Israel Journal of Mathematics

, Volume 6, Issue 2, pp 133–149 | Cite as

Extensions of the Hausdorff-Young theorem

  • M. M. Rao


In this paper the clasical Hausdorff-Young theorem, which states that iffL p, 1≦p≦2, on the line and\(\hat f\) is its Fourier transform, then\(\left\| {\hat f} \right\|_q \leqq \left\| f \right\|_p \) whereq −1+p −1=1, is extended in two ways for certain Orlicz spacesL Φ. IfL Φ is based on (G, μ), (1) an arbitrary compact topological group with Haar measure, and (2) a locally compact abelian topological group andμ is again the Haar measure, then the above inequality is extended to these cases. Various other related results and remarks are also included.


Compact Group Haar Measure Orlicz Space Compact Abelian Group Sublinear Operator 
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. 1.
    A. P. Calderón and A. Zygmund,A note on the interpolation of sublinear operations, Amer. J. Math.,78 (1956), 282–288.zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    G. H. Hardy and J. E. Littlewood,Some new properties of Fourier constants, Math. Annalen,97 (1927), 159–209.CrossRefMathSciNetGoogle Scholar
  3. 3.
    H. Helson,Conjugate series in several variables, Pacific J. Math.,9 (1959), 513–523.zbMATHMathSciNetGoogle Scholar
  4. 3a.
    E. Hewitt,Fourier transforms of the class L p, Ark. für Mat.,2 (1954), 571–574.zbMATHCrossRefMathSciNetGoogle Scholar
  5. 4.
    I. I. Hirschman, Jr.,A maximal problem in harmonic analysis-II, Pacific J. Math.,9 (1959), 525–540.zbMATHMathSciNetGoogle Scholar
  6. 5.
    M. A. Krasnosel’kii and Ya. B. Rutickii,Convex Functions and Functions and Orlicz Spaces, (Translation) P. Noordhoff Ltd., Groningen, 1961.Google Scholar
  7. 6.
    R. A. Kunze,L p-Fourier transforms on locally compact unimodular groups, Trans. Amer. Math. Soc.,89 (1958), 519–540.CrossRefMathSciNetGoogle Scholar
  8. 7.
    L. H. Loomis,Abstract Harmonic Analysis, D. Van Nostrand Company, New Jersey, 1953.zbMATHGoogle Scholar
  9. 8.
    W. A. J. Luxemburg,Banach Function Spaces, Univ. of Delft, 1955.Google Scholar
  10. 9.
    M. M. Rao,Interpolation, ergodicity, and martingales, J. Math. Mech.,16 (1966), 543–567.zbMATHMathSciNetGoogle Scholar
  11. 10.
    M. M. Rao,Theory of lower bounds for risk functions in estimation, Math. Annalen,143 (1961), 379–398.zbMATHCrossRefGoogle Scholar
  12. 11.
    M. M. Rao,Smoothness of Orlicz spaces, Indag. Math.27 (1965), 671–690.Google Scholar
  13. 12.
    W. J. Riordan, Unpublished P.h.D. thesis, University of Chicago, (1957). (See also abstract in Notices of Amer. Math. Soc., 5 (1958), 590.)Google Scholar
  14. 13.
    W. Rudin,Fourier Analysis on Groups, Interscience, New York, 1962.zbMATHGoogle Scholar
  15. 14.
    E. C. Titchmarsh,A contribution to the theory of Fourier transforms, Proc. London Math. Soc. (2),23 (1923), 279–289.Google Scholar
  16. 15.
    A. Zygmund,Trigonometric Series, Vols. I, II (2nd ed.), Cambridge Univ, Press, 1959.Google Scholar

Copyright information

© Hebrew University 1968

Authors and Affiliations

  • M. M. Rao
    • 1
  1. 1.Carnegie-Mellon UniversityPittsburgh

Personalised recommendations