A purely metric characterization of supercritical Sobolev spaces

Abstract

In this paper we provide a purely metric and derivative-free characterization of continuous mappings in the local supercritical Sobolev space. More specifically, we show that a continuous mapping $f\colon\mathbb{R}^n\to\mathbb{R}^m$ lies in the local Sobolev space $W^{1,p}_\text{loc}(\mathbb{R}^n\colon\mathbb{R}^m)$ if and only if $f$ is a $(p,1-n/p)$-compactly H\"older mapping, where $p>n$. In fact, we prove the following more general result: whenever $X$ is a complete $Q$-Ahlfors regular metric space supporting a $p$-Poincar\'e inequality with $p>Q$, and $V$ is any Banach space, the compactly H\"older class $CH^{p,1-Q/p}(X\colon V)$ consists precisely of the continuous representatives of the local Haj\l asz and Newtonian Sobolev spaces $M^{1,p}_\text{loc} (X \colon V)$ and $N^{1,p}_\text{loc}(X\colon V)$. We also establish quantitative comparisons between the corresponding seminorms.