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 s<t.Note that for every >0 the RHS of (0.4) can be estimated by ^^\ *f^t 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