Quasihyperbolic boundary conditions and capacity: Hölder continuity of quasiconformal mappings

Abstract.

We prove that quasiconformal maps onto domains which satisfy a quasihyperbolic boundary condition are globally Hölder continuous in the internal metric. The primary improvement here over existing results along these lines is that no assumptions are made on the source domain. We reduce the problem to the verification of a capacity estimate in domains satisfing a quasihyperbolic boundary condition, which we establish using a combination of a chaining argument involving the Poincaré inequality on Whitney cubes together with Frostman's theorem. We also discuss related results where the quasihyperbolic boundary condition is slightly weakened; in this case the Hölder continuity of quasiconformal maps is replaced by uniform continuity with a modulus of continuity which we calculate explicitly.