Heat Kernel Estimates Of Non-Local Dirichlet Forms On Doubling Spaces And Parabolic Harnack Inequality

The heat kernel theory,generalizing the heat equation theory on Euclidean spaces,is widely applied to partial differential equations,geometric analysis,fractals,graghs and stochastic processes.An elementary part of heat kernel theory is the upper and lower estimates,as well as their weak forms such as parabolic Harnack inequalities.A general setting consists of a metric measure space and a regular Dirichlet form,where heat kernel estimates follow generally two approaches,either analytic or probabilistic.The analytic approach is based on the abstract parabolic maximal principle and construction of auxiliary functions,while the probabilistic method concerns stopping times of a Hunt process,especially Lévy systems and the Meyer decomposition.The best estimates have been obtained analytically for Dirichlet forms that are either strongly local(sub-Gaussian estimate)or purely jump-type with regular scalings(stable-like estimate).However,so far only the probabilistic method works for jump-type forms on doubling spaces.To overcome the difficulty for the analytic approach in this case,we modify DaviesGaffney method,which was studied for local forms by Andres and Barlow,essentially in how we rescale caloric functions in different regions and what functions are rescaled in this procedure.The modified method deduces a survival estimate,while the on-diagonal upper estimate follows from the mean-value inequality.Together with truncation techniques,these conclusions yield the best upper estimate,which is the main result of our work.The lower estimate inherits the known analytic method for regular forms but combines the diffusion analogue,where new techniques are involved in order to obtain the lemma of growth and compute the energy of mixed forms.We also obtain necessity of the conditions,for example,two-sided jump kernel bounds,the generalized capacity condition,the Faber-Krahn inequality,the reverse doubling property,etc.Finally,this thesis also contains probabilistic elements,improving the existence criterion for jump kernels.