A (φₙ, φ)-Poincar\'e inequality on John domain

Given a bounded domain Ω ⊂ Rⁿ with n≥2, let φ is a Young function satisfying the doubling condition with the constant K_φ<2ⁿ. If Ω is a John domain, we show that Ω supports a (φₙ, φ)-Poincar\'e inequality. Conversely, assume additionally that Ω is simply connected domain when $n=2$ or a bounded domain which is quasiconformally equivalent to some uniform domain when n≥3. If Ω supports a (φₙ, φ)-Poincar\'e inequality, we show that it is a John domain.