A \((\phi_\frac{n}{s}, \phi)\)-Poincaré inequality on John domains

Let \(\Omega\) $\mathrm{\Omega }$ be a bounded domain in \(\mathbb{R}^n\) ${\mathbb{R}}^n$ with \(n\ge2\) $n\ge 2$ and \(s\in(0,1)\) $s\in (0,1)$ . Assume that \(\phi \colon [0, \infty) \to [0, \infty)\) $\varphi :[0,\infty )\to [0,\infty )$ is a Young function obeying the doubling condition with the constant \(K_\phi< 2^{\frac{n}{s}}\) $K_{\varphi }<2^{\frac{n}{s}}$ . We demonstrate that \(\Omega\) $\mathrm{\Omega }$ supports a \((\phi_\frac{n}{s}, \phi)\) $({\varphi }_{\frac{n}{s}},\varphi )$ -Poincaré inequality if it is a John domain. Alternatively, assume further that \(\Omega\) $\mathrm{\Omega }$ is a bounded domain that is quasiconformally equivalent to a uniform domain (for \(n\geq3\) $n\ge 3$ ) or a simply connected domain (for \(n=2\) $n=2$ ), then we show that \(\Omega\) $\mathrm{\Omega }$ is a John domain if a \((\phi_\frac{n}{s}, \phi)\) $({\varphi }_{\frac{n}{s}},\varphi )$ -Poincaré inequality holds.