  For $d \in \N$ and $\Omega \ne \emptyset$ an open set in $\R^d$, we consider
the eigenfunctions $\Phi$ of the Dirichlet Laplacian $-\Delta_\Omega$ of
$\Omega$. 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 its $L_2$-norm and spectral data. 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. We
furthermore show that all eigenfunctions of $-\Delta_\Omega$ which are
associated with a discrete eigenvalue of $H_\Omega$, belong to $L_1(\Omega)$.
