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)

MSC:

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

Keywords:

Dirichlet form; Gaussian estimate

Citations:

Zbl 0854.35016; Zbl 0806.53041