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On harmonic quasiconformal quasi-isometries. (English) Zbl 1211.30040
The present paper is devoted to the study of harmonic mappings in \({\mathbb R}^n\), \(n\geq 2\), and its connection with the Lipschitz condition by the relation to quasihyperbolic metrics. Some applications to classes of quasiconformal mappings in \({\mathbb R}^n\) are also given.
More in detail, given a domain \(G\subset {\mathbb R}^n\) and a “weight” function \(\rho\), we define
\[ l_{\rho}(\gamma)=\int\limits_{\gamma}\rho(x)ds\quad\text{and}\quad d_{\rho}(a, b)=\inf\limits_{\gamma}l_{\rho}(\gamma), \]
where the infimum is taken over all rectifiable curves joining \(a\) and \(b\), and set
\[ k_G:=d_{\rho}\quad\text{where}\quad \rho(x)=\frac{1}{d(x, \partial G)}. \]
The functions \(k_G\) defined as above are called quasihyperbolic metrics. A mapping \(f\) between two metric spaces \((M, d_M)\) and \((N, d_N)\) is said to be a quasi-isometry, or bi-Lipschitz, if and only if there exists a positive constant \(a\geq 1\) such that
\[ a^{-1}d_M(x, y)\leq d_N\big(f(x), f(y)\big)\leq ad_M(x, y). \]
We say that a mapping \(f\) has the weak uniform boundless property if and only if there exists a constant \(c>0\) such that the condition \(d_M(x, y)< 1/2\) implies that \(d_N\big(f(x), f(y)\big)\leq c\). Let \({\mathbb D}\) denote the unit disk in the plane. Theorem 2.7 states that every harmonic mapping
\[ h:\left({\mathbb D}, k_{\mathbb D}\right)\rightarrow \left(h_{\mathbb D}, k_{h({\mathbb D})}\right) \] satisfying to the weak uniform boundless property is Lipschitz. If, in addition, \(f\) is quasiconformal, then \(f\) is bi-Lipschitz. Some space analogue of Theorem 2.7 is also proved, see Theorem 2.8. Suppose that \(G\) is a proper open subset of the space \({\mathbb R}^n\) and \(h:\left(G, k_{G}\right)\rightarrow \left(h_{G}, k_{h(G)}\right)\) is a harmonic mapping. Then \(h\) is a Lipschitz mapping if and only if \(h\) satisfies the weak uniform boundless property. All results are applicable to many modern classes of plane and space mappings.

MSC:
30C65 Quasiconformal mappings in \(\mathbb{R}^n\), other generalizations
30F15 Harmonic functions on Riemann surfaces
26A16 Lipschitz (Hölder) classes
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