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

Abstract

. 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 ..