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 −∆Ω of Ω. If Φ is associated with an eigenvalue below the essential spectrum of −∆Ω we provide estimates for the L1-norm of Φ in terms of its L2-norm and spectral data. These L1estimates 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 −∆Ω which are associated with a discrete eigenvalue of HΩ, belong to L1(Ω)