Skip to main content
SpringerLink
Log in
Book cover

Fractals and Spectra pp 49–123Cite as

Function spaces on ℝn

  • Chapter
  • First Online:

Part of the book series: Modern Birkhäuser Classics ((MBC))

Abstract

This chapter deals mainly with the function spaces B spq and F spq and their special cases on ℝn. These spaces have been treated in detail in [Tri83], [Tri92] and, as far as some more recent results are concerned, in [ET96] and [RuS96]. The aim of this chapter is twofold. Firstly, to make this and the following chapters selfcontained we give all the necessary definitions and collect those assertions which are needed later on. Secondly, beginning with Section 12, we discuss and establish some new properties in detail. This applies especially to so-called harmonic and local characterizations, as well as atomic and subatomic representations both of scalar-valued and vector-valued function spaces of the above type.

This is a preview of subscription content, 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   39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   54.99
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

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Triebel, H., Theory of function spaces II. Basel, Birkhäuser, 1992

    Book  MATH  Google Scholar 

  2. Falconer, K.J., Fractal geometry. Chichester, Wiley, 1990

    MATH  Google Scholar 

  3. Deliu, A. and Jawerth, B., Geometrical dimension versus smoothness. Constr. Approx. 8 (1992), 211–222

    Article  MATH  MathSciNet  Google Scholar 

  4. Solomyak,M., Piecewise-polynomial approximation of functions from Sobolev spaces, revisited. Preprint, 1996

    Google Scholar 

  5. Sickel, W., Spline representations of functions in Besov-Triebel-Lizorkin spaces on ℝn. Forum Math. 2 (1990), 451–475

    Article  MATH  MathSciNet  Google Scholar 

  6. Triebel, H., Higher analysis. Leipzig, Barth, 1992

    MATH  Google Scholar 

  7. Torchinsky, A., Real-variable methods in harmonic analysis. San Diego, Academic Press, 1986

    MATH  Google Scholar 

  8. Schmeisser, H.-J., Vector-valued Sobolev and Besov spaces. Teubner-Text. Math. 96, Leipzig, Teubner, 1987, 4–44

    Google Scholar 

  9. Frazier, M. and Jawerth, B., A discrete transform and decomposition of distribution spaces. J. Funct. Analysis 93 (1990), 34–170

    Article  MATH  MathSciNet  Google Scholar 

  10. Frazier, M., Jawerth, B. and Weiss, G., Littlewood-Paley theory and the study of function spaces. CBMS-AMS Regional Conf. Ser. 79, 1991

    Google Scholar 

  11. Daubechies, I., Ten lectures on wavelets. Philadelphia, SIAM, 1992

    MATH  Google Scholar 

  12. Diestel, J. and Uhl, J.J., Vector measures. Math. Surveys and Monographs 15. Providence, AMS, 1977

    Google Scholar 

  13. Triebel, H. and Winkelvoss, H., Intrinsic atomic characterizations of function spaces on domains. Math. Z. 221 (1996), 647–673

    MathSciNet  Google Scholar 

  14. Triebel, H., Theory of function spaces. Basel, Birkhäuser, 1983

    Book  Google Scholar 

  15. Edmunds, D.E. and Triebel, H., Function spaces, entropy numbers, differential operators. Cambridge Univ. Press 1996

    Google Scholar 

  16. Haroske, D. and Triebel, H., Entropy numbers in weighted function spaces and eigenvalue distributions of some degenerate pseudodifferential operators I. Math. Nachr. 167 (1994), 131–156

    Article  MATH  MathSciNet  Google Scholar 

  17. Frazier, M. and Jawerth, B., Decomposition of Besov spaces. Indiana Univ. Math. J. 34 (1985), 777–799

    Article  MATH  MathSciNet  Google Scholar 

  18. Runst, Th. and Sickel, W., Sobolev spaces of fractional order, Nemytzkij operators and nonlinear partial differential equations. Berlin, de Gruyter, 1996

    Google Scholar 

  19. Schwartz, L., Distributions a valeurs vectorielles. I. Ann. Inst. Fourier 7 (1957), 1–142. II. Ann. Inst. Fourier 8 (1958), 1–209

    MATH  Google Scholar 

  20. Dunford, N. and Schwartz, J.T., Linear operators, I. New York, Interscience Publ., 1958

    MATH  Google Scholar 

  21. Falconer, K.J., The geometry of fractal sets. Cambridge Univ. Press, 1985

    Google Scholar 

  22. Schmeisser, H.-J. and Triebel, H., Topics in Fourier analysis and function spaces. Chichester, Wiley, 1987

    Google Scholar 

  23. Sickel, W. and Triebel, H., Hölder inequalities and sharp embeddings in function spaces of B \(^{s_{pq}}\) and F \(^{s_{pq}}\) type. Z. Anal. Anwendungen 14 (1995), 105–140

    MATH  MathSciNet  Google Scholar 

  24. Torres, R.H., Boundedness results for operators with singular kernels on distribution spaces. Memoirs AMS 90, 442. Providence, AMS, 1991

    MathSciNet  Google Scholar 

  25. Meyer, Y., Wavelets and operators. Cambridge Univ. Press, 1992

    MATH  Google Scholar 

  26. Bui, H.- Q., Paluszyński, M. and Taibleson, M.H., A maximal function characterization of weighted Besov-Lipschitz and Triebel-Lizorkin spaces. Studia Math. 111 (1996), 219–246

    Google Scholar 

  27. Schott, Th., Function spaces with exponential weights I. Math. Nachr.

    Google Scholar 

  28. Farkas,W., Atomic and subatomic decompositions in anisotropic function spaces. Preprint, Jena, 1997

    Google Scholar 

  29. Triebel, H., Interpolation theory, function spaces, differential operators. Amsterdam, North-Holland, 1978 (Sec. ed. Heidelberg, Barth, 1995)

    Google Scholar 

  30. Amann, H., Linear and quasilinear parabolic problems. I. Basel, Birkhäuser, 1995

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Mathematisches Institut, Friedrich-Schiller-Universität Jena, 07737, Jena, Germany

    Hans Triebel

Authors
  1. Hans Triebel
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hans Triebel .

Rights and permissions

Reprints and permissions

Copyright information

© 1997 Birkhäuser Verlag

About this chapter

Cite this chapter

Triebel, H. (1997). Function spaces on ℝn . In: Fractals and Spectra. Modern Birkhäuser Classics. Springer, Basel. https://doi.org/10.1007/978-3-0348-0034-1_3

Download citation

  • DOI: https://doi.org/10.1007/978-3-0348-0034-1_3

  • Published:

  • Publisher Name: Springer, Basel

  • Print ISBN: 978-3-0348-0033-4

  • Online ISBN: 978-3-0348-0034-1

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation