A real-analytic quasiconformal extension of a quasisymmetric function

Abstract

for all rcal x and t, t+0. It is well-known that every p-quasisymmetric function can be extended to a K-quasiconformal mapping f : H * H, where I/ denotes the upper half-plane. This was first shown by Beurling and Ahlfors [2] who gave an explicit construction for an extension I This extension is in the class C1(I1). In the present note we shall introduce an analytic kernel into the integerals which define / and thus obtain a real-analytic solution of the boundary value problem. The construction follows closely the one in [2]. Special attention, however, is paid to the estimation of the maximal dilatation K. The elementary but laborious computations in [2] leading to the estimate K<qz fot q>l seem to contain some slips. In fact, the same computations yield a somewhat sharper bound.

The harmonic and quasiconformal extension operators

“…mappings for every K ≥ 1. Initiated by Beurling and Ahlfors [BA] and continued after by Kelingos [Ke] and others (see [AK], [Fe1], [Fe2], [FS], [Go], [HH], [Hi1], [Hi2], [Ln1], [Ln2], [KZ], [Kr2], [Pa7], [PZ1], [PZ2], [Tu2], [Za10]) research of this topic appears to be one of the most fascinating branches of qc.-theory with application to the theory of Teichmüller space.…”

“…cf. [AH1], [AK], [BA], [Go], [Hi1], [Hi2], [KZ], [Ke], [Ln1], [Ln2], [PZ1], [RZ1], [RZ2], [SZ], [Tu3].…”