Skip to main content
SpringerLink
Log in

Spectral multiplier theorems and averaged R-boundedness

  • Research Article
  • Published:
Semigroup Forum Aims and scope Submit manuscript

Abstract

Let A be a 0-sectorial operator with a bounded \(H^\infty (\Sigma _\sigma )\)-calculus for some \(\sigma \in (0,\pi ),\) e.g. a Laplace type operator on \(L^p(\Omega ),\, 1< p < \infty ,\) where \(\Omega \) is a manifold or a graph. We show that A has a \(\mathcal {H}^\alpha _2(\mathbb {R}_+)\) Hörmander functional calculus if and only if certain operator families derived from the resolvent \((\lambda - A)^{-1},\) the semigroup \(e^{-zA},\) the wave operators \(e^{itA}\) or the imaginary powers \(A^{it}\) of A are R-bounded in an \(L^2\)-averaged sense. If X is an \(L^p(\Omega )\) space with \(1 \le p < \infty \), R-boundedness reduces to well-known estimates of square sums.

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. Alexopoulos, G.: Spectral multipliers on Lie groups of polynomial growth. Proc. Am. Math. Soc. 120(3), 973–979 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bergh, J., Löfström, J.: Interpolation spaces. An introduction. In: Grundlehren der Mathematischen Wissenschaften, No. 223, pp. x+207. Springer-Verlag, Berlin-New York (1976)

  3. Bonami, A., Clerc, J.-L.: Sommes de Cesàro et multiplicateurs des développements en harmoniques sphériques. Trans. Am. Math. Soc. 183, 223–263 (1973)

    MATH  Google Scholar 

  4. Bourgain, J.: Vector valued singular integrals and the \(H^1\)-BMO duality. In: Probability Theory and Harmonic Analysis (Cleveland, Ohio, 1983). Monogr. Textbooks Pure Appl. Math., vol. 98, pp. 1–19. Dekker, New York (1986)

  5. Cowling, M., Doust, I., McIntosh, A., Yagi, A.: Banach space operators with a bounded \(H^\infty \) functional calculus. J. Aust. Math. Soc. Ser. A 60(1), 51–89 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  6. Clément, P., de Pagter, B., Sukochev, F., Witvliet, H.: Schauder decomposition and multiplier theorems. Studia Math. 138, 135–163 (2000)

    MathSciNet  MATH  Google Scholar 

  7. Chen, P., Ouhabaz, E.M., Sikora, A., Yan, L.: Restriction estimates, sharp spectral multipliers and endpoint estimates for Bochner–Riesz means. Preprint available on arXiv:1202.4052

  8. Diestel, J., Jarchow, H., Tonge, A.: Absolutely summing operators. In: Cambridge Studies in Advanced Mathematics, vol. 43, pp. xvi+474. Cambridge University Press, Cambridge (1995)

  9. Duelli, M., Weis, L.: Spectral projections, Riesz transforms and \(H^\infty \)-calculus for bisectorial operators. In: Nonlinear Elliptic and Parabolic Problems, Prog. Nonlinear Differential Equations Appl., vol. 64, pp. 99–111. Birkhäuser, Basel (2005)

  10. Duong, X.T.: From the \(L^1\) norms of the complex heat kernels to a Hörmander multiplier theorem for sub-Laplacians on nilpotent Lie groups. Pac. J. Math. 173(2), 413–424 (1996)

    Article  MATH  Google Scholar 

  11. Duong, X.T., Ouhabaz, E.M., Sikora, A.: Plancherel-type estimates and sharp spectral multipliers. J. Funct. Anal. 196(2), 443–485 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  12. Haak, B.H., Kunstmann, P.C.: Admissibility of unbounded operators and wellposedness of linear systems in Banach spaces. Int. Equ. Oper. Theory 55(4), 497–533 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  13. Haase, M.: Functional calculus for groups and applications to evolution equations. J. Evolut. Equ. 7(3), 529–554 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  14. Haase, M.: The functional calculus for sectorial operators. In: Operator Theory: Advances and Applications, vol. 169, pp. xiv+392. Birkhäuser Verlag, Basel (2006)

  15. Haase, M.: A transference principle for general groups and functional calculus on UMD spaces. Math. Ann. 345, 245–265 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  16. Haase, M.: Transference principles for semigroups and a theorem of Peller. J. Funct. Anal. 261(10), 2959–2998 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  17. Hytönen, T., Veraar, M.: \(R\)-boundedness of smooth operator-valued functions. Integral Equ. Oper. Theory 63(3), 373–402 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  18. Hörmander, L.: The analysis of linear partial differential operators. I. Distribution theory and Fourier analysis, 2nd edn. In: Grundlehren der Mathematischen Wissenschaften (Fundamental Principles of Mathematical Sciences), vol. 256, pp. xii+440. Springer-Verlag, Berlin (1990)

  19. Kalton, N., Weis, L.: The \(H^\infty \)-Functional Calculus and Square Function Estimates. Preprint available on arXiv:1411.0472

  20. Kriegler, C: Spectral multipliers, \(R\)-bounded homomorphisms, and analytic diffusion semigroups. Ph.D. thesis. http://digbib.ubka.uni-karlsruhe.de/volltexte/1000015866

  21. Kriegler, C.: Spectral multipliers for wave operators. Submitted, Preprint available on arXiv:1210.4261

  22. Kriegler, C.: Hörmander functional calculus for Poisson estimates. Int. Equ. Oper. Theory 80(3), 379–413 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  23. Kriegler, C., Weis, L.: Paley–Littlewood decomposition for sectorial operators and interpolation spaces. Math. Nachr. 289(11–12), 1488–1525 (2016)

  24. Kriegler, C., Weis, L.: Spectral multiplier theorems via \(H_i\) calculus and \(R\)-bounds (Submitted). Preprint available on arXiv:1612.04142

  25. Kriegler, C., Le Merdy, C.: Tensor extension properties of \(C(K)\)-representations and applications to unconditionality. J. Aust. Math. Soc. 88(2), 205–230 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  26. Kunstmann, P.C., Uhl, M.: Spectral multiplier theorems of Hörmander type on Hardy and Lebesgue spaces. Preprint available on arxiv:1209.0358

  27. Kunstmann, P.C., Weis, L.: Maximal \(L_p\)-regularity for parabolic equations, Fourier multiplier theorems and \(H^\infty \)-functional calculus. Functional analytic methods for evolution equations. Based on lectures given at the autumn school on evolution equations and semigroups, Levico Terme, Trento, Italy, October 28–November 2, 2001, Lect. Notes Math., vol. 1855, pp. 65–311. Springer, Berlin (2004)

  28. Lebedev, N.: Special functions and their applications. In: Silverman, R.A. (ed.) Unabridged and Corrected Republication, pp. xii+308. Dover Publications, Inc., New York (1972). Revised edition, translated from the Russian

  29. Le Merdy, C.: On square functions associated to sectorial operators. Bull. Soc. Math. France 132(1), 137–156 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  30. Ouhabaz, E.M.: Analysis of heat equations on domains. In: London Mathematical Society Monographs Series, vol. 31, pp. xiv+284. Princeton University Press, Princeton, NJ (2005)

  31. Rudin, W.: Principles of Mathematical Analysis, 3rd edn. In: International Series in Pure and Applied Mathematics, pp. x+342. McGraw-Hill Book Co., New York-Auckland-Düsseldorf (1976)

    MATH  Google Scholar 

  32. Runst, T., Sickel, W.: Sobolev spaces of fractional order, Nemytskij operators and nonlinear partial differential equations. In: De Gruyter Series in Nonlinear Analysis and Applications, vol. 3, pp. x+547. Walter de Gruyter & Co., Berlin (1996)

  33. Stempak, K.: Multipliers for eigenfunction expansions of some Schrödinger operators. Proc. Am. Math. Soc. 93(3), 477–482 (1985)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

We thank the anonymous referee for the careful reading of the manuscript.

Author information

Authors and Affiliations

  1. CNRS, LMBP, Université Clermont Auvergne, 63000, Clermont-Ferrand, France

    Christoph Kriegler

  2. Institut für Analysis, Fakultät für Mathematik, Karlsruher Institut für Technologie, Englerstrasse 2, 76131, Karlsruhe, Germany

    Lutz Weis

Authors
  1. Christoph Kriegler
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Lutz Weis
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Christoph Kriegler.

Additional information

Communicated by Markus Haase.

The first named author acknowledges financial support from the Franco-German University (DFH-UFA) and the Karlsruhe House of Young Scientists (KHYS). The second named author acknowledges the support by the DFG through CRC 1173.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kriegler, C., Weis, L. Spectral multiplier theorems and averaged R-boundedness. Semigroup Forum 94, 260–296 (2017). https://doi.org/10.1007/s00233-017-9848-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00233-017-9848-7

Keywords

Search

Navigation