$L_1$-estimates for eigenfunctions of the Dirichlet Laplacian

For d ∈ N and 6 ? an open set in R d , we consider the eigenfunc- tionsof the Dirichlet Laplacian − of . Ifis associated with an eigenvalue below the essential spectrum of − we provide estimates for the L1-norm ofin terms of its L2-norm and spectral data. These L1- estimates are then used in the comparison of the heat content of 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 − which are associated with a discrete eigenvalue of H, belong to L1().