A $({\varphi }_{\frac{n}{s}},\varphi )$-Poincaré inequality in John domain

Abstract

Let $\Omega$ be a bounded domain in ${\mathbb{R}}^n$ with $n\ge 2$ and $s\in (0,1)$. Assume that $\varphi :[0,\infty )\to [0,\infty )$ be a Young function obeying the doubling condition with the constant $K_{\varphi }<2^{\frac{n}{s}}$. We demonstrate that $\Omega$ supports a $({\varphi }_{\frac{n}{s}},\varphi )$-Poincaré inequality if it is is a John domain. Alternately, assume further that $\Omega$ is a bounded domain that is quasiconformally equivalent to some uniform domain when $n\ge 3$ or a simply connected domain when $n=2$. We demonstrate $\Omega$ is a John domain if a $({\varphi }_{\frac{n}{s}},\varphi )$-Poincaré inequality holds.