\(L_1\)-estimates for eigenfunctions of the Dirichlet Laplacian (Q906576)

Let \(\Omega\) be an open subset of \(\mathbb{R}^m\). Let \(\Delta\) be the Laplacian with Dirichlet boundary conditions and let \(\Delta\Phi=\lambda\Phi\) be an eigenfunction. The authors provide bounds for the \(L_1\) norm of \(\Phi\) in terms of the spectral data and the \(L_2\)-norm of \(\Phi\). The estimates are often improvements over the usual estimate \(\|\Phi\|_1^2\leq\text{vol}(\Omega)\|\Phi\|_2^2\) if \(\Omega\) has finite volume. The estimates obtained hold for eigenvalues below the essential spectrum and spectral information is used instead of the volume. These estimates are used to obtain a two-sided estimate for the heat content and for the heat trace. There is an introduction to the problem. Section 1 presents estimates for the \(L_1\) norm of eigenvalues. Section 2 establishes an \(L_1\) estimate. Section 3 provides exponential decay estimates. Section 4 deals with a general result for discrete eigenvalues. The final section treats the heat content and the heat trace.