Analytic semigroups of pseudodifferential operators on vector-valued Sobolev spaces

  • B. Barraza Martínez
  • R. Denk
  • J. Hernández Monzón
Article
First Online:
Received:

Abstract

In this paper we study continuity and invertibility of pseudodifferential operators with non-regular Banach space valued symbols. The corresponding pseudodifferential operators generate analytic semigroups on the Sobolev spaces W p k (ℝ n , E) with k ∈ ℕ0, 1 ≤ p ≤ ∞. Here E is an arbitrary Banach space. We also apply the theory to solve non-autonomous parabolic pseudodifferential equations in Sobolev spaces.

Keywords

pseudodifferential operators vector-valued Sobolev spaces generation of analytic semigroup 

Mathematical subject classification

35S05 47D06 35R20 
This is a preview of subscription content, log in to check access.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    H. Amann. Linear and Quasilinear Parabolic Problems. Volumen I, Birkhäuser Verlag (1995).CrossRefMATHGoogle Scholar
  2. [2]
    H. Amann. Operator-Valued Fourier Multipliers, Vector-Valued Besov Spaces, and Applications. Math. Nachr., 186 (1997), 5–56.CrossRefMATHMathSciNetGoogle Scholar
  3. [3]
    H. Amann. Coagulation-Fragmentation Processes. Arch. Rat. Mech. Anal., 151 (2000), 339–366.CrossRefMATHMathSciNetGoogle Scholar
  4. [4]
    H. Amann. EllipticOperators with Infinite Dimensional State Spaces. J. Evol. Equ. 1 (2001), 143–188.CrossRefMATHMathSciNetGoogle Scholar
  5. [5]
    H. Amann. Vector-Valued Distributions and Fourier Multipliers. Zürich 2003, Download from 〈http://www.math.unizh.ch/amann/files/distributions.pdf.Google Scholar
  6. [6]
    B. Barraza Martínez. Pseudodifferentialoperatoren mit nichtregulären banachraumwertigen Symbolen. Dissertation, Johannes Gutenberg-Universität Mainz, (2009).Google Scholar
  7. [7]
    J. Bergh and J. Löfstrom. Interpolation Spaces, Springer (1976).CrossRefMATHGoogle Scholar
  8. [8]
    J. Bourgain. Vector-valued Hausdorff-Young Inequalities and Applications, in Geometric Aspects of Functional Analysis, (1986/87), 239–247.Google Scholar
  9. [9]
    M. Girardi and L. Weis. Operator-Valued Fourier Multiplier Theorems on Besov Spaces. Math. Nachr. 251 (2003), 34–51.CrossRefMATHMathSciNetGoogle Scholar
  10. [10]
    D. Guidotti. On Elliptic Systems in L 1. Osaka J. Math., 30 (1993), 397–429.MathSciNetGoogle Scholar
  11. [11]
    T. Kato. Linear evolution equations of “hyperbolic” type. J. Fac. Sci. Univ. Tokyo, Sect. I, 17 (1970), 241–258.MATHMathSciNetGoogle Scholar
  12. [12]
    T. Kato. Abstract evolution equations, linear and quasilinear, revisited. In: H. Komatsu (Ed.), “FunctionalAnalysis and Related Topics”, Lecture Notes in Math., Springer (1993), 103–125.Google Scholar
  13. [13]
    Chr. Kiehn. Analytic Semigroup of Pseudodifferential Operators on L p (ℝn, E). Preprint Reihe des Fachbereichs Mathematik Nr 10, Johannes Gutenberg-Universität, (2001).Google Scholar
  14. [14]
    Chr. Kiehn. Analytic Semigroups of Pseudodifferential Operators on Vector-Valued Function Spaces. Shaker Verlag (2003).MATHGoogle Scholar
  15. [15]
    H. Kumano-go. Pseudodifferential Operators,MIT Press, Cambridge, MA (1981).Google Scholar
  16. [16]
    F. Lancien, G. Lancien and Ch. Le Merdy. A Joint Functional Calculus for Sectorial Operators with Commuting Resolvents. Proc. London Math. Soc., 77(3) (1998), 387–414.CrossRefMathSciNetGoogle Scholar
  17. [17]
    A. Lunardi. Analytic Semigroups and Optimal Regularity in Parabolic Problems. Birkhäuser Verlag, Boston (1995).CrossRefMATHGoogle Scholar
  18. [18]
    A. Noll, R. Haller and H. Heck. Mikhlin’s Theorem for Operator-Valued Multipliers in n Variables. Math. Nach., 244 (2002), 110–130.CrossRefMATHMathSciNetGoogle Scholar
  19. [19]
    P. Portal and Ž. Štrkalj. Pseudodifferential operators on Bochner Spaces and an application. Math. Z., 253 (2006), 805–819.CrossRefMATHMathSciNetGoogle Scholar
  20. [20]
    H. Schmeisser. Vector-Valued Sobolev and Besov Spaces. Seminar Analysis of the Karl-Weierstraß-Institute (1986), Band 96.Google Scholar
  21. [21]
    L. Weis. Operator-Valued Fourier Multiplier Theorems and Maximal L p-Regularity. Math. Ann., 319 (2001), 735–758.CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Sociedade Brasileira de Matemática 2014

Authors and Affiliations

  • B. Barraza Martínez
    • 1
  • R. Denk
    • 1
  • J. Hernández Monzón
    • 1
  1. 1.Departamento de MatemáticasUniversidad del NorteBarranquillaColombia

Personalised recommendations