A natural problem in interpolation theory is to find necessary and sufficient conditions for interpolation properties to hold among spaces of the same type. This paper serves to provide a solution for Mϕ spaces which appear in fundamental roles in classical interpolation theorems such as the Marcinkiewicz and Stein-Weiss theorems ([1], [2], [10]) as well as in the class of rearrangement invariant Banach function spaces ([6], [8]).


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  1. [1]
    Calderon, A.P., Spaces between L 1 and L and the theorem of Marcinkiewicz, Studia Math. 26 (1966), 273–299.Google Scholar
  2. [2]
    Edwards, R.E., Fourier Series Vol. II, Holt, Rinehart, and Winston, Inc., New York 1967.Google Scholar
  3. [3]
    Lorentz, G.G., Bernstein Polynomials, University of Toronto Press, Toronto 1953.Google Scholar
  4. [4]
    Lorentz, G.G., and Shimogaki, T., Interpolation theorems for operators in function spaces, J. Functional Analysis 2 (1968), 31–51.CrossRefGoogle Scholar
  5. [5]
    Luxemburg, W.A.J., Banach function spaces, Thesis Delft Institute of Technology, Assen, Netherlands 1955.Google Scholar
  6. [6]
    Semenov, E.M., Imbedding theorems for Banach spaces of measurable functions, Soviet Math. Dokl. 5 (1964), 831–834.Google Scholar
  7. [7]
    Sharpley, R.C., Interpolation of operators in function spaces, Dissertation, University of Texas, Austin, Texas 1972.Google Scholar
  8. [8]
    Sharpley, R.C., Spaces. Λα (X) and interpolation, J. Functional Analysis 11 (1972), 479–513.CrossRefGoogle Scholar
  9. [9]
    Sharpley, R.C., Interpolation of operators for. Λϕ spaces, Bull. Amer. Math. Soc. 80 (1974), 259–261.CrossRefGoogle Scholar
  10. [10]
    Zygmund, A., Trignometric Series Vol. II, Cambridge University Press, New York 1968.Google Scholar

Copyright information

© Springer Basel AG 1974

Authors and Affiliations

  • Robert Sharpley
    • 1
  1. 1.Department of MathematicsOakland UniversityRochesterUSA

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