NASH-TYPE INEQUALITIES AND HEAT KERNELS FOR NON-LOCAL DIRICHLET FORMS

Abstract

We use an elementary method to obtain Nash-type inequalities for non- local Dirichlet forms on d-sets. We obtain two-sided estimates for the corresponding heat kernels if the walk dimensions of heat kernels are less than two; these estimates are obtained by combining probabilistic and analytic methods. Our arguments partly simplify those used in Chen and Kumagi (Heat kernel estimates for stable-like processes on d-sets. Stochastic Process Appl. 108 (2003), 27-62).