On the unique extremality of quasiconformal mappings with dilatation bounds
Let \(R\) and \(R'\) be two hyperbolic Riemann surfaces and \(E'\subset R'\) such that \(R'\setminus E'\) has positive measure. Given a non-negative measurable function \(b\) on \(E'\), consider a quasiconformal mapping \(F:R' \to R\) with complex dilatation \(\mu_F\) such that \(| \mu_F(w)| \leq b(w)\) for a.e. \(w\in E'\). Denote by \(Q(F,E',b)\) the class of all quasiconformal maps \(G: R'\to R\) homotopic to \(F\) (mod \(\partial R'\)) and with \(| \mu_G(w)| \leq b(w)\) for a.e. \(w\in E'\). The characterization of the existence of an extremal quasiconformal mapping in the class \(Q(F,E',b)\) was given by \textit{K. Sakan} [J. Math. Kyoto Univ. 26, 31--37 (1986; Zbl 0604.30023)]. In the article under review the uniqueness of such an extremal mapping is considered. After \textit{F.P. Gardiner} [Mich. Math. J. 29, 237--242 (1982; Zbl 0476.30015)] and E. Reich [Arch. Ration. Mech. Anal. 68, 99--112 (1978; Zbl 0421.30013)] it was conjectured that unique extremality is characterized by certain related linear functional having a unique Hahn-Banach extension, a statement already proved to be true in [\textit{V. Bozin, N. Lakic, V. Markovic, M.J. Mateljevic}, Anal. Math. 75, 299--338 (1998; Zbl 0929.30017)] for the particular case \(E' = \emptyset\). The main result proved shows that the if part of the conjecture is still true for a completely general \(E'\).
scientific article; zbMATH DE number 2158076