Powers Of Generators On Dirichlet Spaces And Applications To Harnack Principles

TL;DR

This paper develops a unified framework for operator powers on Dirichlet spaces, extending results to general metric spaces and applying these to PDEs like Harnack principles.

Contribution

It unifies existing results across geometries and introduces new findings for general metric spaces within Dirichlet space theory.

Findings

Unified approach to operator powers on Dirichlet spaces

New results for general metric measured spaces

Applications to PDEs including Harnack principles

Abstract

We provide a general framework for the realization of powers or functions of suitable operators on Dirichlet spaces. The first contribution is to unify the available results dealing with specific geometries; a second one is to provide new results on rather general metric measured spaces that were not considered before and fall naturally in the theory of Dirichlet spaces. The main tool is using the approach based on subordination and semi-groups by Stinga and Torrea. Assuming more on the Dirichlet space, we derive several applications to PDEs such as Harnack and Boundary Harnack principles.