arXiv subjects

Huaying Wei

Publications and source records attributed to Huaying Wei.

At least 19 recordsLinked to original sources

arXiv2026

Distance-Weighted Norm Equivalences for Analytic Functions on John Domains

Let $\Omega\subset\mathbb C$ be a bounded John domain and set $\delta(z)=\text{dist}(z,\partial\Omega)$. For $1 \text{dim}_A(\partial\Omega)-2$, we establish a norm equivalence between $\int_\Omega |g|^p\delta^\alpha\,dA$ and $\int_\Omega |g'|^p\delta^{\alpha+p}\,dA$ for analytic functions $g$ on $\Omega$, 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 Poincar\'e inequality on John domains. Taking $\alpha=mp-2$ 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 $\int_\Omega |f'|^p\delta^{p-2}\,dA$ and $\int_\Omega |f''|^p\delta^{2p-2}\,dA$ 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

Fractional Besov-Sobolev Spaces on Quasicircles

Let $\Gamma$ be a bounded Jordan curve and $\Omega_i,\Omega_e$ its two complementary components. For $p\in (1, \infty),\,s\in(0,1)$ we define the two spaces $\mathcal{B}_{p,p}^s(\Omega_{i,e})$ as the set of harmonic functions $u$ respectively in $\Omega_i$ and $\Omega_e$ such that $$ \iint_{\Omega_{i,e}} |\nabla u(z)|^p d(z,\Gamma)^{(1-s)p-1} dxdy<+\infty.$$ When it is possible to identify these spaces with spaces of functions on the boundary (trace spaces), we address the question of their equality. When $\Gamma$ is the unit circle, these two spaces coincide with homogeneous fractional Besov-Sobolev spaces and the framework of quasicircles appears to be an appropriate generalization. In this framework, we study the boundedness of the Plemelj-Calder\'on operator and apply the results to show that for some values of $p,s$, if the two spaces coincide, they are restrictions to $\Gamma$ of some weighted Sobolev space. If $\Gamma$ is further assumed to be rectifiable, we define $B_{p,p}^s(\Gamma)$ as the space of functions $f\in L^p(\Gamma)$ such that $$\iint_{\Gamma\times \Gamma}\frac{|f(z)-f(\zeta)|^p}{|z-\zeta|^{1+ps}} |dz||d\zeta|<+\infty.$$ Again, these spaces coincide with the homogeneous fractional Besov-Sobolev spaces for the unit circle. While the chord-arc property is the necessary and sufficient condition for the equality $$\mathcal{B}_{p,p}^s(\Omega_{i})=\mathcal{B}_{p,p}^s(\Omega_{e})=B_{p,p}^s(\Gamma)$$ in the case of $s=1/p,\, p\ge 2$, this is no longer the case for general $s\in (0,1)$. However, we show that equality holds for radial-Lipschitz curves. Finally, we re-interpretate some of our results as some "almost"-Dirichlet principle in the spirit of Maz'ya.

math.CV↗

arXiv2025

Cauchy Integral, Fractional Sobolev Spaces and Chord-Arc Curves

Let $\Gamma$ be a bounded Jordan curve and $\Omega_i,\Omega_e$ its two complementary components. For $s\in(0,1)$ we define $\mathcal{H}^s(\Gamma)$ as the set of functions $f:\Gamma\to \mathbb C$ having harmonic extension $u$ in $\Omega_i\cup \Omega_e$ such that $$ \iint_{\Omega_i\cup \Omega_e} |\nabla u(z)|^2 d(z,\Gamma)^{1-2s} dxdy<+\infty.$$ If $\Gamma$ is further assumed to be rectifiable we define $H^s(\Gamma)$ as the space of measurable functions $f:\Gamma\to \mathbb C$ such that $$\iint_{\Gamma\times \Gamma}\frac{|f(z)-f(\zeta)|^2}{|z-\zeta|^{1+2s}} d\sigma(z)d\sigma(\zeta)<+\infty.$$ When $\Gamma$ is the unit circle these two spaces coincide with the homogeneous fractional Sobolev space defined via Fourier series. For a general rectifiable curve these two spaces need not coincide and our first goal is to investigate the cases of equality: while the chord-arc property is the necessary and sufficient condition for equality in the classical case of $s=1/2$, this is no longer the case for general $s\in (0,1)$. We show however that equality holds for Lipschitz curves. The second goal involves the Plemelj-Calder\'on problem. ......

math.CV↗

arXiv2024

Conformal weldings in the Loewner equation and Weil--Petersson quasislit-disks

A simple arc $\Gamma = \gamma(0, T]$, growing into the unit disk $\mathbb D$ from its boundary, generates a driving term $\xi$ and a conformal welding $\phi$ through the Loewner differential equation. When $\Gamma$ is the slit of a Weil--Petersson quasislit-disk $\mathbb D\setminus\Gamma$, the Loewner transform and its inverse $\Gamma \leftrightarrow \xi$ have been well understood due to Y. Wang's work. We investigate the maps $\Gamma \leftrightarrow \phi$ in this case, giving a description of $\Gamma$ in terms of $\phi$.

math.CV↗

arXiv2024

p-Dirichlet spaces over chord-arc domains

Let $\Gamma$ be a rectifiable Jordan curve in the complex plane, let $\Omega_i$ and $\Omega_e$ be its interior and exterior domains, respectively, and let $1 < p < \infty$. Let $E$ be the vector space of restrictions to $\Gamma$ of functions in $C^1(\mathbb C)$. We consider the following three seminorms on $E$: (i) $\lVert u\rVert_i=\left(\frac{1}{2\pi}\iint_{\Omega_i}|\nabla U_i(z)|^p\lambda_{\Omega_i}^{2-p}(z)\,dA(z)\right)^{1/p}$, where $U_i$ is the harmonic extension of $u$ to $\Omega_i$ and $\lambda_{\Omega_i}$ is the hyperbolic density of $\Omega_i$; (ii) $\lVert u\rVert_e$, defined analogously on $\Omega_e$; and (iii) $\lVert u\rVert_{B_p(\Gamma)}=\left(\frac{1}{4\pi^2}\iint_{\Gamma\times\Gamma}\frac{|u(z)-u(\zeta)|^p}{|z-\zeta|^2}\,|dz|\,|d\zeta|\right)^{1/p}$. These three seminorms are known to be equivalent when $\Gamma$ is a chord-arc curve. We investigate the converse problem.

math.CV↗

arXiv2024

Dirichlet spaces over chord-arc domains

If $U$ is a $C^{\infty}$ function with compact support in the plane, we let $u$ be its restriction to the unit circle $\mathbb{S}$, and denote by $U_i,\,U_e$ the harmonic extensions of $u$ respectively in the interior and the exterior of $\mathbb S$ on the Riemann sphere. About a hundred years ago, Douglas has shown that \begin{align*} \iint_{\mathbb{D}}|\nabla U_i|^2(z)dxdy&= \iint_{\bar{\mathbb{C}}\backslash\bar{\mathbb{D}}}|\nabla U_e|^2(z)dxdy &= \frac{1}{2\pi}\iint_{\mathbb S\times\mathbb S}\left|\frac{u(z_1)-u(z_2)}{z_1-z_2}\right|^2|dz_1||dz_2|, \end{align*} thus giving three ways to express the Dirichlet norm of $u$. On a rectifiable Jordan curve $\Gamma$ we have obvious analogues of these three expressions, which will of course not be equal in general. The main goal of this paper is to show that these $3$ (semi-)norms are equivalent if and only if $\Gamma$ is a chord-arc curve.

math.CV↗

arXiv2024

Analytic Besov functions, pre-Schwarzian derivatives, and integrable Teichm\"uller spaces

We study the embedding of integrable Teichm\"uller 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 Teichm\"uller spaces $T_p$ for all $p \geq 1$.

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↗

arXiv2022

Parametrization of the $p$-Weil-Petersson curves: holomorphic dependence

Similarly to the Bers simultaneous uniformization, the product of the $p$-Weil-Petersson Teichmüller spaces for $p \geq 1$ provides the coordinates for the space of $p$-Weil-Petersson embeddings $γ$ of the real line $\mathbb R$ into the complex plane $\mathbb C$. We prove the biholomorphic correspondence from this space to the $p$-Besov space of $u=\log γ'$ on $\mathbb R$ for $p>1$. From this fundamental result, several consequences follow immediately which clarify the analytic structures concerning parameter spaces of $p$-Weil-Petersson curves. In particular, it follows that the correspondence of the Riemann mapping parameters to the arc-length parameters keeping the images of curves is a homeomorphism with bi-real-analytic dependence of change of parameters. This is a counterpart to a classical theorem of Coifman and Meyer for chord-arc curves.

math.CV↗

arXiv2022

A description of $A_{\infty }$-weights for VMO

We present a new characterization of Muckenhoupt $A_{\infty}$-weights whose logarithm is in $\operatorname{VMO}(\mathbb{R})$ in terms of vanishing Carleson measures on $\mathbb{R}_+^2$ and vanishing doubling weights on $\mathbb{R}$. This also gives a novel description of strongly symmetric homeomorphisms on the real line by using a geometric quantity.

math.CV↗

arXiv2022

Strongly symmetric homeomorphisms on the real line with uniform continuity

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.

math.CV↗

arXiv2022

Chordal Loewner chains and Teichmüller spaces on the half-plane

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 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 the 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.

math.CV↗

arXiv2022

The $p$-Weil-Petersson Teichmüller space and the quasiconformal extension of curves

We consider the correspondence between the space of $p$-Weil-Petersson curves $γ$ on the plane and the $p$-Besov space of $u=\log γ'$ on the real line for $p >1$. We prove that the variant of the Beurling-Ahlfors extension defined by using the heat kernel yields a holomorphic map for $u$ on a domain of the $p$-Besov space to the space of $p$-integrable Beltrami coefficients. This in particular gives a global real-analytic section for the Teichmüller projection from the space of $p$-integrable Beltrami coefficients to the $p$-Weil-Petersson Teichmüller space.

math.CV↗

arXiv2021

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

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.

math.CV↗

arXiv2021

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

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.

math.CV↗

arXiv2021

A real-variable construction with applications to BMO-Teichmüller theory

With the use of real-variable techniques, we construct a weight function $ω$ on the interval $[0, 2π)$ that is doubling and satisfies $\log ω$ is a BMO function, but which is not a Muckenhoupt weight ($A_\infty$). Applications to the BMO-Teichmüller space and the space of chord-arc curves are considered.

math.CV↗

arXiv2020

Beurling-Ahlfors extension by heat kernel, ${\rm A}_\infty$-weights for VMO, and vanishing Carleson measures

We investigate a variant of the Beurling-Ahlfors extension of quasisymmetric homeomorphisms of the real line that is given by the convolution of the heat kernel, and prove that the complex dilatation of such a quasiconformal extension of a strongly symmetric homeomorphism (i.e. its derivative is an ${\rm A}_\infty$-weight whose logarithm is in VMO) induces a vanishing Carleson measure on the upper half-plane.

math.CV↗

arXiv2019

Symmetric and strongly symmetric homeomorphisms on the real line with non-symmetric inversion

We show an example of a symmetric homeomorphism $h$ of the real line $\mathbb{R}$ onto itself such that $h^{-1}$ is not symmetric. This implies that the set of all symmetric self-homeomorphisms of $\mathbb{R}$ does not constitute a group under the composition. We also deal with strongly symmetric self-homeomorphisms of $\mathbb{R}$ along the same line. These results reveal the difference of the sets of such self-homeomorphisms of the real line from those of the unit circle.

math.CV↗