. #656, Springer-Verlag (1977), 95–118.

    Google Scholar 

  22. Z. Nehari, On bounded bilinear forms, Ann. of Math., 65 (1957), 153–162.

    Article  MathSciNet  MATH  Google Scholar 

  23. J. L. Rubio de Francia, Weighted norm inequalities and vector valued inequalities, in Harmonic Analysis (Ed.: F. Ricci & G. Weiss), Lecture Notes in Math. #908, Springer-Verlag (1982), 86–101.

    Google Scholar 

  24. J. L. Rubio de Francia, Factorization and Ap weights, preprint.

    Google Scholar 

  25. C. Sadosky, Some applications of majorized Toeplitz kernels, Proc. Seminar Topics in Harmonic Anal. (Ed.: L. de Michele & F. Ricci), Milano, 1982. (In press.)

    Google Scholar 

  26. A. Weron, Prediction theory in Banach spaces, in Probability Winter School (Eds.: Z. Ciesielski, K. Urbanik, W. A. Woyczynski), Lecture Notes in Math. #472, Springer-Verlag (1975), 207–228.

    Google Scholar 

  27. A. Zygmund, Trigonometric Series, Second Edition, Cambridge University Press, Cambridge, 1959.

    MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Universidad Central de Venezuela, Venezuela

    Mischa Cotlar

  2. Howard University, USA

    Cora Sadosky

Authors
  1. Mischa Cotlar
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  2. Cora Sadosky
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Editor information

Giancarlo Mauceri Fulvio Ricci Guido Weiss

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© 1983 Springer-Verlag

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Cotlar, M., Sadosky, C. (1983). Vector valued inequalities of Marcinkiewicz-Zygmund and Grothendieck type for Toeplitz forms. In: Mauceri, G., Ricci, F., Weiss, G. (eds) Harmonic Analysis. Lecture Notes in Mathematics, vol 992. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0069164

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  • DOI: https://doi.org/10.1007/BFb0069164

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-12299-9

  • Online ISBN: 978-3-540-39885-1

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