Suppose $ \Omega \subset \mathbb{R}^d$ be a domain, and let $ \rho(x) = \mathrm{dist} (x, \partial \Omega)$ be the distance function to the boundary of $ \Omega$ . I want to know for which domains $ \rho$ satisfies a Harnack type inequality. Harnack inequality says that $ \sup _{x \in B} \rho (x) \leq C \inf _{x \in B} \rho(x)$ on a ball $ B= B(a,r), a \in \Omega$ , and $ C$ is a constant depend on $ B$ . It is known that harmonic functions satisfy Harnack inequality. Is it enough if $ \Omega$ satisfy regularity property (e.g.,if it is a Lipschitz or a NTA-domain)? What about boundary Harnack inequality?