Intrinsically H\"older sections in metric spaces

TL;DR

This paper introduces the concept of intrinsically H"older graphs in metric spaces, establishing foundational properties and theorems such as compactness, regularity, and extension results, with new insights even in the Lipschitz case.

Contribution

It defines intrinsically H"older graphs in metric spaces and proves key properties including compactness, regularity, and extension theorems, introducing new structures like vector spaces and convex sets.

Findings

Established Ascoli-Arzelà compactness theorem for these graphs

Proved Ahlfors-David regularity results

Developed extension theorems for intrinsically H"older graphs

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.