H\"older maps from Euclidean spaces to Carnot groups

TL;DR

This paper explores conditions under which Euclidean spaces can be homeomorphically mapped to Carnot groups with a certain Hölder continuity, providing alternative proofs and suggesting directions for sharper bounds.

Contribution

It offers new proofs of existing results on Hölder equivalence between Euclidean spaces and Carnot groups and proposes a potential approach to determine optimal bounds.

Findings

Alternative proofs of Hölder equivalence results

Indication of a method to achieve sharp bounds

Discussion on conditions for Hölder homeomorphisms

Abstract

We give alternative proofs of (unsharp) results of Gromov's on his H\''older equivalence problem: for which $\alpha$ does there exist a $C^\alpha$-homeomorphism of an open set of Euclidean space to an open set of a given Carnot group? We indicate a possible route to sharp bounds.