On the essential self-adjointness of singular sub-Laplacians

TL;DR

This paper establishes a broad criterion for the essential self-adjointness of sub-Laplacians on complete sub-Riemannian manifolds with singular measures, ensuring well-posedness of associated differential operators.

Contribution

It introduces a new criterion for essential self-adjointness of sub-Laplacians on singular sub-Riemannian manifolds, extending previous results to more general settings.

Findings

Intrinsic sub-Laplacian is essentially self-adjoint on equiregular components.

Self-adjointness holds under mild regularity and absence of characteristic points.

Results apply to sub-Riemannian manifolds with singular measures.

Abstract

We prove a general essential self-adjointness criterion for sub-Laplacians on complete sub-Riemannian manifolds, defined with respect to singular measures. As a consequence, we show that the intrinsic sub-Laplacian (i.e. defined w.r.t. Popp's measure) is essentially self-adjoint on the equiregular connected components of a sub-Riemannian manifold. This result holds under mild regularity assumptions on the singular region, and when the latter does not contain characteristic points.