Skip to main content
SpringerLink
Log in

L 1-estimates for eigenfunctions and heat kernel estimates for semigroups dominated by the free heat semigroup

  • Published:
Journal of Evolution Equations Aims and scope Submit manuscript

Abstract

We investigate selfadjoint positivity preserving C 0-semigroups that are dominated by the free heat semigroup on \({\mathbb{R}^d}\). Major examples are semigroups generated by Dirichlet Laplacians on open subsets or by Schrödinger operators with absorption potentials. We show explicit global Gaussian upper bounds for the kernel that correctly reflect the exponential decay of the semigroup. For eigenfunctions of the generator that correspond to eigenvalues below the essential spectrum, we prove estimates of their L 1-norm in terms of the L 2-norm and the eigenvalue counting function. This estimate is applied to a comparison of the heat content with the heat trace of the semigroup.

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. Arendt W., Warma M.: Dirichlet and Neumann boundary conditions: What is in between?. J. Evol. Equ. 3(1), 119–135 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  2. M. van den Berg, R. Hempel and J. Voigt, L 1-estimates for eigenfunctions of the Dirichlet Laplacian, to appear in J. Spectr. Theory.

  3. M. van den Berg and E. B. Davies, Heat flow out of regions in \({\mathbb{R}^m}\), Math. Z. 202 (1989), no. 4, 463–482.

  4. Coulhon T., Sikora A.: Gaussian heat kernel upper bounds via the Phragmén-Lindelöf theorem. Proc. Lond. Math. Soc. 96(2), 507–544 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  5. Davies E. B.: Uniformly elliptic operators with measurable coefficients. J. Funct. Anal. 132(1), 141–169 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  6. M. Keller, D. Lenz, H. Vogt and R. Wojciechowski, Note on basic features of large time behaviour of heat kernels, to appear in J. Reine angew. Math., DOI: 10.1515/crelle-2013-0070.

  7. Li P., Yau S. T.: On the Schrödinger equation and the eigenvalue problem. Comm. Math. Phys. 88(3), 309–318 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  8. Manavi A., Vogt H., Voigt J.: Domination of semigroups associated with sectorial forms. J. Operator Theory 54(1), 9–25 (2005)

    MATH  MathSciNet  Google Scholar 

  9. Ouhabaz E.M.: Comportement des noyaux de la chaleur des opérateurs de Schrödinger et applications à certaines équations paraboliques semi-linéaires. J. Funct. Anal. 238(1), 278–297 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  10. E. M. Ouhabaz and F.-Y. Wang, Sharp estimates for intrinsic ultracontractivity on \({C^{1,\alpha}}\)-domains. Manuscripta Math. 122 (2007), no. 2, 229–244.

Download references

Author information

Authors and Affiliations

  1. Fachbereich 3 – Mathematik, Universität Bremen, 28359, Bremen, Germany

    Hendrik Vogt

Authors
  1. Hendrik Vogt
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hendrik Vogt.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Vogt, H. L 1-estimates for eigenfunctions and heat kernel estimates for semigroups dominated by the free heat semigroup. J. Evol. Equ. 15, 879–893 (2015). https://doi.org/10.1007/s00028-015-0285-3

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00028-015-0285-3

Mathematics Subject Classification

Keywords

Search

Navigation