Conformally invariant inequalities in domains in Euclidean space

We study conformally invariant integral inequalities for real-valued functions defined on domains $\Omega$ in $n$-dimensional Euclidean space. The domains considered are of hyperbolic type, that is, they admit a hyperbolic radius $R=R(x, \Omega)$ satisfying the Liouville non-linear differential equation and vanishing on the boundary of the domain. We prove several inequalities which hold for all smooth compactly supported functions $u$ defined on a given domain of hyperbolic type. Here are two of them:
\begin{gather*} \int|\nabla u|^2R^{2-n}\, dx \geqslant n (n-2)\int|u|^2R^{-n}\, dx, \\ \int|(\nabla u, \nabla R)|^p R^{p-s}\, dx\geqslant \frac{2^pn^p}{p^p}\int|u|^pR^{-s}\, dx, \end{gather*}
where $n\geqslant 2$, $1\leqslant p< \infty$ and $1+n/2 \leqslant s <\infty$. We also study the relations between Euclidean and hyperbolic characteristics of domains.