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Extension of internally bilipschitz maps in John disks. (English) Zbl 1100.30003
A domain $$D \subset \mathbb R^2$$ is called a $$c$$-John domain if any two points $$z_1, z_2 \in D$$ may be joined by a rectifiable arc $$\alpha \subset D$$ such that $\min_{j = 1, 2} \text{length}(\alpha[z_j, z]) \leq c \text{dist}(z, \partial D) .$ A $$c$$-John disk is a simply connected $$c$$-John domain with at least two boundary points. The internal distance between two points $$z_1, z_2 \in D$$ is $$\lambda_D(z_1, z_2) = \inf \text{length}(\alpha)$$, where the infimum is taken over all rectifiable arcs $$\alpha \subset D$$ joining $$z_1$$ and $$z_2$$. A homeomorphism $$f \colon D \rightarrow D'$$ is said to be internally $$L$$-bilipschitz if $\frac1L \lambda_D(z_1, z_2) \leq \lambda_{D'}(f(z_1), f(z_2)) \leq L \lambda_D(z_1, z_2)$ for all points $$z_1, z_2 \in D$$. The symbol $$\partial_r D$$ denotes the set of all rectifiably accesible points of $$\partial D$$. After a discussion of internal distance on the boundary, a three point property for John disks is proved. The main theorem of the paper is a result about extension of internally bilipschitz maps in John disks, reminiscent of a result of Gehring for quasidisks; Suppose $$D$$ is a Jordan $$c$$-John disk and that $$f \colon \partial D \rightarrow \partial D'$$ is a homeomorphism. If $$f \colon (f^{-1}(\partial_r D'), \lambda_D) \rightarrow (\partial_r D', \lambda_{D'})$$ is $$L$$-bilipschitz, then $$D'$$ is a $$c'$$-John domain, and $$f$$ extends as an $$M$$-bilipschitz map $$\tilde f \colon (\overline D, \lambda_D) \rightarrow (\overline{D'}, \lambda_{D'})$$. $$M$$ and $$c'$$ depend only on $$L$$ and $$c$$.

##### MSC:
 30C20 Conformal mappings of special domains 30C62 Quasiconformal mappings in the complex plane
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