Analysis on local Dirichlet spaces. II: Upper Gaussian estimates for the fundamental solutions of parabolic equations. (English) Zbl 0854.35015

[Part I, cf. J. Reine Angew. Math. 456. 173-196 (1994; Zbl 0806.53041).]
The author studies the behaviour of solutions of the parabolic equation \(L_t u= \partial_t u\) on \(\mathbb{R}\times X\), where \(X\) is a locally compact Hausdorff space and for each \(t\in \mathbb{R}\), \(L_t\) is the operator associated to a regular Dirichlet form on \(L^2(X, m)\). The main results are an integral upper Gaussian estimate for the fundamental solution and a pointwise estimate of similar type. The conditions under which these results are proved are very general: among the operators \(L_t\) included are the Laplace-Beltrami operator on a Riemannian manifold, weighted uniformly elliptic operators, and subelliptic operators.
[For part III, see the review below].
Reviewer: J.Urbas (Bonn)


35B45 A priori estimates in context of PDEs
35A08 Fundamental solutions to PDEs
58J35 Heat and other parabolic equation methods for PDEs on manifolds