Skip to main content
SpringerLink
Log in

An abstract approach to optimal decay of functions and operator semigroups

  • Published:
Israel Journal of Mathematics Aims and scope Submit manuscript

Abstract

We provide a new and significantly shorter optimality proof of recent quantified Tauberian theorems, both in the setting of vector-valued functions and of C0-semigroups, and in fact our results are also more general than those currently available in the literature. Our approach relies on a novel application of the open mapping theorem.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. W. Arendt, C. J. K. Batty, M. Hieber and F. Neubrander, Vector-valued Laplace Transforms and Cauchy Problems, Monographs in Mathematics, Vol. 96, Birkhauser, Basel, 2011.

    Book  Google Scholar 

  2. C. J. K. Batty, A. Borichev and Y. Tomilov, L p-tauberian theorems and L p -rates for energy decay, Journal of Functional Analysis 270 (2016), 1153–1201.

    Article  MathSciNet  Google Scholar 

  3. C. J. K. Batty, R. Chill and Y. Tomilov, Fine scales of decay of operator semigroups, Journal of the European Mathematical Society 18 (2016), 853–929.

    Article  MathSciNet  Google Scholar 

  4. C. J. K. Batty and T. Duyckaerts, Non-uniform stability for bounded semi-groups in Banach spaces, Journal of Evolution Equations 8 (2008), 765–780.

    Article  MathSciNet  Google Scholar 

  5. A. Borichev and Y. Tomilov, Optimal polynomial decay of functions and operator semigroups, Mathematische Annalen 347 (2010), 455–478.

    Article  MathSciNet  Google Scholar 

  6. R. Chill and D. Seifert, Quantified versions of Ingham’s theorem, Bulletin of the London Mathematical Society 48 (2016), 519–532.

    Article  MathSciNet  Google Scholar 

  7. A. Debrouwere and J. Vindas, On the non-triviality of certain spaces of analytic functions. Hyperfunctions and ultrahyperfunctions of fast growth, Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas 112 (2018), 473–508.

    Article  MathSciNet  Google Scholar 

  8. G. Debruyne and J. Vindas, Complex Tauberian theorems for Laplace transforms with local pseudofunction boundary behavior, Journal d’Analyse Mathematique 138 (2019), 799–833.

    Article  MathSciNet  Google Scholar 

  9. G. Debruyne and J. Vindas, Optimal Tauberian constant in Ingham’s theorem for Laplace transforms, Israel Journal of Mathematics 228 (2018), 557–586.

    Article  MathSciNet  Google Scholar 

  10. G. Debruyne and J. Vindas, Note on the absence of remainders in the Wiener—Ikehara theorem, Proceedings of the American Mathematical Society 146 (2018), 5097–5103.

    Article  MathSciNet  Google Scholar 

  11. W. Desch and J. Priiss, Counterexamples for abstract linear Volterra equations, Journal of Integral Equations and Applications 5 (1993), 29–45.

    Article  MathSciNet  Google Scholar 

  12. T. H. Ganelius, Tauberian Remainder Theorems, Lecture Notes in Mathematics, Vol. 232, Springer, Berlin—New York, 1971.

    Book  Google Scholar 

  13. A. E. Ingham, On Wiener’s method in Tauberian theorems, Proceedings of the London Mathematical Society 38 (1933), 458–480.

    MathSciNet  MATH  Google Scholar 

  14. J. Karamata, Weiterfiihrung der N. Wienerschen Methode, Mathematische Zeitschrift 38 (1934), 701–708.

    Article  MathSciNet  Google Scholar 

  15. J. Korevaar, Tauberian Theory. A Century of Developments, Grundlehren der Mathematische Wissenschaften, Vol. 329, Springer, Berlin, 2004.

  16. P. Painlevé, Sur les lignes singulières des fonctions analytiques, Annales de la Faculté des Sciences de Toulouse. Mathématiques, Série 12 (1888), B1–B130.

    MATH  Google Scholar 

  17. A. Peyerimhoff, Über das Anwachsen der C κ-Mittel von Laplace-Integralen auf vertikalen Geraden, Mathematische Annalen 128 (1954), 138–143.

    Article  MathSciNet  Google Scholar 

  18. J. Rozendaal, D. Seifert and R. Stahn, Optimal rates of decay for operator semigroups on Hilbert spaces, Advances in Mathematics 346 (2019), 359–388.

    Article  MathSciNet  Google Scholar 

  19. W. Rudin, Lectures on the Edge-of-the-Wedge Theorem, CBMS Regional Conference Series in Mathematics, Vol. 6, American Mathematical Society, Providence, RI, 1971.

  20. R. Stahn, Decay of C 0-semigroups and local decay of waves on even (and odd) dimensional exterior domains, Journal of Evolution Equations 18 (2018), 1633–1674.

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics: Analysis, Logic and Discrete Mathematics, Ghent University, Krijgslaan 281, B 9000, Ghent, Belgium

    Gregory Debruyne

  2. St. John’s College, St. Giles, Oxford, 0X1 3JP, UK

    David Seifert

Authors
  1. Gregory Debruyne
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. David Seifert
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to David Seifert.

Additional information

G.D. gratefully acknowledges support by Ghent University, through a BOF Ph.D. grant

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Debruyne, G., Seifert, D. An abstract approach to optimal decay of functions and operator semigroups. Isr. J. Math. 233, 439–451 (2019). https://doi.org/10.1007/s11856-019-1918-y

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11856-019-1918-y

Search

Navigation