Pointwise inequalities for Sobolev functions on generalized cuspidal domains

Abstract

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