Skip to main content

Operator Valued Fourier Multipliers

  • Chapter

Part of the book series: Progress in Nonlinear Differential Equations and Their Applications ((PNLDE,volume 35))

Abstract

Results on Fourier multipliers are important tools in the study of partial differential equations. They represent a major step, for example, when establishing a priori estimates for solutions of parabolic evolution equations of Agmon-Douglis-Nirenberg type [1]. Among the results known, the classical theorem of Mikhlin, which guarantees the boundedness of translations-invariant operators on L p(ℝn), is of special importance.

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

Buying options

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   129.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   169.99
Price excludes VAT (USA)
  • Durable hardcover 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. Agmon, S., Douglis, A., Nirenberg, L.: “Estimates near boundary for solutions of elliptic partial differential equations satisfying general boundary conditions”, Comm. Pure Appl. Math. 12 (1959), 623–727.

    Article  MathSciNet  MATH  Google Scholar 

  2. Amann, H.: Linear and Quasilinear Parabolic Problems. Birkhäuser, Basel, 1995.

    Book  MATH  Google Scholar 

  3. Amann, H.: “Operator-valued Fourier multipliers, vector-valued Besov spaces, and applications”, Math. Nachr. 186 (1997), 5–56.

    Article  MathSciNet  MATH  Google Scholar 

  4. Benedek, A., Calderón, A.P., Panzone, R.: “Convolution operators on Banach space valued functions”, Proc. Nat. Acad. Sci. USA 48 (1962), 356–365.

    Article  MATH  Google Scholar 

  5. Bergh, J., Löfström, J.: Interpolation Spaces. An Introduction. Springer, Berlin, 1976.

    Book  MATH  Google Scholar 

  6. Bourgain, J.: “Extension of a result of Benedick, Calderön and Panzone”, Ark. Mat. 22 (1983), 91–95.

    Article  MathSciNet  Google Scholar 

  7. Burkholder, D.L.: “A geometrical condition that implies the existence of certain singular integrals of Banach-space-valued functions”. In: Beckner, W., Calderón, A.P., Fefferman, R. and Jones, P.W.,(eds) Conf. Harm. Anal, in Honour of A. Zygmund, Chicago 1981, 270–286, Belmont, Cal., Wadsworth, 1983.

    Google Scholar 

  8. Christ, M.: Lectures on Singular Integral Operators. Conf. Board Math. Sci., Reg. Conf. Series Math., 77, Amer. Math. Soc, Providence, R.I., 1990.

    Google Scholar 

  9. Dore, G. Venni, A.: “On the closedness of the sum of two closed operators”, Math. Z. 196 (1987), 189–201.

    Article  MathSciNet  MATH  Google Scholar 

  10. Grisvard, P.: Elliptic Problems in Nonsmooth Domains. Pitman, Boston, 1985.

    MATH  Google Scholar 

  11. Hieber, M.: “Integrated semigroups and differential operators on L p spaces”, Math. Ann. 291 (1991), 1–16.

    Article  MathSciNet  Google Scholar 

  12. Hörmander, L.: “Estimates for translation invariant operators in L p spaces”, Acta Math. 104 (1960), 93–140.

    Article  MathSciNet  MATH  Google Scholar 

  13. Lunardi, A.: Analytic Semigroups and Optimal Regularity in Parabolic Equations. Birkhäuser, Basel, 1995.

    Book  Google Scholar 

  14. McConnell, T.: “On Fourier multiplier transformations of Banach-space valued functions”, Trans. Amer. Math. Soc. 285 (1984), 739–757.

    Article  MathSciNet  MATH  Google Scholar 

  15. Prüss, J.: Evolutionary Integral Equations and Applications. Birkhäuser, Basel, 1993.

    Book  MATH  Google Scholar 

  16. Rubio de Francia, J.L., Ruiz, F.J., Torrea, J.L.: “Calderón-Zygmund theory for operator-valued kernels”, Adv. Math. 62 (1986), 7–48.

    Article  MATH  Google Scholar 

  17. Schulze, B.-W.: Pseudo-Differential Operators on Manifolds with Singularities. North-Holland, Amsterdam, 1991.

    MATH  Google Scholar 

  18. Schwartz, J.: “A remark on inequalities of Calderon-Zygmund type for vector-valued functions”, Comm. Pure Appl. Math. XIV (1961), 785–799.

    Article  Google Scholar 

  19. Stein, E.M.: Topics in Harmonic Analysis Related to Littlewood-Paley Theory. Princeton University Press, Princeton, 1970.

    MATH  Google Scholar 

  20. Stein, E.M.: Harmonic Analysis: Real-Variables Methods, Orthogonality and Oscillatory Integrals. Princeton University Press, Princeton, 1993.

    Google Scholar 

  21. Triebel, H.: Interpolation Theory, Function Spaces, Differential Operators. North-Holland, Amsterdam, 1978.

    Google Scholar 

  22. Zimmermann, F.: “On vector-valued Fourier multiplier theorems”, Studia Math. 93 (1989), 201–222.

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 1999 Springer Basel AG

About this chapter

Cite this chapter

Hieber, M. (1999). Operator Valued Fourier Multipliers. In: Escher, J., Simonett, G. (eds) Topics in Nonlinear Analysis. Progress in Nonlinear Differential Equations and Their Applications, vol 35. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8765-6_17

Download citation

  • DOI: https://doi.org/10.1007/978-3-0348-8765-6_17

  • Publisher Name: Birkhäuser, Basel

  • Print ISBN: 978-3-0348-9764-8

  • Online ISBN: 978-3-0348-8765-6

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics