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

2022/04/04 by Michael Hinz, Hinz, Michael, Jun Masamune +3
Mathematics · #28A78 #31C12 #31C15 #31C25 #47B25 #47D07 #58J05 #58J35 #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Functional Analysis (math.FA) #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations #Numerical methods in inverse problems

paper · pdf · doi:10.48550/arxiv.2204.01378

openalex publication_date 2022/04/04 · openalex created_date 2022/05/05 · openalex updated_date 2026/07/28

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.

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