Skip to main content
SpringerLink
Log in

A remark on the behaviour ofL p-multipliers and the range of operators acting onL p-spaces

  • Published:
Israel Journal of Mathematics Aims and scope Submit manuscript

Abstract

In this paper, new results are obtained concerning the uniform approximation property (UAP) inL p-spaces (p≠2,1,∞). First, it is shown that the “uniform approximation function” does not allow a polynomial estimate. This fact is rather surprising since it disproves the analogy between UAP-features and the presence of “large” euclidian subspaces in the space and its dual. The examples are translation invariant spaces on the Cantor group and this extra structure permits one to replace the problem with statements about the nonexistence of certain multipliers in harmonic analysis. Secondly, it is proved that the UAP-function has an exponential upper estimate (this was known forp=1, ∞). The argument uses Schauder’s fix point theorem. Its precise behaviour is left unclarified here. It appears as a difficult question, even in the translation invariant context.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. J. Bourgain and L. Tzafriri,Invertibility of large submatrices with applications to the geometry of Banach spaces, Israel J. Math.57 (1987), 137–224.

    Article  MATH  MathSciNet  Google Scholar 

  2. T. Figiel, W.B. Johnson and G. Schechtman,Factorizations of natural embeddings of l np into Lr, I, Studia Math.89 (1988), 79–103.

    MATH  MathSciNet  Google Scholar 

  3. E. Gluskin, N. Tomczak-Jaegermann and L. Tzafriri,Subspaces of l Np of small codimension, Israel J. Math., this issue.

  4. W.B. Johnson and G. Pisier,The proportional UAP characterizes weak Hilbert spaces, to appear.

  5. W.B. Johnson and G. Schechtman,On subspaces of L 1 with maximal distance to Euclidian spaces, Proc. Workshop on Banach Space Theory (Bor-Luh-Lin, ed.), Univ. Iowa Press, 1981, pp. 83–96.

  6. H.E. Lacey,The Isometric Theory of Classical Banach Spaces, Springer-Verlag, Berlin, 1974.

    MATH  Google Scholar 

  7. V. Mascioni,On the duality of the uniform approximation property in Banach spaces, Illinois J. Math.35 (1991), 191–197.

    MATH  MathSciNet  Google Scholar 

  8. I.P. Nathanson,Constructive Function Theory, Vol. II, F. Ungar Publ. Co., New York, 1965.

    Google Scholar 

  9. G. Pisier,Holomorphic semigroups and the geometry of Banach spaces, Ann. of Math.115 (1982), 375–392.

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Institute des Hautes Etudes Sciences, 35 Route de Chartres, 91440, Bures-Sur-Yvette, France

    Jean Bourgain

Authors
  1. Jean Bourgain
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Bourgain, J. A remark on the behaviour ofL p-multipliers and the range of operators acting onL p-spaces. Israel J. Math. 79, 193–206 (1992). https://doi.org/10.1007/BF02808215

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02808215

Keywords

Search

Navigation