A $(φ_n, φ)$-Poincaré inequality on John domain

Abstract: Given a bounded domain $\Omega \subset {\mathbb R}^{n}$ with $n\ge2$, let $\phi $ is a Young function satisfying the doubling condition with the constant $K_\phi<2^{n}$. If $\Omega$ is a John domain, we show that $\Omega $ supports a $(\phi_{n}, \phi)$-Poincar\'e inequality. Conversely, assume additionally that $\Omega$ is simply connected domain when $n=2$ or a bounded domain which is quasiconformally equivalent to some uniform domain when $n\ge3$. If $\Omega$ supports a $(\phi_n, \phi)$-Poincar\'e inequality, we show that it is a John domain.