Israel Journal of Mathematics

, Volume 54, Issue 2, pp 129–138 | Cite as

Extensions of lipschitz maps into Banach spaces

  • William B. Johnson
  • Joram Lindenstrauss
  • Gideon Schechtman
Article

Abstract

It is proved that ifYX are metric spaces withY havingn≧2 points then any mapf fromY into a Banach spaceZ can be extended to a map\(\hat f\) fromX intoZ so that\(\left\| {\hat f} \right\|_{lip} \leqq c log n\left\| f \right\|_{lip} \) wherec is an absolute constant. A related result is obtained for the case whereX is assumed to be a finite-dimensional normed space andY is an arbitrary subset ofX.

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References

  1. 1.
    P. Enflo,On the non-existence of uniform homeomorphisms between L p spaces, Ark. Mat.8 (1969), 195–197.MathSciNetGoogle Scholar
  2. 2.
    W. B. Johnson and J. Lindenstrauss,Extensions of Lipschitz mappings into a Hilbert space, Contemp. Math.26 (1984), 189–206.MATHMathSciNetGoogle Scholar
  3. 3.
    J. Lindenstrauss,On non-linear projections in Banach spaces, Mich. Math. J.11 (1964), 263–287.MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    M. B. Marcus and G. Pisier,Characterizations of almost surely continuous p-stable random Fourier series and strongly stationary processes, Acta Math.152 (1984), 245–301.MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    E. M. Stein,Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, 1970.MATHGoogle Scholar
  6. 6.
    J. H. Wells and L. R. Williams,Embeddings and Extensions in Analysis, Springer-Verlag, 1975.Google Scholar
  7. 7.
    H. Whitney,Analytic extensions of differentiable functions defined in closed sets, Trans. Am. Math. Soc.36 (1934), 63–89.MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Hebrew University 1986

Authors and Affiliations

  • William B. Johnson
    • 1
    • 2
  • Joram Lindenstrauss
    • 3
    • 1
  • Gideon Schechtman
    • 4
    • 1
  1. 1.Ohio State UniversityColumbusUSA
  2. 2.Texas A&M UniversityCollege StationUSA
  3. 3.The Hebrew University of JerusalemJerusalemIsrael
  4. 4.The Weizmann Institute of ScienceRehovotIsrael

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