L₁-estimates for eigenfunctions of the Dirichlet Laplacian

For d ∈ N and Ω ≠ ∅ an open set in R^d, we consider the eigenfunctions φ of the Dirichlet Laplacian -Δ_Ω of Ω. If φ is associated with an eigenvalue below the essential spectrum of -Δ_Ω we provide estimates for the L₁-norm of Φ in terms of its L₂-norm and spectral data. These L₁-estimates are then used in the comparison of the heat content of Ω 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 -Δ_Ω which are associated with a discrete eigenvalue of H_Ω, belong to L₁(Omega).