A purely metric characterization of supercritical Sobolev spaces

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 $f\text{\hspace{0.17em}⁣}:{\mathbb{R}}^n\to {\mathbb{R}}^m$ lies in the local Sobolev space $W_{\text{loc}}^{1,p}({\mathbb{R}}^n\text{\hspace{0.17em}⁣}:{\mathbb{R}}^m)$ if and only if $f$ is a $(p,1-n\mathrm{/}p)$-compactly Hölder 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 $p$-Poincaré inequality with $p>Q$, and $V$ is any Banach space, the compactly Hölder class $C{H}^{p,1-Q\mathrm{/}p}(X\text{\hspace{0.17em}⁣}:V)$ consists precisely of the continuous representatives of the local Hajłasz and Newtonian Sobolev spaces $M_{\text{loc}}^{1,p}(X\text{\hspace{0.17em}⁣}:V)$ and $N_{\text{loc}}^{1,p}(X\text{\hspace{0.17em}⁣}:V)$. We also establish quantitative comparisons between the corresponding seminorms.