论文标题
准信息约旦域
Quasiconformal Jordan domains
论文作者
论文摘要
我们将经典的carathéodory扩展定理扩展到Quasiconformal jordan域$（y，d_ {y}）$。我们说，如果完成的$ \ of $（y，d_ {y}）$具有有限的hausdorff $ 2 $ - mmeasure，y，d _ {y}）$是一个准空间的jordan域，则是一个准文献约旦域。 $ \ mathbb {s}^{1} $，并且存在同构$ ϕ \ colon \ colon \ mathbb {d} \ rightArrow（y，d_ {y}）$，在几何意义上是quasicemonformal。 我们表明，$ ϕ $具有连续的，单调的和溢流的扩展名$φ\ colon \ colon \ overline {\ mathbb {d}}} \ rightarrow \ rightarrow \ overline {y} $。在这种一般性中，这是最好的。此外，我们发现$φ$的必要条件是形式的同构同构。我们提供了足够的条件，可以限制$φ$至$ \ mathbb {s}^{1} $是quasisymmetry，而$ \ partial y $ a bi-lipschitz等于平面中的Quasicircle。
We extend the classical Carathéodory extension theorem to quasiconformal Jordan domains $( Y, d_{Y} )$. We say that a metric space $( Y, d_{Y} )$ is a quasiconformal Jordan domain if the completion $\overline{Y}$ of $( Y, d_{Y} )$ has finite Hausdorff $2$-measure, the boundary $\partial Y = \overline{Y} \setminus Y$ is homeomorphic to $\mathbb{S}^{1}$, and there exists a homeomorphism $ϕ\colon \mathbb{D} \rightarrow ( Y, d_{Y} )$ that is quasiconformal in the geometric sense. We show that $ϕ$ has a continuous, monotone, and surjective extension $Φ\colon \overline{ \mathbb{D} } \rightarrow \overline{ Y }$. This result is best possible in this generality. In addition, we find a necessary and sufficient condition for $Φ$ to be a quasiconformal homeomorphism. We provide sufficient conditions for the restriction of $Φ$ to $\mathbb{S}^{1}$ being a quasisymmetry and to $\partial Y$ being bi-Lipschitz equivalent to a quasicircle in the plane.
