Sharp boundary estimates for elliptic operators

We prove sharp L-2 boundary decay estimates for the eigenfunctions of certain second order elliptic operators acting in a bounded region, and of their first space derivatives, using only the Hardy inequality These imply L-2 boundary decay properties of the heat kernel and spectral density. We deduce bounds on the rate of convergence of the eigenvalues when the region is slightly reduced in size. It is remarkable that several of the bounds do not involve the space dimension