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Sobolev Spaces with only Trivial Isometries

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Abstract

We will give some conditions for Sobolev spaces on bounded Lipschitz domains to admit only trivial isometries.

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References

  1. Adams, R. and Founier, J.: Sobolev Spaces, Academic Press 2003.

  2. Banach, S.: Theorie des operations lineaires, Warsaw, 1932.

  3. Bellenot, S. F.: Banach spaces with trivial isometries, Israel J. Math. 56 (1986), 89–96.

    Google Scholar 

  4. Bellenot, S. F., Isometries of James' space, Contemp. Math. 85 (1989), 1–18.

    Google Scholar 

  5. Casazza, P. G. and Shura, T. J.: Tsirelson's Space, Lecture Notes in Math. Vol. 1363, Springer-Verlag, Berlin, New York, 1988.

  6. Davis, W. J.: Separable Banach spaces with only trivial isometries, Rev. Roumaine Math. Pures Appl. 16 (1971), 1051–1054.

    Google Scholar 

  7. Fleming, R. J. and Jamison, J. E.: Isometries on Banach spaces: a survey, In: Analysis, Geometry and Groups: a Riemann Legacy Volume 1, Hardronic Press, Palm Harbor, FL, 1993, 52–123.

  8. Gordon, Y. and Lewis, D. R.: Isometries of diagonally symmetric Banach spaces, Israel J. Math. 28(1–2) (1977), 45–67.

    Google Scholar 

  9. Gordon, Y. and Loewy, R.: Uniqueness of (Δ) bases and isometries of Banach spaces, Math. Ann. 241(2) (1979), 159–180.

    Google Scholar 

  10. Hardin, C. D. Jr.: Isometries on subspaces of L p, Indiana Univ. Math J. 30 (1981), 449–465.

  11. Jarosz, K.: Any Banach space has an equivalent norm with trivial isometries, Israel J. Math. 64 (1988), 49–56.

    Google Scholar 

  12. Koldobsky, A. and König, H.: Aspects of Isometric Theory of Banach Spaces, Handbook of the Geometry of Banach Spaces, Vol. 1, Elsevier Science B. V., 2001.

  13. Lamperti, J.: On the isometries of certain function spaces, Pacific J. Math 8 (1958), 459–466.

    Google Scholar 

  14. Plotkin, A. I.: Isometric operators in spaces of summable analytic and harmonic functions, Dokl. Akad. Nauk SSSR 185 (1969), 995–997.

    Google Scholar 

  15. Plotkin, A. I., Isometric operators on subspaces of L p, Dokl. Akad. Nauk SSSR 193 (1970), 537–539.

  16. Plotkin, A. I.: Continuation of L p isometries, Zap. Naucn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 22 (1971), 103–129.

  17. Plotkin, A. I.: Isometric Operators in L p Spaces of Analytic and Harmonic Functions, Investigations on Linear Operators and the Theory of Functions, III, Zap. Naucn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 30 (1972), 130–145.

  18. Plotkin, A. I.: An algebra that is generated by translation operators and L p norms, Functional Analysis, No. 6: Theory of Operators in Linear Spaces, Ulyanovsk. Gos. Ped. Inst., Ulyanovsk (1976), 112–121.

  19. Rudin, W.: L p -isometries and equimeasurability, Indiana Univ. Math J. 25 (1976), 215–228.

    Google Scholar 

  20. Sersouri, A.: On James' type spaces, Trans. Amer. Math. Soc. 310 (1988), 715–745.

    Google Scholar 

  21. Semenov, P. V. and Skorik, A. I.: Isometries of James type spaces, Math Notes 38 (1986), 804–808.

    Google Scholar 

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Correspondence to Geoff Diestel.

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Diestel, G., Koldobsky, A. Sobolev Spaces with only Trivial Isometries. Positivity 10, 135–144 (2006). https://doi.org/10.1007/s11117-005-4703-6

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  • DOI: https://doi.org/10.1007/s11117-005-4703-6

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