Weighted heat traces of the Dirichlet Laplacian on Lipschitz domains

Abstract

We prove an asymptotic expansion of a weighted heat trace of the Dirichlet Laplacian on a bounded Lipschitz domain. The weight is given by a power of the distance to the boundary times a bounded function that is continuous near the boundary. Depending on the exponent of the weight, we obtain one-term or two-term asymptotics. We present two applications, namely a version of one-term asymptotics of weighted Riesz means for arbitrary orders, as well as convergence results for a family of measures that describes localization of Laplace eigenfunctions.