A proper subdomain D of \(R^ n\) is called a Hölder domain if for a fixed y in D, the quasi-hyperbolic metric \(k_ D(x,y)\) is bounded by a constant plus a constant multiple of the logarithm of the Euclidean distance from x to the boundary of D. For simply connected planar domains D, it is known that these domains are characterized by the fact that the Riemann mapping function of the unit disk onto D satisfies a Hölder condition with some positive exponent.
For y in D fixed, we prove that \(\exp (\tau k_ D(x,y))\) is integrable over D for some \(\tau >0\). One corollary of this is that the boundaries of these domains have Hausdorff dimension less than n. Other applications pertain to Poincaré domains and to averaging domains. Our method involves extending some recent results of Carleson-Jones and Jones- Makarov on simply connected planar domains to multiply connected domains in \(R^ n\) by using the quasi-hyperbolic metric.