On Douady–Earle extension and the contractibility of the VMO-Teichmüller space

Tang, Shuan; Wei, Huaying; Shen, Yuliang

doi:10.1016/j.jmaa.2016.04.073

Cited by 14 publications

(12 citation statements)

References 10 publications

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“…By (V 0 ) ⇒ (V 1 ) in Section 1, any h ∈ SS(S) can be extended to a quasiconformal mapping of D onto itself such that its complex dilatation belongs to M 0 (D). The Douady-Earle extension E(h) is such an extension, and moreover, it is a bi-Lipschitz diffeomorphism under the hyperbolic metric (see [8,27]). The second condition in (10) implies E(h)(0) = 0, and E(h)(∞) = ∞ by reflection across S.…”

Section: ])mentioning

confidence: 99%

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The VMO-Teichmüller space and the variant of Beurling-Ahlfors extension by heat kernel

Wei¹,

Matsuzaki²

2021

Preprint

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We give a real-analytic section for the Teichmüller projection onto the VMO-Teichmüller space by using the variant of Beurling-Ahlfors extension by heat kernel introduced by Fefferman, Kenig and Pipher in 1991. Based on this result, we prove that the VMO-Teichmüller space can be endowed with a real Banach manifold structure that is real-analytically equivalent to its complex Banach manifold structure. We also obtain that the VMO-Teichmüller space admits a real-analytic contraction mapping.

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Section: ])mentioning

confidence: 99%

“…In 1986, Douady and Earle [9] proved that this is also true for the conformally barycentric extension. In 2016, Tang, Wei and Shen [27] showed that the conformally barycentric extension yields a continuous section for the VMO-Teichmüller space T v . However, the existence of the real-analytic section for T v is not known yet in the literature.…”

mentioning

confidence: 99%

The VMO-Teichmüller space and the variant of Beurling-Ahlfors extension by heat kernel

Wei¹,

Matsuzaki²

2021

Preprint

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“…Concerning the topology of T c and p(T c ), we immediately see the following. Noting the fact that T v is contractible shown in [24], the connectedness problem on T c can be also passed to this quotient.…”

Section: Foliated Structure Of the Chord-arc Curve Subspacementioning

confidence: 99%

“…For any σ ∈ T v = Möb(S)\SS, the complex dilatation of the Douady-Earle extension of σ is denoted by µ. Then, µ ∈ M 0 (D) by [24,Theorem 3.7] (see also [19]). We take an increasing sequence of positive numbers r n < 1 (n = 1, 2, .…”

Section: The Quotient Bers Embedding Of the Bmo Teichmüller Spacementioning

confidence: 99%

BMO Teichmüller spaces and their quotients with complex and metric structures

Wei,

Matsuzaki

2018

Preprint

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The paper presents some recent results on the BMO Teichmüller space, its subspaces and quotient spaces. We first consider the chord-arc curve subspace and prove that every element of the BMO Teichmüller space is represented by its finite composition. Moreover, we show that these BMO Teichmüller spaces have affine foliated structures induced by the VMO Teichmüller space. By which, their quotient spaces have natural complex structures modeled on the quotient Banach space. Then, a complete translation-invariant metric is introduced on the BMO Teichmüller space and is shown to be a continuous Finsler metric in a special case.

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“…For the ease of calculation, the Carleson measure on ∆ can be defined in the following equivalent way. A positive measure µ on ∆ is a Carleson measure if there is a constant C such that µ(S) Ch(11) for every sector S = {re iθ : 1 − h r < 1, |θ − θ 0 | h}.…”

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confidence: 99%

Ahlfors-regular curves and Carleson measures

Wei

Zinsmeister

2019

Sci. China Math.

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We study the relation between the boundary of a simply connected domain Ω being Ahlfors-regular and the invariance of Carleson measures under the push-forward operator induced by a conformal mapping from the unit disk ∆ onto the domain Ω. As an application, we characterize the chord-arc curve with small norm and the asymptotically smooth curve in terms of the complex dilatation of some quasiconformal reflection with respect to the curve.A Carleson measure µ is called a vanishing Carleson measure if lim r→0 µ(Ω∩D(z, r))/r = 0 uniformly for z ∈ ∂Ω. We denote by CM(Ω) and CM 0 (Ω) the set of all Carleson measures and vanishing Carleson measures on Ω, respectively. It is easy to see that CM(Ω) is a Banach space with the Carleson norm ·

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On Douady–Earle extension and the contractibility of the VMO-Teichmüller space

Cited by 14 publications

References 10 publications

The VMO-Teichmüller space and the variant of Beurling-Ahlfors extension by heat kernel

The VMO-Teichmüller space and the variant of Beurling-Ahlfors extension by heat kernel

BMO Teichmüller spaces and their quotients with complex and metric structures

Ahlfors-regular curves and Carleson measures

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