Fine properties of Newtonian functions and the Sobolev capacity on metric measure spaces

TL;DR

This paper explores the regularity and capacity properties of Newtonian functions in metric measure spaces, establishing quasi-continuity, boundedness, and continuity under certain integrability conditions, with special focus on borderline cases.

Contribution

It provides new results on the regularity, quasi-continuity, and capacity of Newtonian functions, especially in borderline integrability cases, extending the theory of Sobolev spaces to metric measure spaces.

Findings

Newtonian functions are quasi-continuous assuming density of continuous functions.

Sobolev capacity is shown to be an outer capacity.

Under high integrability, Newtonian functions are bounded and continuous.

Abstract

Newtonian spaces generalize first-order Sobolev spaces to abstract metric measure spaces. In this paper, we study regularity of Newtonian functions based on quasi-Banach function lattices. Their (weak) quasi-continuity is established, assuming density of continuous functions. The corresponding Sobolev capacity is shown to be an outer capacity. Assuming sufficiently high integrability of upper gradients, Newtonian functions are shown to be (essentially) bounded and (H\"older) continuous. Particular focus is put on the borderline case when the degree of integrability equals the "dimension of the measure". If Lipschitz functions are dense in a Newtonian space on a proper metric space, then locally Lipschitz functions are proven dense in the corresponding Newtonian space on open subsets, where no hypotheses (besides being open) are put on these sets.