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 .
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Michiel van den Berg, Rainer Hempel, Jürgen Voigt, -estimates for eigenfunctions of the Dirichlet Laplacian. J. Spectr. Theory 5 (2015), no. 4, pp. 829–857DOI 10.4171/JST/115