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

Abstract

This note is a follow up on our recent paper with L. Malý (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-Carathéodory 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-Carathéodory 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-Carathéodory property.