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Interpolation spaces between the Lipschitz class
and the space of continuous functions


Authors: Michael Cwikel and Mieczyslaw Mastylo
Journal: Proc. Amer. Math. Soc. 124 (1996), 1103-1109
MSC (1991): Primary 46M35, 46E15, 46E35
DOI: https://doi.org/10.1090/S0002-9939-96-03171-1
MathSciNet review: 1307507
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Abstract: It is shown that the complex interpolation spaces $[C([0,1]), \linebreak Lip_{1}([0,1])]_{\theta }$ and $[C([0,1]),Lip_{1}([0,1])]^{\theta }$ do not coincide with $Lip_{ \theta }([0,1])$ or $lip_{\theta }([0,1])$ and also that the couple $(C,Lip_{1})$ is not a Calderón couple. Similar results are also obtained for the couples $(C,Lip_{\alpha })$ and $(Lip_{\alpha },Lip_{1})$ when $\alpha \in (0,1)$.


References [Enhancements On Off] (What's this?)

  • 1. J. Bergh, On the relation between the two complex methods of interpolation, Indiana Univ. Math. J. 28 (1979), 775--778. MR 80f:46062
  • 2. J. Bergh and J. Löfström, Interpolation spaces. An introduction, Grundlehren Math. Wiss., vol. 223, Springer, Berlin, Heidelberg, and New York, 1976. MR 58:2349
  • 3. Y. Brudnyi and N. Krugljak, Interpolation functors and interpolation spaces, North-Holland, Amsterdam, New York, Oxford, and Tokyo, 1991. MR 93b:46141
  • 4. Y. Brudnyi and A. Shteinberg, Calderón couples of Lipschitz spaces, J. Functional Analysis 131 (1995), 459--498. CMP 95:16
  • 5. ------, Calderón constants of finite dimensional couples, Israel J. Math. (to appear).
  • 6. T. Bychkova, Couples without C-properties, Investigations in the Theory of Functions of Several Variables, Yaroslavl, 1990, pp. 18--51.
  • 7. A. P. Calderón, Intermediate spaces and interpolation, the complex method, Studia Math. 24 (1964), 113--190. MR 29:5097
  • 8. M. Cwikel, Complex interpolation, a discrete definition and reiteration, Indiana Univ. Math. J. 27 (1978), 1005--1009. MR 80h:46118
  • 9. ------, Monotonicity properties of interpolation spaces II, Ark. Mat. 19 (1981), 123--136. MR 83a:46082
  • 10. ------, $K$-divisibility of the $K$-functional and Calderón couples, Ark. Mat. 22 (1984), 39--62. MR 85m:46074
  • 11. M. Cwikel and Y. Sagher, Relations between real and complex interpolation spaces, Indiana Univ. Math. J. 36 (1987), 905--912. MR 88m:46083
  • 12. T. Holmstedt, Interpolation of quasi-normed spaces, Math. Scand. 26 (1970), 177--199. MR 54:3440
  • 13. V. I. Ovchinnikov, Interpolation properties of fractional BMO space, Abstracts of Proceedings of the 13th All-Union School in Operator Theory in Functional Spaces, Kuibyshev, 1988.
  • 14. J. Peetre, Exact interpolation theorems for Lipschitz continuous functions, Richerche Mat. 18 (1969), 239--259. MR 42:841

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Additional Information

Michael Cwikel
Affiliation: Department of Mathematics, Technion Israel Institute of Technology, Haifa, 32000 Israel
Email: mcwikel@techunix.technion.ac.il

Mieczyslaw Mastylo
Affiliation: Faculty of Mathematics and Computer Science, A. Mickiewicz University, Matejki 48/49, 60-769 Poznań, Poland
Email: mastylo@math.amu.edu.pl

DOI: https://doi.org/10.1090/S0002-9939-96-03171-1
Keywords: Lipschitz class, complex interpolation space, Calder\'{o}n couple
Received by editor(s): September 2, 1994
Additional Notes: The research of the first author was supported by the Fund for Promotion of Research at the Technion. The research of the second author was supported in part by a Lady Davis Fellowship at the Technion.
Communicated by: Dale Alspach
Article copyright: © Copyright 1996 American Mathematical Society

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