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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.

30C65 Quasiconformal mappings in \(\mathbb{R}^n\), other generalizations
30F15 Harmonic functions on Riemann surfaces
26A16 Lipschitz (Hölder) classes
Full Text: DOI EuDML
[1] doi:10.1017/CBO9780511546600
[2] doi:10.1016/S1874-5709(05)80014-9
[7] doi:10.1016/j.jmaa.2006.12.069 · Zbl 1116.30010
[9] doi:10.1007/s00209-007-0270-9 · Zbl 1151.30014
[22] doi:10.1155/S1025583499000326 · Zbl 0946.30010
[25] doi:10.1307/mmj/1029003136 · Zbl 0574.30027
[26] doi:10.2298/FIL0901085M · Zbl 1199.30128
[29] doi:10.1017/CBO9780511618789
[30] doi:10.1007/BF02786713 · Zbl 0349.30019
[36] doi:10.1016/S1874-5709(05)80005-8
[40] doi:10.1007/BF02567058 · Zbl 0777.31003
[42] doi:10.1007/BF02792546 · Zbl 0601.30025
[43] doi:10.2140/pjm.1998.182.359 · Zbl 0892.58017
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