Extension method in Dirichlet spaces with sub-Gaussian estimates and applications to regularity of jump processes on fractals

We investigate regularity properties of some non-local equations defined on Dirichlet spaces equipped with sub-gaussian estimates for the heat kernel associated to the generator. We prove that weak solutions for homogeneous equations involving pure powers of the generator are actually H\"older continuous and satisfy an Harnack inequality. Our methods are based on a version of the Caffarelli-Silvestre extension method which is valid in any Dirichlet space and our results complement the existing literature on solutions of PDEs on classes of Dirichlet spaces such as fractals.