Intrinsically Hölder sections in metric spaces

Abstract

We introduce a notion of intrinsically Hölder graphs in metric spaces. Following a recent paper of Le Donne and the author, we prove some relevant results as the Ascoli-Arzelà compactness Theorem, Ahlfors-David regularity and the Extension Theorem for this class of sections. In the first part of this note, thanks to Cheeger theory, we define suitable sets in order to obtain a vector space over $\mathbb{R}$ or $\C,$ a convex set and an equivalence relation for intrinsically Hölder graphs. These last three properties are new also in the Lipschitz case. Throughout the paper, we use basic mathematical tools.

We use (1) as the main definition. In Ascoli-Arzelá we can use the second definition because we consider compact subset. In Proposition 1.5 Y must be bounded. arXiv admin note: substantial text overlap with arXiv:2205.02086