Criteria for optimal global integrability of Hajłasz–Sobolev functions

Abstract

The author establishes some geometric criteria for a domain of ${\mathbb R}^n$ with $n\ge2$ to support a $(pn/(n-ps),p)_s$-Hajłasz–Sobolev–Poincaré imbedding with $s\in(0,1]$ and $p\in(n/(n+s), n/s)$ or an $s$-Hajłasz–Trudinger imbedding with $s\in(0,1]$.