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Conformal capacity and the quasihyperbolic metric. (English) Zbl 0862.30021

By examining certain capacity conditions we establish a connection between two classes of domains in \(\mathbb{R}^n\) introduced by Gehring and Martio. We present new characterizations, for uniform domains and for domains which satisfy a quasihyperbolic boundary condition, which are valid in the category of domains quasiconformally equivalent to uniform domains. We obtain analogs of (the Hölder continuity of quasiconformal mappings) results of Gehring and Martio by replacing the quasihyperbolic boundary condition with our capacity conditions. We exhibit sharp pointwise growth estimates which hold for both Harnack and monotone functions in certain domains. We produce examples which illustrate our ideas.

MSC:

30C65 Quasiconformal mappings in \(\mathbb{R}^n\), other generalizations
31B15 Potentials and capacities, extremal length and related notions in higher dimensions
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