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

Abstract: Let $\Omega$ be a bounded domain in $\mathbb{R}^n$ with $n\ge2$ and $s\in(0,1)$. Assume that $\phi : [0, \infty) \to [0, \infty)$ be a Young function obeying the doubling condition with the constant $K_\phi<2^{\frac{n}{s}}$. We demonstrate that $\Omega $ supports a $(\phi_\frac{n}{s}, \phi)$-Poincar\'e 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 $n\ge3$ or a simply connected domain when $n=2$. We demonstrate $\Omega$ is a John domain if a $(\phi_\frac{n}{s}, \phi)$-Poincar\'e inequality holds.