Boundary Harnack Principle and the Quasihyperbolic Boundary Condition

Abstract

We discuss the global boundary Harnack principle for domains in Rⁿ (n ≥ 2) satisfying some conditions related to the quasihyperbolic metric. For this purpose, we reformulate the global boundary Harnack principle and the global Carleson estimate in terms of the Green function. Based on our other result asserting the equivalence between the global boundary Harnack principle and the global Carleson estimate, we obtain four equivalent con ditions. Using the box argument for the estimate of the harmonic measure, we obtain the global Carleson estimate in terms of the Green function (and thus the global boundary Harnack principle) for a domain satisfying a con dition on the quasihyperbolic metric and the capacity density condition, as well as for a Holder domain whose boundary is locally given by the graph of a Holder continuous function in Rⁿ⁻¹. Our argument is purely analytic and elementary, unlike the probabilistic approach of Bass-Burdzy-Banuelos.