$L_1$-estimates for eigenfunctions of the Dirichlet Laplacian

Abstract

For $d\in \mathbb{N}$ and $\Omega \ne \varnothing$ an open set in ${\mathbb{R}}^d$, 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 $t^{\prime }>0$, 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 $L_1(\Omega )$.