Hardy's inequality and Green function on metric measure spaces

We prove an abstract form of Hardy's inequality for local and non-local regular Dirichlet forms on metric measure spaces, using the Green operator of the Dirichlet form in question. Under additional assumptions such as the volume doubling, the reverse volume doubling, and certain natural estimates of the Green function, we obtain the classical form of Hardy's inequality containing distance to a reference point or set. (C) 2021 Elsevier Inc. All rights reserved.

Automated summary

This paper proves an abstract form of Hardy's inequality for local and non-local regular Dirichlet forms on metric measure spaces, using the Green operator. With additional assumptions such as volume doubling, reverse volume doubling, and certain estimates of the Green function, the classic form of Hardy's inequality containing distance to a reference point or set is obtained.