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Jiaolong Chen

Publications and source records attributed to Jiaolong Chen.

At least 19 recordsLinked to original sources
arXiv2026

Hardy-Littlewood type phenomena and the Girela-Peláez conjecture for the Möbius invariant Laplacian operator

The purpose of this paper is twofold. First, we investigate the Hardy-Littlewood type phenomena for Dirichlet solutions to the Möbius invariant Laplace equation on the unit ball in $\mathbb{R}^n$. Our work extends and improves several key results due to Pavlovć [Rev. Mat. Iberoam. 23: 831-845, 2007] and Chen et al. [J. Geom. Anal. 34: 23 pp, 2024]. In particular, we give a complete answer to a question raised by Makoto Masumoto. Second, motivated by Aikawa's work, we study the boundedness of the operator norm of $P_α$, where $P_α[φ]$ is the Dirichlet solution of such equation for the boundary data $φ$. By using alternative proof techniques, we obtain an equivalent characterization of the boundedness of the operator norm of $P_α$. Finally, we show that the Girela-Peláez conjecture holds positively for more general classes of functions induced by the Möbius invariant Laplacian operator.

math.FA↗
arXiv2025

Schwarz-Pick type lemma and Landau type theorem for $α$-harmonic mappings

The aim of this paper is twofold. First, we obtain a Schwarz-Pick type lemma for the $α$-harmonic mapping $u=P_α[ϕ]$, where $ϕ\in L^{p}(\mathbb{S}^{n-1},\mathbb{R} )$ and $p\in[1,\infty]$. We get an explicit form of the sharp function $\mathbf{C}_{α, q}(x)$ in the inequality $|\nabla u(x)| \leq \mathbf{C}_{α, q}(x)\|ϕ\|_{L^p(\mathbb{S}^{n-1}, \mathbb{ R} )}$. Second, we prove a Landau type theorem for $u=P_α[ϕ]$, where $ϕ\in L^{\infty}(\mathbb{S}^{n-1},\mathbb{R}^{n})$. These results generalize and extend the corresponding results due to Kalaj (Complex Anal. Oper. Theory, 2024) and Khalfallah et al. (Mediterr. J. Math., 2021).

math.AP↗
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↗
arXiv2020

Dirichlet-type energy of mappings between two concentric annuli

Let $\mathbb{A}$ and $\mathbb{A_{*}}$ be two non-degenerate spherical annuli in $\mathbb{R}^{n}$ equipped with the Euclidean metric and the weighted metric $|y|^{1-n}$, respectively. Let $\mathcal{F}(\mathbb{A},\mathbb{A_{*}})$ denote the class of homeomorphisms in $\mathcal{W}^{1,n-1}(\mathbb{A},\mathbb{A_{*}})$. For $n=3$, the second author \cite{kalaj2018} proved that the minimizers of the Dirichlet-type energy $\mathcal{E}[h]=\int_{\mathbb{A}} \frac{\|Dh(x)\|^{n-1}}{|h(x)|^{n-1}}dx$ are certain generalized radial diffeomorphisms, where $h\in \mathcal{F}(\mathbb{A},\mathbb{A_{*}})$. For the case $n\geq 4$, he conjectured that the minimizers are also certain generalized radial diffeomorphisms between $\mathbb{A}$ and $\mathbb{A_{*}}$. The main aim of this paper is to consider this conjecture. First, we investigate the minimality of the following combined energy integral: $$ \mathbb{E}[a,b][h] =\int_{\mathbb{A}}\frac{a^{2}ρ^{n-1}(x)\|DS(x)\|^{n-1}+b^{2}|\nabla ρ(x)|^{n-1}}{|ρ(x)|^{n-1}}dx, $$ where $h=ρS\in \mathcal{F}(\mathbb{A},\mathbb{A_{*}})$, $ρ=|h|$ and $a,b>0$. The obtained result is a generalization of \cite[Theorem 1.1]{kalaj2018}. As an application, we show that the above conjecture is almost true for the case $n\geq 4$, i.e., the minimizer of the energy integral $\mathcal{E}[h]$ does not exist but there exists a minimizing sequence which belongs to the generalized radial mappings.

math.AP↗
arXiv2020

Khavinson problem for hyperbolic harmonic mappings in Hardy space

\begin{abstract} In this paper, we partly solve the generalized Khavinson conjecture in the setting of hyperbolic harmonic mappings in Hardy space. Assume that $u=\mathcal{P}_Ω[ϕ]$ and $ϕ\in L^{p}(\partialΩ, \mathbb{R})$, where $p\in[1,\infty]$, $\mathcal{P}_Ω[ϕ]$ denotes the Poisson integral of $ϕ$ with respect to the hyperbolic Laplacian operator $Δ_{h}$ in $Ω$, and $Ω$ denotes the unit ball $\mathbb{B}^{n}$ or the half-space $\mathbb{H}^{n}$. For any $x\in Ω$ and $l\in \mathbb{S}^{n-1}$, let $\mathbf{C}_{Ω,q}(x)$ and $\mathbf{C}_{Ω,q}(x;l)$ denote the optimal numbers for the gradient estimate $$ |\nabla u(x)|\leq \mathbf{C}_{Ω,q}(x)\|ϕ\|_{ L^{p}(\partialΩ, \mathbb{R})} $$ and gradient estimate in the direction $l$ $$|\langle\nabla u(x),l\rangle|\leq \mathbf{C}_{Ω,q}(x;l)\|ϕ\|_{ L^{p}(\partialΩ, \mathbb{R})}, $$ respectively. Here $q$ is the conjugate of $p$. If $q=\infty$ or $q\in[\frac{2K_{0}-1}{n-1}+1,\frac{2K_{0}}{n-1}+1]\cap [1,\infty)$ with $K_{0}\in\mathbb{N}=\{0,1,2,\ldots\}$, then $\mathbf{C}_{\mathbb{B}^{n},q}(x)=\mathbf{C}_{\mathbb{B}^{n},q}(x;\pm\frac{x}{|x|})$ for any $x\in\mathbb{B}^{n}\backslash\{0\}$, and $\mathbf{C}_{\mathbb{H}^{n},q}(x)=\mathbf{C}_{\mathbb{H}^{n},q}(x;\pm e_{n})$ for any $x\in \mathbb{H}^{n}$, where $e_{n}=(0,\ldots,0,1)\in\mathbb{S}^{n-1}$. However, if $q\in(1,\frac{n}{n-1})$, then $\mathbf{C}_{\mathbb{B}^{n},q}(x)=\mathbf{C}_{\mathbb{B}^{n},q}(x;t_{x})$ for any $x\in\mathbb{B}^{n}\backslash\{0\}$, and $\mathbf{C}_{\mathbb{H}^{n},q}(x)=\mathbf{C}_{\mathbb{H}^{n},q}(x;t_{e_{n}})$ for any $x\in \mathbb{H}^{n}$. Here $t_{w}$ denotes any unit vector in $\mathbb{R}^{n}$ such that $\langle t_{w},w\rangle=0$ for $w\in \mathbb{R}^{n}\setminus\{0\}$. \end{abstract}

math.AP↗
arXiv2020

Optimal estimates for hyperbolic harmonic mappings in Hardy space

Assume that $p\in(1,\infty]$ and $u=P_{h}[ϕ]$, where $ϕ\in L^{p}(\mathbb{S}^{n-1},\mathbb{R}^{n})$. Then for any $x\in \mathbb{B}^{n}$, we obtain the sharp inequalities $$ |u(x)|\leq \frac{\mathbf{C}_{q}^{\frac{1}{q}}(x)}{(1-|x|^2)^{\frac{n-1 }{p}}} \|ϕ\|_{L^{p}} \quad\text{and}\quad |u(x)|\leq \frac{\mathbf{C}_{q}^{\frac{1}{q}} }{(1-|x|^2)^{\frac{n-1 }{p}}} \|ϕ\|_{L^{p} } $$ for some function $\mathbf{C}_{q}(x)$ and constant $\mathbf{C}_{q}$ in terms of Gauss hypergeometric and Gamma functions, where $q$ is the conjugate of $p$. This result generalize and extend some known result from harmonic mapping theory ([5, Theorems 1.1 and 1.2] and [1, Proposition 6.16]).

math.CA↗
arXiv2020

Koebe and Caratheódory type boundary behavior results for harmonic mappings

We study the behavior of the boundary function of a harmonic mapping from global and local points of view. Results related to the Koebe lemma are proved, as well as a generalization of a boundary behavior theorem by Bshouty, Lyzzaik and Weitsman. We also discuss this result from a different point of view, from which a relation between the boundary behavior of the dilatation at a boundary point and the continuity of the boundary function of our mapping can be seen.

math.CV↗
arXiv2020

Schwarz lemma for hyperbolic harmonic mappings in the unit ball

Assume that $p\in[1,\infty]$ and $u=P_{h}[ϕ]$, where $ϕ\in L^{p}(\mathbb{S}^{n-1},\mathbb{R}^n)$ and $u(0) = 0$. Then we obtain the sharp inequality $|u(x)|\le G_p(|x|)\|ϕ\|_{L^{p}}$ for some smooth function $G_p$ vanishing at $0$. Moreover, we obtain an explicit form of the sharp constant $C_p$ in the inequality $\|Du(0)\|\le C_p\|ϕ\|_{L^{p}}$. These two results generalize and extend some known result from harmonic mapping theory (\cite[Theorem 2.1]{kalaj2018}) and hyperbolic harmonic theory (\cite[Theorem 1]{bur}).

math.CV↗
arXiv2017

Generalized Bloch spaces, Integral means of hyperbolic harmonic mappings in the unit ball

In this paper, we investigate the properties of hyperbolic harmonic mappings in the unit ball $\mathbb{B}^{n}$ in $\IR^n$ $(n\geq 2)$. Firstly, we establish necessary and sufficient conditions for a hyperbolic harmonic mapping to be in the Bloch space $\mathcal{B}(\mathbb{B}^{n})$ and the generalized Bloch space $\mathcal{L}_{\infty,ω}\mathcal{B}_{α,\mathrm{a}}^{0}(\mathbb{B}^{n})$, respectively. Secondly, we discuss the relationship between the integral means of hyperbolic harmonic mappings and that of their gradients. The obtained results are the generalizations of Hardy and Littlewood's related ones in the setting of hyperbolic harmonic mappings. Finally, we characterize the weak uniform boundedness property of hyperbolic harmonic mappings in terms of the quasihyperbolic metric.

math.CV↗
arXiv2016

Geometric properties of $\log$-polyharmonic mappings

In this paper, a class of $\log$-polyharmonic mappings $\mathcal{L}_p\mathcal{H}$ together with its subclass $\mathcal{L}_p\mathcal{H}(G)$ in the unit disk $\mathbb{D}=\{z: |z|<1\}$ is introduced, and several geometrical properties such as the starlikeness, convexity and univalence are investigated. In particular, we consider the Goodman-Saff conjecture and prove that the conjecture is true in $\mathcal{L}_p\mathcal{H}(G)$.

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

Lipschitz conditions, triangular ratio metric, and quasiconformal maps

The triangular ratio metric is studied in subdomains of the complex plane and Euclidean $n$-space. Various inequalities are proven for it. The main results deal with the behavior of this metric under quasiconformal maps. We also study the smoothness of metric disks with small radii.

math.CA↗
arXiv2014

On lengths, areas and Lipschitz continuity of polyharmonic mappings

In this paper, we continue our investigation of polyharmonic mappings in the complex plane. First, we establish two Landau type theorems. We also show a three circles type theorem and an area version of the Schwarz lemma. Finally, we study Lipschitz continuity of polyharmonic mappings with respect to the distance ratio metric.

math.CV↗
arXiv2013

Coefficient estimates and the Fekete-Szeg\H o problem for certain classes of polyharmonic mappings

We give coefficient estimates for a class of close-to-convex harmonic mappings, and discuss the Fekete-Szegő problem of it. We also introduce two classes of polyharmonic mappings $\mathcal{HS}_{p}$ and $\mathcal{HC}_{p}$, consider the starlikeness and convexity of them, and obtain coefficient estimates on them. Finally, we give a necessary condition for a mapping $F$ to be in the class $\mathcal{HC}_{p}$.

math.CV↗
arXiv2013

Landau's theorem for polyharmonic mappings

In this paper, we first investigate coefficient estimates for bounded polyharmonic mappings in the unit disk $\mathbb{D}$. Then, we obtain two versions of Landau's theorem for polyharmonics mapping $F$, and for the mappings of the type $L(F)$, where $L$ is the differential operator of Abdulhadi, Abu Muhanna and Khuri. Examples and numerical estimates are given.

math.CV↗
arXiv2013

Starlikeness and convexity of polyharmonic mappings

In this paper, we first find an estimate for the range of polyharmonic mappings in the class $HC_{p}^{0}$. Then, we obtain two characterizations in terms of the convolution for polyharmonic mappings to be starlike of order $α$, and convex of order $β$, respectively. Finally, we study the radii of starlikeness and convexity for polyharmonic mappings, under certain coefficient conditions.

math.CV↗