Analysis on local Dirichlet spaces. II. Upper Gaussian estimates for the fundamental solutions of parabolic equations

expl --lJL| 4K (t-S) K (t-s) uniformly for all points (s, x) and (t, y) eRxX with s0 the RHS of (0. 4) can be estimated by ^^\ *fᵗ S) / with a constant C = C (). For parabolic divergence form operators on R N this type of estimate is due to E. B. Davies 6 improving previous results by D. G. Aronson 1. For Laplace-Beltrami operators on Riemannian manifolds it is due to P. Li and S. T. Yau 23 (whose result was improved by E. B. Davies, L. Saloff-Coste, N. Varopoulos and many others). Finally, for Hrmander type and general subelliptic operators on R N this Gaussian estimate is due to D. Jerison and A. Sanchez-Calle 18 and to S. Kusuoka and D