Intrinsically Hölder Sections in Metric Spaces

We introduce the notion of intrinsically Holder graphs in metric spaces that generalized the one of ¨ intrinsically Lipschitz sections. This concept is relevant because it has many properties similar to Holder maps but ¨ is profoundly different from them. We prove some relevant results as the Ascoli-Arzela 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 Holder graphs. These last three properties are new also in the Lipschitz case.