Stability of Kac regularity under domination of quadratic forms

TL;DR

This paper proves that Kac regularity of domains remains stable under quadratic form domination and explores applications to Dirichlet forms, Cheeger energies, and Schrödinger operators, also characterizing Sobolev spaces with boundary conditions.

Contribution

It establishes the stability of Kac regularity under quadratic form domination and provides new characterizations of Sobolev spaces with boundary conditions in metric measure spaces.

Findings

Kac regularity is stable under quadratic form domination.

Characterization of Sobolev spaces with Dirichlet boundary conditions.

Applications to measure perturbations, Cheeger energies, and Schrödinger operators.

Abstract

A domain is called Kac regular for a quadratic form on $L^2$ if the closure of all functions vanishing almost everywhere outside a closed subset of the domain coincides with the set of all functions vanishing almost everywhere outside the domain. It is shown that this notion is stable under domination of quadratic forms. As applications measure perturbations of quasi-regular Dirichlet forms, Cheeger energies on metric measure spaces and Schr\"odinger operators on manifolds are studied. Along the way a characterization of the Sobolev space with Dirichlet boundary conditions on domains in infinitesimally Riemannian metric measure spaces is obtained.