Skip to main content
SpringerLink
Log in
SpringerLink
Log in

Counter-Examples in Interpolation Space Theory from Harmonic Analysis

  • Chapter
Linear Operators and Approximation II / Lineare Operatoren und Approximation II

Abstract

That the classical interpolation theorems of Riesz-Thorin and Marcinkiewicz are of great importance in harmonic analysis has long since been realized. The functional analytic generalization of these interpolation theorems — abstract interpolation space theory — also is now proving to be of importance in harmonic analysis (cf., for instance, [1]). In this paper we reverse the flow in the sense that a circle of ideas from Lp-multiplier theory and tensor products of Banach spaces is used to provide answers to a number of questions that arise naturally in the general theory of abstract interpolation spaces. While some of these questions have been answered before (though not always published), the unified approach used in this paper introduces techniques which may well be of wider utility in interpolation space theory. We shall be interested in both the real interpolation spaces (X o, X 1)θq;J, (X o, X 1)θ, q;k of Peetre and the complex interpolation spaces (X o,X 1)θ of Lions and Calderon. For all unexplained notation and terminology see [2], [5], [6] or [11].

Supported in part by N.S.F. Grant GP-38865. AMS (MOS) subject classification (1970) Primary 43A22, 46E35 Secondary 43A15, 47B10.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 54.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 69.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Bennet, C. — Gilbert, J.E., Homogeneous algebras on the circle. II. Multipliers, Ditkin Conditions. Ann. l’Inst. Fourier (Grenoble) XXII (1972), 21–50.

    Article  Google Scholar 

  2. Butzer, P.L. — Berens, H., Semi-Groups of Operators and Approximation. Berlin/Heidelberg/New York, Springer-Verlag 1967.

    Book  Google Scholar 

  3. Eymard, P., Algèbres A P et convoluteurs de L p. Séminaire Bourbaki #367 (1969/70).

    Google Scholar 

  4. Figa-Talamanca, A. — Rider, D., A theorem of Littlewood and lacunary series for compact groups. Pacific J. Math. 16 (1966), 505–514.

    Article  Google Scholar 

  5. Gilbert, J.E., L p-colvolution operators and tensor products of Banach spaces. I., submitted for publication.

    Google Scholar 

  6. Gilbert, J.E., Tensor products of Banach spaces and weak type (p,p) multipliers, to appear in J. Approx. Theory.

    Google Scholar 

  7. ——, Restriction and extension results connected with L —, convolution operator theory, in preparation.

    Google Scholar 

  8. Hewitt, E. — Ross, K.A., Abstract Harmonic Analysis, Vol. II. Berlin/ Heidelberg/New York, Springer-Verlag 1970.

    Book  Google Scholar 

  9. Hewitt, E. — Ross, K.A., Edwards, R.E., Lacunarity for compact groups. I. Indiana Univ. Math. J. 21 (1972), 787–806.

    Article  Google Scholar 

  10. Lindenstrauss, J. — Pelczynski, Absolutely summing operators in L p — spaces and applications. Studia Math., 29 (1968), 275–326.

    Google Scholar 

  11. Lions, J. — L. — Magenes E., Problèmes aux limites non-homogenes et applications. Vol. I. Paris, Dunod, 1968.

    Google Scholar 

  12. Lions, J., Magenes E., Peetre, J., Sur une classe d’espaces d’interpolation. Publ. Math. I.H.E.S. 19 (1964), 5–68.

    Google Scholar 

  13. Peetre, J., Zur Interpolation von Operatorenräumen. Archiv der Math. XXI (1970), 601–608.

    Article  Google Scholar 

  14. ——, Approximation of linear operators. Preprint.

    Google Scholar 

  15. Pietsch, A., Absolutely p-summing operators in L r — spaces. Bull. Soc. Math. France Mem. 31–32 (1972), 285–315.

    Google Scholar 

  16. Pietsch, A., Ideale von S-Operatoren in Banachraumen. Studia Math. 38 (1970), 59–69.

    Google Scholar 

  17. Saphar, P., Produits tensoriels d’espaces de Banach et classes d’applications linéaires. Studia Math. 38 (1970), 71–100.

    Google Scholar 

  18. Stafney, J.D., Analytic interpolation of certain multiplier spaces. Pacific J. Math. 32 (1970), 241–248.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, The University of Texas at Austin, Austin, Texas, 78712, USA

    John E. Gilbert

Authors
  1. John E. Gilbert
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Lehrstuhl A für Mathematik, RWTH Aachen, 51, Aachen, West Germany

    P. L. Butzer

  2. Bolai Intézete, József Attila Tudományegyetem, Aradi vértanúk tere 1, Szeged, Hungary

    B. Szőkefalvi-Nagy

Rights and permissions

Reprints and permissions

Copyright information

© 1974 Springer Basel AG

About this chapter

Cite this chapter

Gilbert, J.E. (1974). Counter-Examples in Interpolation Space Theory from Harmonic Analysis. In: Butzer, P.L., Szőkefalvi-Nagy, B. (eds) Linear Operators and Approximation II / Lineare Operatoren und Approximation II. International Series of Numerical Mathematics / Internationale Schriftenreihe zur Numerischen Mathematik / Série Internationale D’Analyse Numérique, vol 25. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-5991-2_11

Download citation

  • DOI: https://doi.org/10.1007/978-3-0348-5991-2_11

  • Publisher Name: Birkhäuser, Basel

  • Print ISBN: 978-3-0348-5992-9

  • Online ISBN: 978-3-0348-5991-2

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation