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

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