The unique extremal QC mapping and uniqueness of Hahn-Banach extensions. (English) Zbl 1035.46501

Mateljević, M.; Marković, V.

Mat. Vesn. 48, No. 3-4, 107-112 (1996).

Let \(M\) be the unit ball in the space \(L^\infty\) of essentially bounded complex valued measurable functions defined on the unit disc \(\Delta\) and for \(\varphi\in M\), let \(\Lambda_\varphi\) be the corresponding linear functional on the space \(L_a^1\) of analytic functions on \(\Delta\) in the space of \(L^1\)-integrable functions.
An outline of proof of the main steps of the proof of the following theorem is given: Let \(\varphi\in M\). If \(|\varphi|=k<1\) a.e. on \(\Delta\) and \(\|\Lambda_\varphi\|=\|\varphi\|_\infty\), then \(\varphi\) is the unique complex dilatation.

Reviewer: Miroljub Jevtić (Beograd)

MSC:

46E15	Banach spaces of continuous, differentiable or analytic functions
30C62	Quasiconformal mappings in the complex plane
30C70	Extremal problems for conformal and quasiconformal mappings, variational methods

Keywords:

quasi-conformal mapping; dilatation