6533b86ffe1ef96bd12ce673
RESEARCH PRODUCT
Cone conditions and quasiconformal mappings
Bruce PalkaRaimo Näkkisubject
Quasiconformal mappingPure mathematicsGeometric measure theoryGeometric function theoryBounded functionHölder conditionConformal mapBall (mathematics)Modulus of continuityMathematicsdescription
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 Holder 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 Holder 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.
year | journal | country | edition | language |
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1988-01-01 |