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arXiv · 2605.22674

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

Anders Björn·Jana Björn
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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.

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Anders Björn, Jana Björn. 2026-05-21. Quasicontinuity of $N^{1,\infty}$ functions and the Vitali-Carath\'eodory property on general metric spaces. https://arxiv.org/abs/2605.22674

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