EXPONENTIAL INTEGRABILITY OF THE QUASI-HYPERBOLIC METRIC ON HOLDER DOMAINS

A proper subdomain D of R(n) is called a Holder 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 Holder 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 Poincare 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.