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Xiantao Wang

Publications and source records attributed to Xiantao Wang.

At least 19 recordsLinked to original sources
arXiv2026

Dimension-free inner uniform estimates for quasigeodesics

In this paper, we establish a dimension-free inner uniform estimate for quasigeodesics. More precisely, we prove that a $c_0$-quasigeodesic in a $δ$-Gromov hyperbolic $c$-John domain in $\mathbb{R}^n$ is $b$-inner uniform, for some constant $b$ depending only on $c_0$, $δ$ and $c$, but not on the dimension $n$. The proof relies crucially on the techniques introduced by Guo-Huang-Wang in their recent work [arXiv:2502.02930, 2025]. In particular, we actually show that the above result holds in general Banach spaces, which answers affirmatively an open question of J. Väisälä in [Analysis, 2004] and partially addresses the open question of Bonk-Heinonen-Koskela in [Asterisque, 2001]. As a byproduct of our main result, we obtain that a $c_0$-quasigeodesic in a $δ$-Gromov hyperbolic $c$-John domain in $\mathbb{R}^n$ is a $b$-cone arc with a dimension-free constant $b=b(c_0,δ,c)$. This resolves an open problem of J. Heinonen in [Rev. Math. Iberoam., 1989].

math.CV↗
arXiv2026

The dimension-free Gehring-Hayman inequality for quasigeodesics

A well-known theorem of J. Heinonen and S. Rohde in 1993 states that if $D\subset \mathbb{R}^n$ is quasiconformally equivalently to an uniform domain, then the Gehring-Hayman inequality holds in $D$: quasihyperbolic geodesics in $D$ minimizes the Euclidean length among all curves in $D$ with the same end points, up to a universal dimension-dependent multiplicative constant. In this paper, we develop a new approach to strengthen the above result in the following three aspects: 1) obtain a dimension-free multiplicative constant in the Gehring-Hayman inequality; 2) relax the class of quasihyperbolic geodesics to more general quasigeodesics; 3) relax the quasiconformal equivalence to more general coarsely quasihyperbolic equivalence. As a byproduct of our general approach, we are able to prove that the above improved Gehring-Hayman inequality indeed holds in Banach spaces. This answers affirmatively an open problem raised by J. Heinonen and S. Rohde in 1993 and reformulated by J. Väisälä in 2005.

math.CV↗
arXiv2025

Characterizations of quasihyperbolic John domains and uniform domains in metric spaces

In a recent work of Zhou and Ponnusamy [Ann. Sc. Norm. Super. Pisa Ci. Sci. 2025], the authors studied the following natural question: find sufficient and necessary conditions for a domain $Ω$ in a metric space $X$ to be quasihyperbolic John. It was proved that Gromov hyperbolic John domains are quasihyperbolic John, quantitatively. As an application, they obtained a characterization of uniform domains in Ahlfors regular spaces. In a recent work, using a deep improved characterization of Gromov hyperbolicity, Guo, Huang and Wang [arXiv 2025] proved the quantitative equivalence bteween inner uniformity and the quasihyperbolic John condition in metric doubling spaces. However, the proof does not yield a similar characterization for uniform domains. In this article, we find a new elementary approach to successfully extend the above characterization to uniform domains: a domain $Ω$ in a doubling length space $X$ is uniform if and only if it is linearly locally connected (LLC) and satisfies the ball separation condition, if and only if it is LLC-1 and quasihyperbolic John, quantitatively. This substantially improved the corresponding results of Zhou and Ponnusamy. Our new approach also allows us to give an alternative proof of the inner uniformity result of Guo-Huang-Wang without using the improved characterization on Gromov hyperbolicity.

math.CV↗
arXiv2025

Gromov hyperbolicity III: an improved geometric characterization and its applications

In the seminal work of Balogh-Buckley [Invent. Math. 2003], the authors asked the following fundamental open problem: for proper subdomains in the Euclidean space $\mathbb{R}^n$, does the ball separation condition alone imply the Gehring-Hayman inequality? In this paper, via a completely new measure-independent approach, we establish the following geometric characterization of Gromov hyperbolicity in a fairly general setting: The Gromov hyperbolicity of a proper subdomain in a doubling metric space is quantitatively equivalent to the geometric ball separation condition, with explicit dependence on the coefficients. In the special case of Euclidean spaces, it affirmatively solves the above Balogh-Buckely problem. Our result also significantly improves the main result of Koskela-Lammi-Manojlović [Ann. Sci. Éc. Norm. Supér. 2014]. As applications, we obtain the quasiconformal invariance of ball separation condition, a geometric characterization of inner uniformity in terms of ball separation condition, and the Gromov hyperbolicity of quasihyperbolic John length spaces.

math.CV↗
arXiv2025

Quantitative correspondence between quasi-symmetric mappings on complete metric spaces and rough quasi-isometric mappings on their hyperbolic fillings

In this paper, we establish a quantitative correspondence between power quasi-symmetric mappings on complete metric spaces and rough quasi-isometric mappings on their hyperbolic fillings. In particular, we prove that the exponents in the power quasi-symmetric mappings coincide with the coefficients in the rough quasi-isometric mappings. This shows that the obtained correspondence is both sharp and consistent. In this way, we generalize the corresponding result by Björn, Björn, Gill, and Shanmugalingam (J. Reine Angew. Math., 2017) from the setting of rooted trees to that of hyperbolic fillings.

math.CV↗
arXiv2025

Sharp Riesz conjugate functions theorems for quasiregular mappings

One of the celebrated results by Riesz \cite{Rie} is the Riesz conjugate functions theorem for analytic functions in the complex plane $\mathbb{C}$. The study on the Riesz conjugate functions theorem for functions in higher dimensional spaces has attracted much attention. Fefferman and Stein \cite{FS-1972} established the Riesz conjugate functions theorem for the Cauchy-Riemann systems in the upper half real space $\mathbb{R}^{n+1}_{+}$. Astala and Koskela \cite{AS-2} investigated the Riesz conjugate functions theorem for quasiconformal mappings of the unit ball $\mathbf{B}^{n}$ in $\mathbb{R}^n$, and posed an open problem which is as follows: Does there exist a quasiconformal analog for the Riesz theorem on conjugate functions? The purpose of this paper is to develop some methods to study this topic further, in particular, Astala-Koskela's open problem. First, we prove a sharp Riesz conjugate functions theorem for a class of quasiregular mappings of $\mathbf{B}^{n}$ for all $n\geq 2$ which satisfy the so-called Heinz's nonlinear differential inequality. As a direct consequence of this result, we find that the answer to Astala-Koskela's open problem is affirmative for harmonic quasiregular mappings of $\mathbf{B}^{n}$ for all $n\geq 2$. Second, we obtain a sharp Riesz conjugate functions theorem for invariant harmonic $K$-quasiregular mappings of $\mathbf{B}^{n}$ for all $n\geq 2$ which shows that the answer to Astala-Koskela's open problem is affirmative for these mappings. At last, we introduce the family of $κ$-pluriharmonic mappings of the unit ball $\mathbb{B}^n$ in $\mathbb{C}^n$, and establish a sharp Riesz conjugate functions theorem for these mappings for all $n\geq 1$. Consequently, we generalize and improve all main results by Liu and Zhu \cite{L-Z}.

math.FA↗
arXiv2025

Geometric characterizations of inner uniformity through Gromov hyperbolicity

In this paper, we study the characterization of inner uniformity of bounded domains $G$ in $\IR^n$, and prove that the following three conditions are equivalent: $(1)$ $G$ is inner uniform; $(2)$ $G$ is Gromov hyperbolic and its inner metric boundary is naturally quasisymmetrically equivalent to the Gromov boundary; $(3)$ $G$ is Gromov hyperbolic and linearly locally connected with respect to the inner metric. The equivalence between the conditions $(1)$ and $(2)$, and the implication from $(2)$ to $(3)$ affirmatively answer three questions raised by Bonk, Heinonen, and Koskela in 2001.

math.CV↗
arXiv2024

Application of Prompt Learning Models in Identifying the Collaborative Problem Solving Skills in an Online Task

Collaborative problem solving (CPS) competence is considered one of the essential 21st-century skills. To facilitate the assessment and learning of CPS competence, researchers have proposed a series of frameworks to conceptualize CPS and explored ways to make sense of the complex processes involved in collaborative problem solving. However, encoding explicit behaviors into subskills within the frameworks of CPS skills is still a challenging task. Traditional studies have relied on manual coding to decipher behavioral data for CPS, but such coding methods can be very time-consuming and cannot support real-time analyses. Scholars have begun to explore approaches for constructing automatic coding models. Nevertheless, the existing models built using machine learning or deep learning techniques depend on a large amount of training data and have relatively low accuracy. To address these problems, this paper proposes a prompt-based learning pre-trained model. The model can achieve high performance even with limited training data. In this study, three experiments were conducted, and the results showed that our model not only produced the highest accuracy, macro F1 score, and kappa values on large training sets, but also performed the best on small training sets of the CPS behavioral data. The application of the proposed prompt-based learning pre-trained model contributes to the CPS skills coding task and can also be used for other CSCW coding tasks to replace manual coding.

cs.HC↗
arXiv2023

Locally biHölder continuous mappings and their induced embeddings between Besov spaces

In this paper, we introduce a class of homeomorphisms between metric spaces, which are locally biHölder continuous mappings. Then an embedding result between Besov spaces induced by locally biHölder continuous mappings between Ahlfors regular spaces is established, which extends the corresponding result of Björn-Björn-Gill-Shanmugalingam (J. Reine Angew. Math. 725: 63-114, 2017). Furthermore, an example is constructed to show that our embedding result is more general. We also introduce a geometric condition, named as uniform boundedness, to characterize when a quasisymmetric mapping between uniformly perfect spaces is locally biHölder continuous.

math.FA↗
arXiv2021

Borderline case of traces and extensions for weighted Sobolev spaces

In this paper, we study the traces and the extensions for weighted Sobolev spaces on upper half spaces when the weights reach to the borderline cases. We first give a full characterization of the existence of trace spaces for these weighted Sobolev spaces, and then study the trace parts and the extension parts between the weighted Sobolev spaces and a new kind of Besov-type spaces (on hyperplanes) which are defined by using integral averages over selected layers of dyadic cubes.

math.FA↗
arXiv2021

On Lipschitz continuity of solutions of hyperbolic Poisson's equation

In this paper, we investigate solutions of the hyperbolic Poisson equation $Δ_{h}u(x)=ψ(x)$, where $ψ\in L^{\infty}(\mathbb{B}^{n}, \mathbb{R}^n)$ and \[ Δ_{h}u(x)= (1-|x|^2)^2Δu(x)+2(n-2)(1-|x|^2)\sum_{i=1}^{n} x_{i} \frac{\partial u}{\partial x_{i}}(x) \] is the hyperbolic Laplace operator in the $n$-dimensional space $\mathbb{R}^n$ for $n\ge 2$. We show that if $n\geq 3$ and $u\in C^{2}(\mathbb{B}^{n},\mathbb{R}^n) \cap C(\overline{\mathbb{B}^{n}},\mathbb{R}^n )$ is a solution to the hyperbolic Poisson equation, then it has the representation $u=P_{h}[ϕ]-G_{ h}[ψ]$ provided that $u\mid_{\mathbb{S}^{n-1}}=ϕ$ and $\int_{\mathbb{B}^{n}}(1-|x|^{2})^{n-1} |ψ(x)|\,dτ(x)<\infty$. Here $P_{h}$ and $G_{h}$ denote Poisson and Green integrals with respect to $Δ_{h}$, respectively. Furthermore, we prove that functions of the form $u=P_{h}[ϕ]-G_{h}[ψ]$ are Lipschitz continuous.

math.AP↗
arXiv2018

Bi-Lipschitz characteristic of quasiconformal self-mappings of the unit disk satisfying bi-harmonic equation

Suppose that $f$ is a $K$-quasiconformal self-mapping of the unit disk $\mathbb{D}$, which satisfies the following: $(1)$ the biharmonic equation $Δ(Δf)=g$ $(g\in \mathcal{C}(\overline{\mathbb{D}}))$, (2) the boundary condition $Δf=φ$ ($φ\in\mathcal{C}(\mathbb{T})$ and $\mathbb{T}$ denotes the unit circle), and $(3)$ $f(0)=0$. The purpose of this paper is to prove that $f$ is Lipschitz continuos, and, further, it is bi-Lipschitz continuous when $\|g\|_{\infty}$ and $\|φ\|_{\infty}$ are small enough. Moreover, the estimates are asymptotically sharp as $K\to 1$, $\|g\|_{\infty}\to 0$ and $\|φ\|_{\infty}\to 0$, and thus, such a mapping $f$ behaves almost like a rotation for sufficiently small $K$, $\|g\|_{\infty}$ and $\|φ\|_{\infty}$.

math.CV↗
arXiv2018

Schwarz type lemma, Landau type theorem and Lipschitz type space of solutions to biharmonic equations

The purpose of this paper is to study the properties of the solutions to the biharmonic equations: $Δ(Δf)=g$, where $g:$ $\overline{\mathbb{D}}\rightarrow\mathbb{C}$ is a continuous function and $\overline{\mathbb{D}}$ denotes the closure of the unit disk $\mathbb{D}$ in the complex plane $\mathbb{C}$. In fact, we establish the following properties for those solutions: Firstly, we establish the Schwarz type lemma. Secondly, by using the obtained results, we get a Landau type theorem. Thirdly, we discuss their Lipschitz type property.

math.CV↗
arXiv2017

Several properties of $α$-harmonic functions in the unit disk

The aim of this paper is to obtain the Schwarz-Pick type inequality for $α$-harmonic functions $f$ in the unit disk and get estimates on the coefficients of $f$. As an application, a Landau type theorem of $α$-harmonic functions is established.

math.CV↗
arXiv2015

On the Lipschitz continuity of certain quasiregular mappings between smooth Jordan domains

We first investigate the Lipschitz continuity of $(K, K')$-quasiregular $C^2$ mappings between two Jordan domains with smooth boundaries, satisfying certain partial differential inequalities concerning Laplacian. Then two applications of the obtained result are given: As a direct consequence, we get the Lipschitz continuity of $ρ$-harmonic $(K, K')$-quasiregular mappings, and as the other application, we study the Lipschitz continuity of $(K,K')$-quasiconformal self-mappings of the unit disk, which are the solutions of the Poisson equation $Δw=g$. These results generalize and extend several recently obtained results by Kalaj, Mateljević and Pavlović.

math.CV↗
arXiv2015

Semisolidity and locally weak quasisymmetry of homeomorphisms in metric spaces

In this paper, we investigate the relationship between semisolidity and locally weak quasisymmetry of homeomorphisms in quasiconvex and complete metric spaces. Our main objectives are to (1) generalize the main result in [X. Huang and J. Liu, Quasihyperbolic metric and quasisymmetric mappings in metric spaces, Trans. Amer. Math. Soc. 367 (2015), 6225-6246] together with other related results, and (2) give a complete answer to the open problem given in [X. Huang and J. Liu, Quasihyperbolic metric and quasisymmetric mappings in metric spaces, Trans. Amer. Math. Soc. 367 (2015), 6225-6246]. As an application, we prove that the composition of two locally weakly quasisymmetric mappings is a locally weakly quasisymmetric mapping and that it is quasiconformal.

math.CV↗
arXiv2015

On the subinvariance of uniform domains in metric spaces

Suppose that $X$ and $Y$ are quasiconvex and complete metric spaces, that $G\subset X$ and $G'\subset Y$ are domains, and that $f: G\to G'$ is a homeomorphism. Our main result is the following subinvariance property of the class of uniform domains: Suppose both $f$ and $f^{-1}$ are weakly quasisymmetric mappings and $G'$ is a quasiconvex domain. Then the image $f(D)$ of every uniform subdomain $D$ in $G$ under $f$ is uniform. The subinvariance of uniform domains with respect to freely quasiconformal mappings or quasihyperbolic mappings is also studied with the additional condition that both $G$ and $G'$ are locally John domains.

math.CV↗