Carleson measures and chord-arc curves

This paper focuses on the structure of the subspace \(T_c\) of the BMO Teichmüller space \(T_b\) corresponding to chord-arc curves, which contains the VMO Teichmüller space \(T_v\). We prove that \(T_c\) is not a subgroup with respect to the group structure of \(T_b\), but it is preserved under the inverse operation and the left and the right translations by any element of \(T_v\). Moreover, we show that \(T_b\) has a fiber structure induced by \(T_v\), and the complex structure of \(T_b\) can be projected down to the quotient space \(T_v \backslash T_b\). Then, we see that \(T_c\) consists of fibers of this projection, and its quotient space also has the induced complex structure.

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“…′ | for the Poincaré density ρ Ω on Ω. As λ µ ∈ CM 0 (D), we have that | µ| 2 ρ Ω ∈ CM 0 (Ω) by [29,Theorem 3.2].…”

mentioning

confidence: 88%

Space of chord-arc curves and BMO/VMO Teichmüller space

Matsuzaki

Wei

2022

Ann. Fenn. Math.

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“…. By ( * ) and welldefinedness of the pull-back operator from CM 0 (Ω * ) into CM 0 (D * ) (see [26,Theorem 3…”

Section: The Quotient Bers Embedding Of the Bmo Teichmüller Spacementioning

confidence: 99%

“…Here, the constant M is the norm of the pull-back operator as in [26,Theorem 3.4]. We can also obtain such a constant M by a direct computation of the Carleson norm of λ σ(ϕ) .…”

Section: The Carleson Metric and Its Quotientmentioning

confidence: 99%

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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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“…The injectivity of the quotient Bers embedding has been also proved in a different setting and in a different method. See a recent work of Wei and Zinsmeister [29].…”

Section: Introductionmentioning

confidence: 99%

Injectivity of the quotient Bers embedding of Teichmüller spaces

Matsuzaki¹

2019

Ann. Acad. Sci. Fenn. Math.

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The Bers embedding of the Teichmüller space is a homeomorphism into the Banach space of certain holomorphic automorphic forms. For a subspace of the universal Teichmüller space and its corresponding Banach subspace, we consider whether the Bers embedding can project down between their quotient spaces. If this is the case, it is called the quotient Bers embedding. Injectivity of the quotient Bers embedding is the main problem in this paper. Alternatively, we can describe this situation as the universal Teichmüller space having an affine foliated structure induced by this subspace. We give several examples of subspaces for which the injectivity holds true, including the Teichmüller space of circle diffeomorphisms with Hölder continuous derivative. As an application, the regularity of conjugation between representations of a Fuchsian group into the group of circle diffeomorphisms is investigated.2010 Mathematics Subject Classification. Primary 30F60, Secondary 37E30.

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Carleson measures and chord-arc curves

Cited by 6 publications

References 36 publications

Space of chord-arc curves and BMO/VMO Teichmüller space

Space of chord-arc curves and BMO/VMO Teichmüller space

BMO Teichmüller spaces and their quotients with complex and metric structures

Injectivity of the quotient Bers embedding of Teichmüller spaces

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