Intrinsically H\"older sections in metric spaces

Abstract

We introduce a notion of intrinsically H\"older graphs in metric spaces. Following a recent paper of Le Donne and the author, we prove some relevant results as the Ascoli-Arzel\`a 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 $\R$ or \C, a convex set and an equivalence relation for intrinsically H\"older graphs. These last three properties are new also in the Lipschitz case. Throughout the paper, we use basic mathematical tools.Comment: We use (1) as the main definition. In Ascoli-Arzel\'a 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:2205.0208