Pointwise inequalities for Sobolev functions on generalized cuspidal domains

Abstract

Let $\mathrm{\Omega }\subset {\mathbb{R}}^{n-1}$ be a bounded star-shaped domain and ${\mathrm{\Omega }}_{\psi }$ be an outward cuspidal domain with base domain $\mathrm{\Omega }$. We prove that for $1<p\le \mathrm{\infty }$, $W^{1,p}({\mathrm{\Omega }}_{\psi })=M^{1,p}({\mathrm{\Omega }}_{\psi })$ if and only if $W^{1,p}(\mathrm{\Omega })=M^{1,p}(\mathrm{\Omega })$.