Analysis on local Dirichlet spaces. II: Upper Gaussian estimates for the fundamental solutions of parabolic equations
[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].
scientific article; zbMATH DE number 880172