Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Uniform homeomorphisms between the unit balls in $ L\sb p$ and $ l\sb p$

Author: Gun-Marie Lövblom
Journal: Proc. Amer. Math. Soc. 123 (1995), 405-409
MSC: Primary 46B99; Secondary 46B25, 46E30
MathSciNet review: 1227523
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ T:B({L_p}) \to B({l_p}),1 \leq p < 2$, be a uniform homeomorphism with modulus of continuity $ {\delta _T}$. It is shown that for any $ \gamma ,0 \leq \gamma < \frac{{2 - p}}{{2p}}$, there exists $ K > 0$ and a sequence $ \{ {\varepsilon _n}\} $ with $ {\varepsilon _n} \to 0$ such that $ \delta _T^{ - 1}({\delta _T}({\varepsilon _n})) \geq K{\varepsilon _n}\vert\log {\varepsilon _n}{\vert^\gamma }$ for all $ {\varepsilon _n}$.

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

  • [1] J. Bourgain, Remarks on the extension of Lipschitz maps defined on discrete sets and uniform homeomorphisms, Lecture Notes in Math., vol. 1267 (J. Lindenstrauss and V. D. Milaman, eds.), Springer-Verlag, New York, 1987, pp. 157-167. MR 907692 (88m:46021)
  • [2] P. Enflo, On the nonexistence of uniform homeomorphisms between $ {L_p}$-spaces, Ark. Mat. 8 (1969), 103-105. MR 0271719 (42:6600)
  • [3] Y. Benyamini, The uniform classification of Banach spaces, University of Texas Functional Analysis Seminar (E. Odell and H. P. Rosenthal, eds.), Longhorn Notes, Univ. of Texas at Austin, Austin, TX, 1985, pp. 15-38. MR 832247
  • [4] J. Lindenstrauss, On nonlinear projections in Banach spaces, Michigan J. Math. 11 (1964), 268-287. MR 0167821 (29:5088)
  • [5] G.-M. Lövblom, Uniform homeomorphisms between unit balls in $ {L_p}$-spaces, Math. Scand. 62 (1988), 294-302. MR 964229 (90a:46033)
  • [6] S. Mazur, Une remarque sur l'homéomorphie des champs fonctionnels, Studia Math. 1 (1929).

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 46B99, 46B25, 46E30

Retrieve articles in all journals with MSC: 46B99, 46B25, 46E30

Additional Information

Article copyright: © Copyright 1995 American Mathematical Society

American Mathematical Society