Asymptotics for the spectral heat function and bounds for integrals of Dirichlet eigenfunctions

We introduce the spectral heat function H associated with the Dirichlet-Lap lace-Beltrami operator Delta(M) on a compact smooth Riemannian manifold M w ith a non-empty smooth boundary. We obtain two-term asymptotics for H witho ut assuming any billiard conditions on M. As a corollary, we obtain estimat es for the integral of the k'th Dirichlet eigenfunction of Delta(M) over M for k --> infinity.