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<p>We consider a univalent analytic function <i>f</i> on the half-plane satisfying the condition that the supremum norm of its (pre-)Schwarzian derivative vanishes on the boundary. Under certain extra assumptions on <i>f</i>, we show that there exists a chordal Loewner chain initiated from <i>f</i> until some finite time, and that this Loewner chain defines a quasiconformal extension of <i>f</i> over the boundary such that its complex dilatation is given explicitly in terms of the (pre-)Schwarzian derivative in some neighborhood of the boundary. This can be regarded as the half-plane version of the corresponding result developed on the disk by Becker, and also as a generalization of the Ahlfors–Weill formula. As an application of this quasiconformal extension, we complete the characterization of an element of the VMO-Teichmüller space on the half-plane using the vanishing Carleson measure condition induced by the (pre-)Schwarzian derivative.</p>

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Chordal Loewner chains and Teichmüller spaces on the half-plane

  • 期刊论文

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  • 发表日期：

    2026年1月31日

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  • Katsuhiko Matsuzaki,
  • Huaying Wei

摘要

We consider a univalent analytic function f on the half-plane satisfying the condition that the supremum norm of its (pre-)Schwarzian derivative vanishes on the boundary. Under certain extra assumptions on f, we show that there exists a chordal Loewner chain initiated from f until some finite time, and that this Loewner chain defines a quasiconformal extension of f over the boundary such that its complex dilatation is given explicitly in terms of the (pre-)Schwarzian derivative in some neighborhood of the boundary. This can be regarded as the half-plane version of the corresponding result developed on the disk by Becker, and also as a generalization of the Ahlfors–Weill formula. As an application of this quasiconformal extension, we complete the characterization of an element of the VMO-Teichmüller space on the half-plane using the vanishing Carleson measure condition induced by the (pre-)Schwarzian derivative.