Removable sets and $L^p$-uniqueness on manifolds and metric measure spaces

TL;DR

This paper investigates how removing small sets from metric measure spaces affects the essential self-adjointness and $L^p$-uniqueness of symmetric diffusion operators, providing characterizations based on capacities and Hausdorff dimension.

Contribution

It offers new criteria for $L^p$-uniqueness after removing sets, including a truncation result for potentials, and applies these to various geometric contexts.

Findings

Self-adjointness depends on the size of the removed set.

Capacities and Hausdorff dimension characterize critical set sizes.

Laplace operators on certain spaces are determined by their regular parts.

Abstract

We study symmetric diffusion operators on metric measure spaces. Our main question is whether or not the restriction of the operator to a suitable core continues to be essentially self-adjoint or $L^p$-unique if a small closed set is removed from the space. The effect depends on how large the removed set is, and we provide characterizations of the critical size in terms of capacities and Hausdorff dimension. As a key tool we prove a truncation result for potentials of nonnegative functions. We apply our results to Laplace operators on Riemannian and sub-Riemannian manifolds and on metric measure spaces satisfying curvature dimension conditions. For non-collapsing Ricci limit spaces with two-sided Ricci curvature bounds we observe that the self-adjoint Laplacian is already fully determined by the classical Laplacian on the regular part.