Analysis on local Dirichlet spaces. II. Upper Gaussian estimates for the fundamental solutions of parabolic equations
Karl‐Theodor Sturm · Osaka City University (Osaka City University) · 1995
P 2 (w) \•expl -ί-lJ£L| 4K(t-S ) K(t-s)uniformly for all points (s,x) and (t,y)eRxX with s 0 the RHS of (0.4) can be estimated bywith a constant C = C(ε).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 Hόrmander type and general subelliptic operators on R N this Gaussian estimate is due to D. and to S. Kusuoka and D.W. Stroock [19].In the particular time-independent case L t = L, (0.4) is just an estimate for the heat kernel for L. From this heat kernel estimate one easily deduces the following Green function estimate: g(χ,y)