Analysis on local Dirichlet spaces. III: The parabolic Harnack inequality
This is a sequel to the author's papers [J. Reine Angew. Math. 456, 173-196 (1994; Zbl 0806.53041); Osaka J. Math. 32, No. 2, 275-312 (1995; see the review above)], in which he 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 result here is that the parabolic Harnack inequality for nonnegative solutions is true if and only if the measure of balls in \(X\) satisfies a doubling property and the Poincaré inequality holds. Some consequences of this result are upper and lower Gaussian estimates for the heat kernel associated with the operator.
scientific article; zbMATH DE number 891063