The Unique Extremal Qc Mapping And Uniqueness Of Hahn-Banach Extensions

. Let be an essentialy bounded complex valued measurable function defined on the unit dise \Delta, and let be the corrensponding linear functional on the space B of analytic L 1 -integrable functions. An outline of proof of main steps of the following is given: If jj is a constant function in \Delta, then the uniqueness of Hahn-Banach extension of from B to L 1 , when k k = kk1 , implies that is the unique complex dilatation. We give a short review of some related results. 1. Introduction Let \Delta = f jzj ! 1 g, \Gamma = @ \Delta, @ = (@ x \Gamma i@ y )=2 and @ = (@ x + i@ y )=2. Like many authors, we shall use "qc mapping" as an abbreviation for "quasiconformal mapping". For a qc mapping F on \Delta, denote by = [F ] the complex dilatation [F ] = @F=@F . We let L 1 = L 1 (\Delta) be the space of essentially bounded complex-valued measurable functions on \Delta, and let M be the open unit ball in L 1 . For any in M there exists a solution f : \Delta ! \Delta ..

Unique extremality in the tangent space of the universal Teichmueller space

The primary purpose of this paper is to announce some results obtained by the authors, including characterizations of unique extremality and examples of unique extremal dilatation of nonconstant modulus. These results bring much light on the question of unique extremality. The examples actually are a surprise and give answer to a longstanding question posed by Teichmüller, Rech and the others