On the local dilatation of quasisymmetric mappings and a theorem of Kurt Friedrichs
Let \(\Omega\) be a bounded plane domain where \(\partial \Omega\) consists of free smooth arcs and curves except for finitely many corners (non-zero angles!). If \(z_ 0\in \Omega\) then there exists a constant \(b=b(\Omega,z_ 0)<1\) such that \[ | \iint_{\Omega}f(z)dxdy| \leq b\iint_{\Omega}| f(z)| dxdy \] holds for all \(f\in A(\Omega)\) (holomorphic and \(L^ 1\)-integrable functions in \(\Omega)\) which are normalized by \(f(z_ 0)=0.\) This inequality due to K. Friedrichs is proved in this paper by means of the theory of extremal quasiconformal mappings and some very elementary computations of local dilatations of quasisymmetric mappings. Then a more general statement is derived which says that there exists such a constant \(b<1\) if and only if \(\partial \Omega\) contains no substantial boundary point with respect to the boundary values induced by the affine mapping \(F_ K(x+iy)=Kx+iy\) (for an arbitrary \(K>1).\) Also bounds from below are given for \(b(\Omega,z_ 0)\), e.g. if \(\Omega\) is simply connected and \(\partial \Omega\) contains a corner of angle \(\gamma\), \(0<\gamma \leq 2\pi\), then \(b(\Omega,z_ 0)\geq | \sin \gamma | \gamma^{-1}\) for every \(z_ 0\in \Omega\). This is a kind of a converse estimation of an inequality of E. Reich.
scientific article; zbMATH DE number 3914671
scientific article