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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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