Symmetrical Extensions of Dirichlet Operators

TL;DR

This paper constructs and analyzes extensions of Dirichlet operators associated with log-concave measures, providing conditions for their self-adjointness in symmetric differential forms.

Contribution

It introduces new extensions of Dirichlet operators for log-concave measures and establishes criteria for their essential self-adjointness.

Findings

Conditions for self-adjointness in one-dimensional case

Criteria for hypersymmetric part self-adjointness

Extension construction for symmetric differential forms

Abstract

There is constructed and considered the extension of classical Diriclet operator corresponding to uniformly log-concave measure in the space of symmetric differential forms. Sufficient conditions for its essential self-adjointness in one-dimensional case as well as for the same of its "sypersymmetric" part in general situations are given.