Annali di Matematica Pura ed Applicata

, Volume 99, Issue 1, pp 247–316 | Cite as

Interpolation of several banach spaces

  • Gunnar Sparr


Peetre's K- and J-methods for interpolation are extended to the situation of more than two spaces. The theory developed is applied to interpolation of Lp-spaces with weights and to spaces of Besov and Sobolev type.


Sobolev Type 
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.


  1. [1]
    T. I. Amanov,Representation and imbedding theorems for the function spaces S p,θ(r) B(R)n andS p*,θ(r) B(0⩻x j⩻2π;j=1,2,...,n), Trudy Mat. Inst. Steklov,77 (1965), pp. 3–34 (=Proc. Steklov Inst. Math.,77 (1967), pp. 1–37) (Russian).zbMATHMathSciNetGoogle Scholar
  2. [2]
    C. A. Berenstein -N. Kerzman,Sur la réitération dans les espaces moyennes, C.R. Acad. Sci. Paris,263, no. 18 (1966), pp. 609–612.MathSciNetzbMATHGoogle Scholar
  3. [3]
    O. V. Besov,Investigation of a family of function spaces in connection with theorems of imbedding and extension, Trudy Mat. Inst. Steklov,60 (1961), pp. 42–71 (=Proc. Steklov Inst. Math.,40 (1964), pp. 85–126) (Russian).zbMATHMathSciNetGoogle Scholar
  4. [4]
    N. Bourbaki,Intégration, Chapitre 7. Mesure de Haar, Hermann, Paris, 1963.Google Scholar
  5. [5]
    P. L. Butzer -H. Berens,Semi-groups of operators and approximation, Springer, Berlin, 1967.zbMATHGoogle Scholar
  6. [6]
    C. Foias -J. L. Lions,Sur certains théorèmes d'interpolation, Acta Sci. Math. (Szeged),22 (1961), pp. 269–282.MathSciNetzbMATHGoogle Scholar
  7. [7]
    P. Grisvard,Commutativité de deux foncteurs d'interpolation et applications, J. Math. Pure Appl.,45 (1966), pp. 143–206 and 207–290.zbMATHMathSciNetGoogle Scholar
  8. [8]
    T. Holmstedt,Interpolation of quasi-normed spaces, Math. Scand.,26 (1970), pp. 177–199.zbMATHMathSciNetGoogle Scholar
  9. [9]
    L. Hörmander,Estimates for translation invariant operators in L p-spaces, Acta Math.,104 (1960), pp. 93–140.zbMATHMathSciNetGoogle Scholar
  10. [10]
    N. Kerzman,Interpolation among n quasinormed spaces, Notes, Buenos Aires, 1966.Google Scholar
  11. [11]
    N. Kerzman,Sur certains ensembles convexes liés à des espaces L p, C.R. Acad. Sci. Paris,263, no. 11 (1966), pp. 365–367.zbMATHMathSciNetGoogle Scholar
  12. [12]
    S. Lang,Analysis, II, Addison-Wesley, Reasing, Mass., 1969.zbMATHGoogle Scholar
  13. [13]
    J. L. Lions -J. Peetre,Sur une classe d'espaces d'interpolation, Publ. Math. I.H.E.S.,19 (1964), pp. 5–68.MathSciNetzbMATHGoogle Scholar
  14. [14]
    S. M. Nikolskij,Imbedding, continuation and approximation theorems for differentiable functions of several variables, Uspehi Mat. Nauk,16, no. 5 (1961), pp. 63–115 (Russian).zbMATHMathSciNetGoogle Scholar
  15. [15]
    S. M. Nikolskij,Functions with dominant mixed derivate satisfying a multiple Hölder condition, Sibirsk. Mat. Z.,4 (1963), pp. 1342–1364 (Russian).zbMATHMathSciNetGoogle Scholar
  16. [16]
    J. Peetre,A theory of interpolation of normed spaces, Notes, Brasilia, 1963 (= otas de matematica,39 (1968)).Google Scholar
  17. [17]
    J. Peetre,Sur le nombre de paramètres dans la dèfinition de certains espaces d'interpolation, Ricerche Mat.,12 (1963), pp. 248–262.zbMATHMathSciNetGoogle Scholar
  18. [18]
    J. Peetre,Espaces d'interpolation, généralisation, applications, Rend. Sem. Mat. Fis. Milano,34 (1964), pp. 133–165.zbMATHMathSciNetCrossRefGoogle Scholar
  19. [19]
    J. Peetre,Espaces d'interpolation et théorème de Soboleff, Ann. Inst. Fourier,16 (1966), pp. 279–317.zbMATHMathSciNetGoogle Scholar
  20. [20]
    J. Peetre,Sur les espaces de Besov, C.R. Acad. Sci. Paris,264 (1967), pp. 281–283.zbMATHMathSciNetGoogle Scholar
  21. [21]
    J. Peetre,On a fundamental lemma in the theory of interpolation spaces, Technical report, Lund, 1968.Google Scholar
  22. [22]
    J. Peetre,A new approach in interpolation spaces, Studia Math.,34 (1970), pp. 23–42.zbMATHMathSciNetGoogle Scholar
  23. [23]
    J. Peetre,Banach couples, Technical report, Lund, 1971.Google Scholar
  24. [24]
    J. Peetre -G. Sparr,Interpolation of normed Abelian groups, Ann. di Mat. pura e appl.,92 (1972), pp. 217–262.MathSciNetzbMATHGoogle Scholar
  25. [25]
    A. Yoshikawa,Sur la théorie d'espaces d'interpolation — les espaces de moyenne de plusieurs espaces de Banach, J. Fac. Sci. Univ. Tokyo,16 (1970), pp. 407–468.zbMATHMathSciNetGoogle Scholar

Copyright information

© Nicola Zanichelli Editore 1974

Authors and Affiliations

  • Gunnar Sparr
    • 1
  1. 1.Lund

Personalised recommendations