Quasiconformal extension of biholomorphic mappings in several complex variables (Q816479)

Let \(\mathbb{C}^{n}\) denote the space of \(n\) complex variables and let \(B=\{z\in \mathbb{C}^{n} \mid \| z\| <1\}\) be the Euclidean unit ball in \(\mathbb{C}^{n}\). Consider \(H(B)\) the set of holomorphic mappings from \(B\) into \(\mathbb{C}^{n}\). A mapping \(v\in H(B)\) is called a Schwarz mapping if \(\| v(z)\| \leq\| z\| \) for all \(z\in B\). Let \(\{f(z,t) , t\in[0,\alpha]\}\) be a family of mappings such that \(f_{t}(z)=f(z,t)\in H(B)\) and \(f_{t}(0)=0\) for each \(t\in [0,\alpha]\). The family \(\{f(z,t) , t\in[0,\alpha]\}\) is called a subordination chain if there exist Schwarz mappings \(v(z,s,t)\) such that \(f(z,s)=f(v(z,s,t),t)\) for \(z\in B , 0\leq s\leq t\leq \alpha\). Let \(\Omega , \Omega'\) be domains in \(\overline{\mathbb{R}^{m}}\). A homeomorphism \(f:\Omega\rightarrow\Omega'\) is said to be \(k\)-quasiconformal (\(k\)-constant, \(k>0\)) if it is differentiable a.e., ACL and \(\| D(f;x)\| ^{m}\leq k| \text{det}D(f;x)| \) a.e. in \(\Omega\), where \(D(f;x)\) denotes the Jacobian matrix of \(f\) and \(\| D(f;x)\| =\sup \{\| D(f;x)(a)\| : \| a\| =1\}\). In this paper the authors give sufficient conditions for a first quasiconformal element \(f(z)=f(z,0)\) of a subordination chain to be extendable to a quasiconformal homeomorphism on a neighbourhood of \(\overline{B}\). They also show that \(f\) can be extended to a quasiconformal homeomorphism of \(\overline{\mathbb{R}^{2n}}\) into itself. Applications are given. Some of these include sufficient conditions for functions of the form \(f(z)=z+\omega(z) , \omega\in H(B)\;, \omega(0)=0 , D \omega(0)=0\) to be extendable to a quasiconformal homeomorphism of \(\overline{\mathbb{R}^{2n}}\) into itself.