Off-diagonal bounds of non-Gaussian type for the Dirichlet heat kernel

The paper considers the heat kernel K-Omega(t, x, y) of the operator -Delta on a proper Euclidean domain Omega, with Dirichlet boundary conditions. A general pointwise lower bound for K-Omega, which is valid for t larger than a suitable t(0)(x, y), is proved (the short-time behaviour being well unde rstood). The resulting non-Gaussian bounds describe simultaneously both the case of bounded domains and the case, modelled on the half-space example, of domains which satisfy a twisted infinite internal cone condition. Bounds for the Green's function are given as well.