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

Abstract

Given a bounded domain $\Omega \subset {\mathbb{R}}^n$ with $n\ge 2$, let $\varphi$ is a Young function satisfying the doubling condition with the constant $K_{\varphi }<2^n$. If $\Omega$ is a John domain, we show that $\Omega$ supports a $({\varphi }_n,\varphi )$-Poincaré 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 $n\ge 3$. If $\Omega$ supports a $({\varphi }_n,\varphi )$-Poincaré inequality, we show that it is a John domain.

arXiv admin note: substantial text overlap with arXiv:2305.04016