Cone conditions and quasiconformal mappings

Abstract

Let f be a quasiconformal mapping of the open unit ball B ⁿ = {x ∈ R ⁿ: | x | < l× in euclidean n-space R ⁿ onto a bounded domain D in that space. For dimension n= 2 the literature of geometric function theory abounds in results that correlate distinctive geometric properties of the domain D with special behavior, be it qualitative or quantitative, on the part of f or its inverse. There is a more modest, albeit growing, body of work that attempts to duplicate in dimensions three and above, where far fewer analytical tools are at a researcher’s disposal, some of the successes achieved in the plane along such lines. In this paper we contribute to that higher dimensional theory some observations relating the behavior of f and f ^-1 to one of the venerable geometric conditions in analysis, the cone condition prominent in potential theory, geometric measure theory, and elsewhere. We first demonstrate that, when D obeys a specific interior cone condition along its boundary, f must satisfy a uniform Hölder condition in B ⁿ. With regard to f ^-1, the dual result one might anticipate — that an exterior cone condition satisfied by D at its boundary would lead to a uniform Hölder estimate for f ^-l in D — is not, in general, true. We show, however, that in the presence of a certain auxiliary condition on D, one which is implied by an exterior cone condition when D is a Jordan domain in the plane, such a cone condition does exert a definite influence on the modulus of continuity of f ^-1.