A $(\phi_n, \phi)$-Poincar\'e inequality on John domain
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초록
Given a bounded domain $\Omega \subset \rn$ with $n\ge2$, let $\phi $ be 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, if $\Omega$ supports a $(\phi_n, \phi)$-Poincar\'e inequality with an additional assumption that it is a simply connected domain when $n=2$ or a bounded domain which is quasiconformally equivalent to some uniform domain when $n\ge3$, then it is a John domain.
Abstract
Given a bounded domain $\Omega \subset \rn$ with $n\ge2$, let $\phi $ be 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, if $\Omega$ supports a $(\phi_n, \phi)$-Poincar\'e inequality with an additional assumption that it is a simply connected domain when $n=2$ or a bounded domain which is quasiconformally equivalent to some uniform domain when $n\ge3$, then it is a John domain.