-estimates for eigenfunctions of the Dirichlet Laplacian

  • Michiel van den Berg

    University of Bristol, UK
  • Rainer Hempel

    Technische Universität Braunschweig, Germany
  • Jürgen Voigt

    Technische Universität Dresden, Germany


For and an open set in , we consider the eigenfunctions of the Dirichlet Laplacian of . If is associated with an eigenvalue below the essential spectrum of we provide estimates for the -norm of in terms of its -norm and spectral data. These -estimates are then used in the comparison of the heat content of at time and the heat trace at times , where a two-sided estimate is established. We furthermore show that all eigenfunctions of which are associated with a discrete eigenvalue of , belong to .

Cite this article

Michiel van den Berg, Rainer Hempel, Jürgen Voigt, -estimates for eigenfunctions of the Dirichlet Laplacian. J. Spectr. Theory 5 (2015), no. 4, pp. 829–857

DOI 10.4171/JST/115