Operator theory in harmonic analysis

  • Bernard Russo
Conference paper
Part of the Lecture Notes in Mathematics book series (LNM, volume 604)


The purpose of this paper is to give an illustration of how operator theoretic results in Hilbert space can be applied to obtain results in classical and abstract harmonic analysis. An inequality for integral operators will be used to give new proofs for the classical Hausdorff-Young Theorems on the unit circle (Fourier coefficients) and on the real line (Fourier integral). These proofs were discovered in the course of the author's investigations into Hausdorff-Young phenomena on non-commutative non-compact groups. The proof for the unit circle is short and will be given in §1. The proof for the real line is more involved and will be given in §2. In §3 a brief summary of the evolution of abstract harmonic analysis is given which includes the statement of the Hausdorff-Young Theorem for unimodular groups. In §4 the question of sharpness in the Hausdorff-Young Theorem in various settings is discussed and recent results of W. Beckner, J. Fournier and the author are described.


Fourier Coefficient Convolution Operator Compact Abelian Group Good Constant Fourier Integral 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. I.
    F. Riesz, Uber orthogonale Funktionensysteme, Nachr. Akad. Wiss. Gottingen Math. Phys. Kl. 1907, 116–122.Google Scholar
  2. II.
    E. Fischer, Sur la convergence en moyenne, C.R. Acad. Sci. Paris 144 (1907) 1022–1024.zbMATHGoogle Scholar
  3. III.
    W. H. Young, On the multiplication of successions of Fourier constants, Proc. Roy. Soc. London, Ser. A 87(1912) 331–339CrossRefzbMATHGoogle Scholar
  4. IV.
    W. H. Young, on The determination of the summability of a function by means of its Fourier constants, Proc. Lon. Math. Soc. (2) 12(1913) 71–88.MathSciNetCrossRefzbMATHGoogle Scholar
  5. V.
    F. Hausdorff, Eine Ausdehuung des Parsevalschen Satzes uber Fourier-reihen, Math. Z. 16(1923) 163–169.MathSciNetCrossRefzbMATHGoogle Scholar
  6. VI.
    M. Riesz, Sur les maxima des formes bilineaires et sur les fonctionnelles lineaires, Acta Math. 49(1926) 465–497.MathSciNetCrossRefzbMATHGoogle Scholar
  7. VII.
    M. Plancherel, Contribution a letude de la representation d une fonction arbitraire par les integrales definies, Rend. Circ. Mat. Palermo 30(1910) 289–335.CrossRefzbMATHGoogle Scholar
  8. VIII.
    E. C. Titchmarsh, A contribution to the theory of Fourier transforms, Proc. London Math. Soc. (2) 23(1924) 279–289.MathSciNetzbMATHGoogle Scholar
  9. 1.
    K. I. Babenko, An inequality in the theory of Fourier integrals, Izv. Akad. Nauk, SSSR Ser. Mat. 25(1961) 531–542. Engl. Transl. AMS Transl. (2) 44 115–128.MathSciNetzbMATHGoogle Scholar
  10. 2.
    W. Beckner, Inequalities in Fourier analysis, Ann. of Math. 102(1975) 159–182.MathSciNetCrossRefzbMATHGoogle Scholar
  11. 3.
    J. Dixmier, "Les algebres d'operateurs dan l'espace Hilbertien", Gauthier Villars, Paris 1957.zbMATHGoogle Scholar
  12. 4.
    J. Dixmier, "Les C*-algebres et leurs representations", Gauthier Villars, Paris 1964.zbMATHGoogle Scholar
  13. 5.
    J. J. F. Fournier, Sharpness in Young's inequality for convolution, (to appear)Google Scholar
  14. 6.
    J. J. F. Fournier and B. Russo, Abstract interpolation and operator valued kernels, (to appear)Google Scholar
  15. 7.
    E. Hewitt and K. A. Ross, "Abstract Harmonic analysis", Springer-Verlag I (1963), II (1970).Google Scholar
  16. 8.
    A. Klein and B. Russo, Young's inequality for semi-direct products, in preparation.Google Scholar
  17. 9.
    R. A. Kunze, Lp-Fourier transforms on locally compact unimodular groups, Trans. Amer. Math. Soc. 89(1958) 519–540.MathSciNetzbMATHGoogle Scholar
  18. 10.
    B. Russo, The norm of the Lp-Fourier transform on unimodular groups, Trans. Amer. Math. Soc. 192(1974) 293–305.MathSciNetzbMATHGoogle Scholar
  19. 11.
    _____, The norm of the Lp-Fourier transform. II, Can. J. Math 28(1976)Google Scholar
  20. 12.
    _____, On the Hausdorff-Young Theorem for integral operators, Pacific. J. Math. (to appear)Google Scholar
  21. 13.
    _____, The norm of the Lp-Fourier transform III, (compact extensions), to appear.Google Scholar
  22. 14.
    A. Weil, L'integration dans les groupes topologiques et ses applications, Actualities Sci. et Ind. 869, 1145 Paris: Hermann & Cie. 1941 and 1951.Google Scholar

Copyright information

© Springer-Verlag 1977

Authors and Affiliations

  • Bernard Russo
    • 1
  1. 1.University of CaliforniaIrvineUSA

Personalised recommendations