A domain \(D\subset {\mathbb{R}}^ d\) of finite volume is said to be a p- Poincaré domain if \(\int_{D}| u-u_ D|^ p dx\leq c\int_{D}| \nabla u|^ p dx\) for some constant c and all \(u\in C^ 1(D)\), where \(u_ D\) stands for the mean value. The paper deals with various geometric conditions on D sufficient for d to be a p- Poincaré domain. One of the main results reads as follows:
If \(D\subset {\mathbb{R}}^ d\) is a Hölder domain, then D is a p-Poincaré domain for all \(p\geq d\). Here D is a Hölder domain if the quasi- hyperbolic distance from a fixed \(x_ 0\in D\) to \(x\in D\) is bounded by a constant multiple of the logarithm of the Euclidean distance of x to \(\partial D\). In two dimensions there is a close connection to Hölder continuous Riemannian mappings from the unit disk onto D. Geometric conditions are considered which guarantee that the Sobolev embedding \(W^{1,p}(D)\to L^ p(D)\) is compact.