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Katsuhiko Matsuzaki

Publications and source records attributed to Katsuhiko Matsuzaki.

At least 19 recordsLinked to original sources
arXiv2026

Weak Asymptotic Symmetry of Quasisymmetric Embeddings on Quasilines

We investigate the relationship between weak asymptotic symmetry ($\mathrm{WAS}$) and asymptotic symmetry ($\mathrm{AS}$) for quasisymmetric maps associated with planar quasilines. To this end, we introduce a formally weaker condition, called equidistant weak asymptotic symmetry ($\mathrm{EWAS}$), and prove that, for quasisymmetric embeddings of $\mathbb{R}$ into $\mathbb{C}$, the three conditions $\mathrm{AS}$, $\mathrm{WAS}$, and $\mathrm{EWAS}$ are equivalent. We then establish that $\mathrm{WAS}$ implies $\mathrm{AS}$ for quasisymmetric homeomorphisms from an arbitrary quasiline onto $\mathbb{R}$. More generally, if $h\colonΓ_1\toΓ_2$ is a quasisymmetric homeomorphism between quasilines and $Γ_1$ is asymptotically conformal, then $\mathrm{WAS}$ implies $\mathrm{AS}$. A formally dual statement holds when $Γ_2$ is asymptotically conformal, provided that $h$ is uniformly continuous. In the compact setting of bounded quasicircles, these results yield an answer to the $\mathrm{WAS}$-$\mathrm{AS}$ problem posed by Brania and Yang.

math.CV↗
arXiv2026

The real analytic structure of the Teichmüller space of circle diffeomorphisms with Zygmund continuous derivatives

We apply the methods of simultaneous uniformization and composition operators on Besov spaces to the Teichmüller space $T^Z$ of circle diffeomorphisms with Zygmund continuous derivatives. As consequences, we obtain the following: (1) a new proof of the correspondence between quasiconformal self-homeomorphisms of the unit disk with complex dilatations of linear decay order and their quasisymmetric extensions to the unit circle with regularity in the Zygmund continuously differentiable class; (2) a real-analytic equivalence of $T^Z$ with the real Banach space of Zygmund continuous functions on the unit circle.

math.CV↗
arXiv2026

A Conway-Coxeter theorem for decorated frieze patterns

Let $m\geq0$ and put $n=m+3$. We introduce a normalized positive Laurent class of decorated frieze patterns and prove a decorated analogue of the Conway-Coxeter classification theorem. Namely, such friezes of width $m$ are in canonical bijection with weighted triangulations of a convex $n$-gon, whose boundary edges are labeled by independent variables $y_1,\ldots,y_n$ and whose diagonals are labeled by $x_1,\ldots,x_m$. While a weighted Conway-Coxeter propagation algorithm constructs the frieze from a weighted triangulation, a main new ingredient is an explicit nonrecursive Laurent formula expressing each quiddity entry directly from the weighted triangles incident to the corresponding vertex. Conversely, specialization to $1$, Laurent positivity, and a decorated cutting-and-gluing procedure recover the weighted triangulation from the frieze.

math.CO↗
arXiv2026

Distance-Weighted Norm Equivalences for Analytic Functions on John Domains

Let $Ω\subset\mathbb C$ be a bounded John domain and set $δ(z)=\text{dist}(z,\partialΩ)$. For $1 \text{dim}_A(\partialΩ)-2$, we establish a norm equivalence between $\int_Ω|g|^pδ^α\,dA$ and $\int_Ω|g'|^pδ^{α+p}\,dA$ for analytic functions $g$ on $Ω$, with a point-evaluation term fixing the additive constant. The estimate of the derivative term is local and holds on every proper planar domain, whereas the converse follows from a distance-weighted Poincaré inequality on John domains. Taking $α=mp-2$ yields the corresponding comparison between the $m$-th and $(m+1)$-st derivatives. For $m\ge2$ the boundary-dimension condition is automatic, so the only dimension-sensitive case is the comparison between $\int_Ω|f'|^pδ^{p-2}\,dA$ and $\int_Ω|f''|^pδ^{2p-2}\,dA$ when $1 1$, an inward-cusp $s$-John domain on which the comparison fails, showing that the ordinary John condition cannot in general be weakened.

math.CV↗
arXiv2026

Graph Weldings Associated with Functions in Zygmund, $\mathrm{BMO}$, $\mathrm{VMO}$, and $H^{1/2}$

Let $f\colon \mathbb{R}\to\mathbb{R}$ be a continuous function. We define the graph welding associated with $f$ as the homeomorphism \[ φ= G^{-1}\circ F\colon \mathbb{R}\to\mathbb{R}. \] Here, $F(x)=x+if(x)$ parametrizes the graph of $f$, and $G$ is a conformal mapping from the upper half-plane $\mathbb{H}$ onto one of the two domains bounded by the graph of $f$, admitting a continuous extension to $\mathbb{R}$. In this paper, we investigate how the regularity of the graph function $f$ influences the analytic properties of the associated graph welding $φ$. In particular, under certain assumptions that $f$ within Zygmund, $\mathrm{BMO}$, $\mathrm{VMO}$, and Hardy spaces, we establish results on absolute continuity, quasisymmetry, symmetry, strong quasisymmetry, strong symmetry, and the Weil--Petersson property of the associated graph welding $φ$. These results clarify the interplay between the regularity of graph functions, the geometry of graph curves, and the boundary behavior of conformal mappings.

math.CV↗
arXiv2026

Boundary Characterizations of Little Bloch and $\mathrm{VMOA}$ Functions on the Half-Plane

We extend Pommerenke's characterizations of boundary curves of conformal mappings in terms of the little Bloch and $\mathrm{VMOA}$ conditions from the unit disk $\mathbb{D}$ to the upper half-plane $\mathbb{H}$. Let $G\colon \mathbb{H}\toΩ$ be a conformal mapping onto an unbounded quasidisk $Ω$ with $G(\infty)=\infty$, and let $g\colon \mathbb{R}\toΓ=\partialΩ$ be its boundary extension. In the non-compact setting, the Euclidean smallness on the boundary curve is not necessarily comparable to the smallness of the parameter on $\mathbb{R}$. To overcome this difficulty, we use relative versions of the asymptotic conformality and the asymptotic smoothness with respect to the parametrization $g$. We prove that $\log G'\in B_0(\mathbb{H})$ is equivalent to the asymptotic conformality of $Γ$ relative to $g$, and also to the asymptotic symmetry of the embedding $g$. We further prove that $\log G'\in \mathrm{VMOA}(\mathbb{H})$ is equivalent to the asymptotic smoothness of $Γ$ relative to $g$, and also to the asymptotic smoothness of $g$. These results provide half-plane analogues of Pommerenke's theorems and clarify the role of the parametrization in the unbounded case.

math.CV↗
arXiv2026

Analytic Besov functions, pre-Schwarzian derivatives, and integrable Teichmüller spaces

We study the embedding of integrable Teichmüller spaces $T_p$ into analytic Besov spaces via pre-Schwarzian derivatives. In contrast to the Bers embedding by Schwarzian derivatives, a significant difference arises between the cases $p>1$ and $p=1$. In this paper we focus on the case $p=1$ and extend previous results obtained for $p>1$. This provides a unified framework for the complex-analytic theory of integrable Teichmüller spaces $T_p$ for all $p \geq 1$.

math.CV↗
arXiv2026

Weil-Petersson curves and Dirichlet finite harmonic functions on Riemann surfaces

On two subsurfaces of a Riemann surface divided by a $p$-Weil-Petersson curve $γ$, we consider the spaces of harmonic functions whose $p$-Dirichlet integrals are finite in the complementary domains of $γ$. By requiring the coincidence of boundary values on $γ$, we establish a correspondence between the harmonic functions in these Banach spaces. We analyze the operator arising from this correspondence via the composition operator acting on the Banach space of $p$-Besov functions on the unit circle.

math.CV↗
arXiv2025

Characterization of Asymptotically Smooth Curves

We construct an explicit example of an asymptotically conformal chord-arc curve that fails to be asymptotically smooth. This implies that a function belonging to both the little Bloch space and BMOA does not necessarily lie in VMOA, and that a strongly quasisymmetric homeomorphism which is symmetric is not necessarily strongly symmetric. We also provide a complete characterization of asymptotically smooth curves in terms of asymptotic conformality and uniform approximability.

math.CV↗
arXiv2025

A fiber bundle over BMO Teichmüller space

We prove that the fiber space consisting of BMOA functions that are the logarithms of derivatives of conformal homeomorphisms of the unit disk onto bounded quasidisks forms a real-analytic disk bundle over the Bers embedding of the BMO Teichmüller space. For the VMO Teichmüller space, we show that the corresponding sub-bundle consisting of VMOA functions is real-analytically trivial.

math.CV↗
arXiv2025

Simultaneous uniformization of chord-arc curves and BMO Teichmüller space

This article surveys and develops the use of simultaneous uniformization for the study of chord-arc curves in the BMO Teichmüller space. The method of simultaneous uniformization provides a unified complex-analytic framework in which chord-arc curves are parametrized by their BMO embeddings and the logarithm of derivatives of these embeddings form a biholomorphic image in the Banach space of BMO functions. We review this correspondence and its consequences, such as the relation to reparametrizations by strongly quasisymmetric homeomorphisms, in a rather self-contained manner in order to highlight the coherence of the approach. The main new contribution of this exposition concerns the Cauchy transform of BMO functions on a chord-arc curve. We show that the Cauchy transform is expressed through the derivative of the biholomorphic map arising from simultaneous uniformization, and consequently depends holomorphically on the variation of the chord-arc curve. This result connects classical singular integral operators with the complex structure of Teichmüller spaces and illustrates the effectiveness of the method. We also outline the parallel theory in the VMO Teichmüller space, which exhibits further structural properties.

math.CV↗
arXiv2025

Integrable Teichmüller spaces for analysis on Weil-Petersson curves

The integrableTeichmüller space $T_p$ for $p \geq 1$ is defined by the $p$-integrability of Beltrami coefficients. We characterize a quasisymmetric homeomorphism $h$ in $T_p$ by the condition that $\log h'$ belongs to the real $p$-Besov space, with a certain modification applied in the case $p=1$. This is done as part of the arguments for establishing a biholomorphic correspondence $Λ$ from the product of $T_p$ for simultaneous uniformization of $p$-Weil-Petersson curves into the $p$-Besov space. In particular, this proves the real-analytic equivalence between $T_p$ and the real $p$-Besov space. Moreover, the Cauchy transform of Besov functions on Weil-Petersson curves can be expressed by the derivative of this holomorphic map $Λ$, and from this, the Calderón theorem in this setting is straightforward. It also follows that the Cauchy transforms on $p$-Weil-Petersson curves holomorphically depend on their embeddings as they vary in the Bers coordinates.

math.CV↗
arXiv2025

Handlebodies of Infinite Genus and Schottky Groups

In this paper, we begin an investigation of infinite genus handlebodies, infinitely generated Schottky groups, and related uniformization questions by giving appropriate definitions for them. There are uncountably many topological types of infinite genus surfaces with non-planar ends. We show that any such surface and any infinite genus handlebody can be topologically uniformized by an infinitely generated classical Schottky group. We next show that an infinite genus Riemann surface with non-planar ends admitting a bounded pants decomposition can be quasiconformally uniformized by a classical Schottky group. In addition, the conformal equivalence class of such a uniformization is unique.If the assumption of bounded pants decomposition is removed we supply examples of such Riemann surfaces that do not admit a quasiconformal uniformization by a Schottky group.

math.GT↗
arXiv2025

Arithmetic Y-frieze patterns of width 3 and 4

We determine all arithmetic Y-Frieze patterns of width $3$ and $4$. As a consequence, for $n=3,4$, we verify the surjectivity of a map $p_n$ which corresponds arithmetic Y-Frieze patterns of width $n$ to Coxeter's Frieze patterns.

math.CO↗
arXiv2025

The complex structure of the Teichmüller space of circle diffeomorphisms in the Zygmund smooth class II

In our previous paper with the same title, we established the complex Banach manifold structure for the Teichmüller space of circle diffeomorphisms whose derivatives belong to the Zygmund class. This was achieved by demonstrating that the Schwarzian derivative map is a holomorphic split submersion. We also obtained analogous results for the pre-Schwarzian derivative map. In this second part of the study, we investigate the structure of the image of the pre-Schwarzian derivative map, viewing it as a fiber space over the Bers embedding of the Teichmüller space, and prove that it forms a real-analytic disk-bundle. Furthermore, we consider the little Zygmund class and establish corresponding results for the closed Teichmüller subspace consisting of mappings in this class. Finally, we construct the quotient space of this subspace in analogy with the asymptotic Teichmüller space and prove that the quotient Bers embedding and pre-Bers embedding are well-defined and injective, thereby endowing it with a complex structure modeled on a quotient Banach space.

math.CV↗
arXiv2024

Bi-Lipschitz quasiconformal extensions

We survey several methods of extending quasisymmetric homeomorphisms of the real line to bi-Lipschitz diffeomorphisms of the upper half-plane with respect to the hyperbolic metric.

math.CV↗
arXiv2023

The $p$-integrable Teichmüller space for $p \geqslant 1$

We verify that the $p$-integrable Teichmüller space $T_p$ admits the canonical complex Banach manifold structure for any $p \geq 1$. Moreover, we characterize a quasisymmetric homeomorphism corresponding to an element of $T_p$ in terms of the $p$-Besov space for any $p>1$.

math.CV↗