A note on the BMO-Teichmüller space

Wei, Huaying

doi:10.1016/j.jmaa.2015.10.059

Cited by 11 publications

(6 citation statements)

References 14 publications

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“…The subset T c of T b consisting of all elements corresponding to chord-arc curves contains T v (see [38]). It was shown in [36] that all strongly symmetric homeomorphisms on the unit circle constitute a topological group, and from its proof, we see that T v is contained in char(T b ). We expect that char(T b ) = T v but it seems that this is not known yet.…”

Section: Discontinuity Of the Correspondence To Riemann Mappingsmentioning

confidence: 63%

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BMO embeddings, chord-arc curves, and Riemann mapping parametrization

Wei¹,

Matsuzaki²

2021

Preprint

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We consider the space of chord-arc curves on the plane passing through the infinity with their parametrization γ on the real line, and embed this space into the product of the BMO Teichmüller spaces. The fundamental theorem we prove about this representation is that log γ ′ also gives a biholomorphic homeomorphism into the complex Banach space of BMO functions. Using these two equivalent complex structures, we develop a clear exposition on the analytic dependence of involved mappings between certain subspaces. Especially, we examine the parametrization of a chord-arc curve by using the Riemann mapping and its dependence on the arc-length parametrization. As a consequence, we can solve a conjecture of Katznelson, Nag, and Sullivan in 1990 by showing that this dependence is not continuous.

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Section: Discontinuity Of the Correspondence To Riemann Mappingsmentioning

confidence: 63%

“…The following result, which says that T b is a partial topological group in the sense of [16], was proved in [36].…”

Section: 4mentioning

confidence: 97%

BMO embeddings, chord-arc curves, and Riemann mapping parametrization

Wei¹,

Matsuzaki²

2021

Preprint

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“…The sets of those are denoted by SQS(S) and SS(S) respectively. These classes were investigated in [13,31,35,40] during their study of BMO theory on Teichmüller spaces. In particular, it is known that SS(S) is a subgroup of SQS(S).…”

Section: Beltrami Coefficients Inducing Carleson Measuresmentioning

confidence: 99%

Strongly symmetric homeomorphisms on the real line with uniform continuity

Wei¹,

Matsuzaki²

2022

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We investigate strongly symmetric homeomorphisms of the real line which appear in harmonic analysis aspects of quasiconformal Teichmüller theory. An element in this class can be characterized by a property that it can be extended quasiconformally to the upper half-plane so that its complex dilatation induces a vanishing Carleson measure. However, differently from the case on the unit circle, strongly symmetric homeomorphisms on the real line are not preserved under either the composition or the inversion. In this paper, we present the difference and the relation between these two cases. In particular, we show that if uniform continuity is assumed for strongly symmetric homeomorphisms of the real line, then they are preserved by those operations. We also show that the barycentric extension of uniformly continuous one induces a vanishing Carleson measure and so do the composition and the inverse of those quasiconformal homeomorphisms of the upper half-plane.

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“…The strongly quasisymmetric homeomorphisms and its Teichmüller space are usually defined and discussed on the unit circle S. Due to the conformal invariance of A ∞ -weights and Carleson measures, we have the same theory on the real line R. On the other hand, a sense-preserving homeomorphism g on S is called strongly symmetric if it is absolutely continuous such that log g ′ belongs to VMO(S), the space of functions of vanishing mean oscillation on S. This class was introduced in [14] when Partyka studied eigenvalues of quasisymmetric automorphisms determined by VMO functions. It was investigated further later in [21,22] during their study of BMO theory of the universal Teichmüller space. Since the logarithmic derivative does not have conformal invariance, the notion of strongly symmetric homeomorphism on the real line is different from the one on the unit circle.…”

Section: Introductionmentioning

confidence: 99%