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Quasihyperbolic metrics, \(\text{Lip}_G\)-extension domains and John disks. (English) Zbl 1254.30070
Summary: Suppose that \(D\) is a Jordan proper subdomain of \(\mathbb R^2\). In this paper, we prove that (1) \(D\) is a John disk if and only if there exists a constant \(c \geq 1\) such that
\[ k_D(x1, x2) \leq cG_D(x1, x2)\qquad\text{for all}\qquad x_1, x_2 \in D; \] (2) \(D\) is a John disk if and only if \(D\) is a \(\text{Lip}_G\)-extension domain. Here \(k_D\) is the quasihyperbolic metric in \(D\),
\[ G_D(x_1, x_2) = \frac{1}{2} \log\bigg(1 + \frac{l(\gamma )}{ d(x_1,\partial D)} \bigg)\bigg(1 + \frac{l(\gamma )}{ d(x_2,\partial D)}\bigg), \]
\(\gamma \subset D\) is the quasihyperbolic geodesic with endpoints \(x_1\) and \(x_2\), and \(l(\gamma )\) is the Euclidean length of \(\gamma \).
MSC:
30F45 Conformal metrics (hyperbolic, Poincaré, distance functions)
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