Skip to main content
SpringerLink
Log in

The Functional Calculus in Sobolev Spaces

  • Chapter
Function Spaces, Differential Operators and Nonlinear Analysis

Part of the book series: Teubner-Texte zur Mathematik ((TTZM,volume 133))

Abstract

Let E be a real-valued function space. The functional calculus problem for E consists in finding necessary and sufficient conditions — as simple as possible — such that a real variable function G acts, via left composition, on E; in other words, that G o f ∈ E for each f ∈ E. We say that the functional calculus is non trivial if for every “reasonable” function (say G ∈ S(ℝn) with G(0) = 0) acts on E.

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 49.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 49.99
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 - Papers summarized in the exposition

  1. Bourdaud, G.. Le calcul fonctionnel dans les espaces de Sobolev. Invent. Math. 104 (1991), 435–446.

    Article  MathSciNet  MATH  Google Scholar 

  2. Bourdaud, G. Le calcul fonctionnel dans lspace de Besov critique. Proc. Amer. Math. Soc. (to appear).

    Google Scholar 

  3. Bourdaud, G. Fonctions qui opèrent sur les espaces de Besov et de Triebel. Annales I.H.P. - Analyse non linéaire - 10 (1993) (to appear).

    Google Scholar 

  4. C.R. Acad. Sci. Paris, t.314, Série 1, 187–190.

    Google Scholar 

  5. Bourdaud, G. Calcul fonctionnel et propriété de Fubini dans les espaces de Triebel -Lizorkin. Preprint, Paris V II (1992).

    Google Scholar 

  6. Bourdaud, G., Kateb, Dalila. Fonctions qui opérent sur certains espaces de Besov. Ann. Inst. Fourier, Grenoble, 40 (1990), 153–162.

    Article  MathSciNet  MATH  Google Scholar 

  7. Bourdaud, G., Kateb, Dalila. Fonctions qui opérent sur les espaces de Besov. Proc. Amer. Math. Soc. 112 (1991), 1067–1076.

    MathSciNet  MATH  Google Scholar 

  8. Bourdaud, G., Kateb, Djalil. Calcul fonctionnel dans l’espace de Sobolev fractionnaire. Math. Z. 210 (1992), 607–613.

    Article  MathSciNet  MATH  Google Scholar 

  9. Bourdaud, G., Meyer, Y. Fonctions qui opérent sur les espaces de Sobolev. J. Funct. Anal. 97 (1991), 351–360.

    MathSciNet  MATH  Google Scholar 

2 - Previous works on the functional calculus

  1. Adams, D.R. On the existence of capacitary strong type estimates in 118“. Ark. Mat. 14 (1976), 125–140.

    Article  MathSciNet  MATH  Google Scholar 

  2. Bourdaud, G. Fonctions qui opèrent sur les espaces de Sobolev. Sém. Anal. Harm., Orsay (1980–1981), 6–17.

    Google Scholar 

  3. Dahlberg, B.E.T. A note on Sobolev spaces. Proc. Symp. Pure Math. 35, 1 (1979), 183–185.

    Article  Google Scholar 

  4. Igari, S. Sur les fonctions qui opèrent sur l’espace Â2. Ann. Inst. Fourier, Grenoble, 15 (1965), 525–536.

    Article  MathSciNet  MATH  Google Scholar 

  5. Janson, S. Harmonic analysis and partial differential equations (El Escorial, 1987), 193–201, Lect. Notes in Math. 1384, Springer 1989.

    Google Scholar 

  6. Marcus, M., Mizel, J. Complete characterization of functions which act via superposition on Sobolev spaces. Trans. Amer. Math. Soc. 251 (1979), 187–218.

    MATH  Google Scholar 

  7. Maz’ya, V. Sobolev spaces. Springer 1985.

    Google Scholar 

  8. Oswald, P. On the boundedness of the mapping f —+ i f i in Besov spaces. Comment. Univ. Carolinae 33 (1992), 57–66.

    MathSciNet  MATH  Google Scholar 

  9. Peetre, J. Interpolation of Lipschitz operators and metric spaces. Mathematica (Cluj) 12 (35) (1970), 325–334.

    MathSciNet  Google Scholar 

  10. Runst, T. Mapping properties of non-linear operators in spaces of Triebel-Lizorkin and Besov type. Anal. Math. 12 (1986), 313–346.

    Article  MathSciNet  MATH  Google Scholar 

  11. Szigeti, F. On the Nemitzki operators in Sobolev spaces. Z. Angew. Math. Mech. 63 (1983), T 332.

    Google Scholar 

3 - Works on wavelets and function spaces

  1. Bourdaud, G. Analyse fonctionnelle dans l’espace Euclidien. Publ. Math. Paris VII, 23 (1987).

    Google Scholar 

  2. Frazier, M., Jawerth,B. Decomposition of Besov spaces. Indiana Univ. Math. J. 84 (1985), 777–799.

    MathSciNet  Google Scholar 

  3. Frazier, M., Jawerth, B.. A discrete transform and applications to distribution spaces. J. Funct. Anal. 93 (1990), 34–170.

    Article  MathSciNet  MATH  Google Scholar 

  4. Goldberg, D. A local version of real Hardy spaces. Duke Math. J. 46 (1979), 27–42.

    Article  MathSciNet  MATH  Google Scholar 

  5. Meyer, Y. Ondelettes et Opèrateurs. Hermann, Paris 1990/91.

    Google Scholar 

  6. Triebel, H. Theory of function spaces. Birkhäuser, Basel, Boston, Stuttgart 1983.

    Google Scholar 

  7. Triebel, H. Local approximation spaces. Z. Anal. Anwendungen 8 (1989), 261–288.

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Univ. Paris VII - Denis Diderot C.N.R.S.U.A. 212, Tour 45 - 55 - 5° étage 2, place Jussieu, 75251, Paris Cedex 05, France

    Gérard Bourdaud

Authors
  1. Gérard Bourdaud
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Friedrich-Schiller-University, Jena, Germany

    Hans-Jürgen Schmeisser  & Hans Triebel  & 

Rights and permissions

Reprints and permissions

Copyright information

© 1993 Springer Fachmedien Wiesbaden

About this chapter

Cite this chapter

Bourdaud, G. (1993). The Functional Calculus in Sobolev Spaces. In: Schmeisser, HJ., Triebel, H. (eds) Function Spaces, Differential Operators and Nonlinear Analysis. Teubner-Texte zur Mathematik, vol 133. Vieweg+Teubner Verlag, Wiesbaden. https://doi.org/10.1007/978-3-663-11336-2_2

Download citation

  • DOI: https://doi.org/10.1007/978-3-663-11336-2_2

  • Publisher Name: Vieweg+Teubner Verlag, Wiesbaden

  • Print ISBN: 978-3-8154-2045-4

  • Online ISBN: 978-3-663-11336-2

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation