# $L_1$-Estimates for Eigenfunctions of the Dirichlet Laplacian
For $d \in {\bf N}$ and $\Omega \ne \emptyset$ an open set in ${\bf R}^d$, we consider the eigenfunctions $\Phi$ of the Dirichlet Laplacian $-\Delta_\Omega$ of $\Omega$. We do {\it not} require $\Omega$ to be of finite volume. % If $\Phi$ is associated with an eigenvalue below the essential spectrum of $-\Delta_\Omega$, we provide estimates for the $L_1$-norm of $\Phi$ in terms of the $L_2$-norm of $\Phi$ and suitable spectral data of $-\Delta_\Omega$. The main idea in obtaining such estimates consists in finding a---sufficiently small---subset $\Omega' \subset \Omega$ where $\Phi$ is localized in the sense that $\Phi$ decays exponentially as one moves away from $\Omega'$. These $L_1$-estimates are then used in the comparison of the heat content of $\Omega$ at time $t>0$ and the heat trace at times $t' > 0$, where a two-sided estimate is established. \vskip.5em This is joint work with Michiel van den Berg (Bristol) and J\"urgen Voigt (Dresden), with improvements by Hendrik Vogt (Dresden).