A $({\phi }_n,\phi )$-Poincaré inequality on John domain

Given a bounded domain $\mathrm{\Omega }\subset {\mathbb{R}}^n$ with $n\ge 2$, let $\phi$ is a Young function satisfying the doubling condition with the constant $K_{\phi }<2^n$. If $\mathrm{\Omega }$ is a John domain, we show that $\mathrm{\Omega }$ supports a $({\phi }_n,\phi )$-Poincaré inequality. Conversely, assume additionally that $\mathrm{\Omega }$ is simply connected domain when $n=2$ or a bounded domain which is quasiconformally equivalent to some uniform domain when $n\ge 3$. If $\mathrm{\Omega }$ supports a $({\phi }_n,\phi )$-Poincaré inequality, we show that it is a John domain.