  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.
