Sobolev functions in the critical case are uniformly continuous in $s$-Ahlfors regular metric spaces when $s \le 1$

TL;DR

This paper proves that functions in the Haj extl{}asz-Sobolev space on s-Ahlfors regular metric spaces are uniformly continuous for s less than or equal to 1, extending understanding of function regularity in these spaces.

Contribution

The paper establishes uniform continuity of Sobolev functions in s-Ahlfors regular metric spaces specifically for the critical case when s ≤ 1, filling a gap in the theory.

Findings

Functions in M^{1,s} are uniformly continuous for s ≤ 1 in s-Ahlfors regular spaces.

The result applies to the critical case s=1, which was previously unresolved.

Provides new insights into the regularity of Sobolev functions in metric measure spaces.

Abstract

We prove that functions in the Haj\l{}asz-Sobolev space $M^{1,s}$ on an $s$-Ahlfors regular metric space are uniformly continuous when $s\le 1$.