Quasicontinuity of $N^{1,\infty}$ functions and the Vitali-Carath\'eodory property on general metric spaces

TL;DR

This paper explores the properties of $N^{1,ty}$ functions on metric spaces, providing examples and characterizations related to quasicontinuity and the Vitali-Carathéodory property.

Contribution

It presents a simple example of a metric space where $L^ty$ has the Vitali-Carathéodory property but $N^{1,ty}$ contains non-quasicontinuous functions, and characterizes when $L^ty$ has this property.

Findings

$L^ty$ can have the Vitali-Carathéodory property without all $N^{1,ty}$ functions being quasicontinuous.

Provided an example of a compact metric space with specific capacity properties.

Characterized conditions under which $L^ty$ possesses the Vitali-Carathéodory property.

Abstract

This note is a follow up on our recent paper with L. Mal\'y (to appear in Rev. Mat. Complut.). We provide a simple example of a compact metric space $\mathcal{P}$ for which $L^\infty(\mathcal{P})$ has the Vitali-Carath\'eodory property, the Sobolev $C_\infty$-capacity is an outer capacity, but the Newtonian space $N^{1,\infty}(\mathcal{P})$ contains functions which are not weakly quasicontinuous. The novelty here is that the Vitali-Carath\'eodory property is satified. We also obtain some related results about quasicontinuous functions in $N^{1,\infty}(\mathcal{P})$ and a characterization of when $L^\infty(\mathcal{P})$ has the Vitali-Carath\'eodory property.