Ein Fortsetzungssatz für quasikonforme Deformationen (Q1070056)

According to \textit{L. V. Ahlfors} [J. Anal. Math. 30, 74-97 (1976; Zbl 0338.30017)] and \textit{H. M. Reimann} [Inv. Math. 33, 247-270 (1976; Zbl 0328.30019)] a continuous function \(f: {\mathbb{R}}^ n\to {\mathbb{R}}^ n\) generates through the equation \(dh/dt=f(h)\) with \(h(x,0)=x,\quad x\in {\mathbb{R}}^ n\) a family h(x,t), \(t\geq 0\), of quasiconformal mappings if some quantity \(\| f\|_ Q\) is finite. Such functions are called quasiconformal deformations. The author introduces for them another seminorm \(\| f\|_ p\) that measures how good f can be approximated on the balls \(| x-x_ 0| <\rho\) of \({\mathbb{R}}^ n\) by affine mappings of the form \(p(x)=Tx+\lambda x+a,\) where \(\lambda\in {\mathbb{R}}\), \(a\in {\mathbb{R}}^ n\) and T a skew-symmetric matrix. The two seminorms \(\| f\|_ Q\) and \(\| f\|_ p\) are equivalent, the latter however has the advantage to allow an immediate generalization to continuous functions \(f: M\to {\mathbb{R}}^ n\) by restricting the x and \(x_ 0\) above to M, thus getting \(\| f\|_{p,M}\). The author shows that for a large class of distinguished subsets M of \({\mathbb{R}}^ n\) (if \(n=2\), any continuum of \({\mathbb{R}}^ 2)\) each continuous \(f: M\to {\mathbb{R}}^ n\) with \(\| f\|_{p,M}<\infty\) allows a continuous extension to \({\mathbb{R}}^ n\) for which \(\| f\|_ Q\) is finite, so that f is a quasiconformal deformation.