Differential properties of mappings, conformal at a point
Let f be a quasiconformal mapping of the ball \(B=B(0,r)\) in \(R^ n\) onto a domain D. Suppose that the coefficient K(x) of quasiconformality of f at x satisfies \(K(x)-1\leq c| x|^{m+\alpha}\) a.e. in B, for some \(m=0,1,...\), and \(0<\alpha <1\). Main theorem: There is a function \(g: R^ n\to R^ n\) whose coordinate functions are polynomials with degrees not greater than \(m+1\) such that \[ | f(x)-g(x)| \leq c'| x|^{m+\alpha +1}\quad and\quad c'=c'(c,r,m,\alpha,n,diam(D)). \] For \(n\geq 3\) the proof uses an approximation result, due to Yu. G. Reshetnyak, and, in fact, the result follows from this for \(m=0\). For \(n=2\) special conformal approximations for f at 0 are used.
scientific article; zbMATH DE number 3962082