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Jian-Feng Zhu

Publications and source records attributed to Jian-Feng Zhu.

16 recordsLinked to original sources
arXiv2026

Contraction properties for holomorphic functions via isoperimetric stability on the Bergman ball

We prove a local contraction property for holomorphic functions that are nearly constant, relating weighted Bergman spaces $A^p_α(\B_n)$ and $A^q_β(\B_n)$. Our approach converts geometric information on weighted superlevel sets into analytic deficit inequalities and rests crucially on a quantitative stability result (of Fuglede type) for the isoperimetric inequality in the Bergman ball. As an application, along the contractive line $q/p=β/α$, we obtain a deficit contraction near the extremizer $f\equiv 1$: if $f=1+ϕ$ with $ϕ$ small and its weighted level sets are nearly spherical (after recentering), then the $A^q_β$-deficit is controlled by the $A^p_α$-deficit, and the same deficit quantitatively controls the deviation of the level sets from spheres.

math.CV↗
arXiv2026

A Lewy theorem for harmonic quasiregular mappings in three-space

Lewy's classical theorem asserts that a one-to-one planar harmonic mapping has nonvanishing Jacobian. We prove a three-dimensional bounded-distortion analogue: if \[ f:Ω\subset \mathbb R^3\to \mathbb R^3 \] is nonconstant, sense-preserving, quasiregular, and harmonic componentwise, then \(J_f>0\) throughout \(Ω\). Thus harmonic quasiconformal mappings between domains in three-space are local harmonic diffeomorphisms. The new point is the Lewy-type differential conclusion \(J_f\neq0\), not merely topological local invertibility, which is already known for sufficiently smooth quasiregular mappings. The proof is by blow-up. A hypothetical zero of \(J_f\) produces a nonconstant homogeneous harmonic polynomial quasiregular mapping \(P:\mathbb R^3\to\mathbb R^3\) of degree \(m>1\). We exclude such homogeneous blow-ups by a second-order trace identity for \(J_P|_{S^2}\): after normalizing the first jet at a positive minimum, the identity gives a negative spherical trace, contradicting the maximum principle. We also derive an affine Liouville theorem for entire harmonic quasiregular mappings in \(\mathbb R^3\).

math.CV↗
arXiv2026

Isoperimetric-type inequalities for pluriharmonic functions on the polydisc

We prove isoperimetric-type inequalities for complex-valued pluriharmonic functions in the unit polydisc $\mathbb{U}^n\subset\mathbb{C}^n$. Denote by $h^p(\mathbb{U}^n)$ and $b^p_{\mathbf q}(\mathbb{U}^n)$, respectively, the pluriharmonic Hardy space and the pluriharmonic weighted Bergman space in $\mathbb{U}^n$, where $\mathbf q=(q_1,\ldots,q_n)>-\mathbf1$. For $m\in\mathbb{N}$, $m\geq2$, write $\mathbf{m-2}=(m-2,\ldots,m-2)$ and let \[ dμ_{\mathbf{m-2}}(z) =\frac{(m-1)^n}{π^n} \prod_{k=1}^n \left[(1-|z_k|^2)^{m-2}\,dx_kdy_k\right], \qquad z_k=x_k+iy_k. \] We prove that if $1 2$, we refine this in the form \[ \|f\|_{b^{mp}_{\mathbf{m-2}}(\mathbb{U}^n)} \leq \left[ \sqrt2\cos\left(\fracπ{4m}\right) \right]^{2n/p} \|f\|_{h^p(\mathbb{U}^n)}. \] Consequently, the obtained constant in the diagonal inclusion tends to $1$ as $p\to\infty$, for fixed $m$ and $n$. When $m=2$ and $n=1$, the latter estimate coincides with best-known planar estimate. Explicit lower bounds at $p=2$, together with the dimension-free upper estimate, show that the optimal diagonal constants converge to $\sqrt2$ as $m\to\infty$, uniformly in the dimension.

math.CV↗
arXiv2026

Hilbert matrix norms on weighted Bergman spaces: even exponents and a counterexample to the beta formula

Let $A_α^p$ be the weighted Bergman space on the unit disk, where $α>-1$. For $f(z)=\sum_{k=0}^{\infty}a_k z^k\in A_α^p$, consider the Hilbert matrix operator $\mathcal{H}f(z)=\sum_{n=0}^{\infty}\left(\sum_{k=0}^{\infty}\frac{a_k}{n+k+1}\right)z^n =\int_0^1\frac{f(t)}{1-tz}\,dt$. For even exponents $p=2m$, we prove that $\|\mathcal{H}\|_{A_α^{2m}\to A_α^{2m}}=B(a,1-a)$, where $a=(α+2)/(2m)$, whenever $0 B(a_0,1-a_0)$. The counterexample is based on the fixed function $f_0(z)=(1-z^2)^{-4/5}=\sum_{k=0}^{\infty}\frac{(4/5)_k}{k!}z^{2k}$. A rigorous interval estimate at $p=1100000$, together with monotonicity in $p$, yields the result on the entire half-line. In particular, the formula fails for every even exponent $p=2m$ with $m\geq 550000$.

math.CV↗
arXiv2026

The Nitsche--Hopf conjecture for minimal graphs

We prove the Nitsche--Hopf conjecture for non-parametric minimal graphs over disks. If \(S\) is a minimal graph over a disk of radius \(R\), and if \(ξ\) is the point above the center, then \[ W(ξ)^2 |K(ξ)|<\frac{π^2}{2R^2}. \] Here \(K\) is the Gaussian curvature and \[ W=\sqrt{1+|\nabla u|^2}=\frac1{n_3} \] is the reciprocal of the vertical component of the upward unit normal. The constant is sharp, as shown by the horizontal tangent-plane extremal sequence of Finn and Osserman. The main difficulty is that the bicentric-quadrilateral comparison theorem for Gaussian curvature controls \(|K|\), but it does not by itself control the normalized quantity \(W^2|K|\): the slope factor \(W\) can be arbitrarily large. We show that the missing information is recovered inside the Scherk-type comparison family from the zero equation for the horizontal harmonic projection. More precisely, in the fixed-arc normalization the point corresponding to the center of the physical disk is a distinguished zero \(z_\circ\) of the harmonic projection. The equation \(f(z_\circ)=0\), written in harmonic-measure coordinates, reduces the sharp Hopf estimate to a scalar derivative inequality at the admissible zero of a monotone function \(G_{A,B}\). We prove this scalar inequality on the full admissible parameter domain by a barrier argument and two explicit Bernstein-polynomial positivity certificates. Combined with the bicentric-quadrilateral comparison theorem of the first author and Melentijević, the Scherk-family estimate gives the sharp normalized Hopf estimate for arbitrary minimal graphs over disks. As a byproduct, we obtain the two-sided bound \[ \frac{π^2}{4}\leq W^2|K|\leq \frac{π^2}{2} \] throughout the normalized Scherk-type comparison family, evaluated at the distinguished point corresponding to the center.

math.CV↗
arXiv2023

$L^p$ norm of truncated Riesz transform and an improved dimension-free $L^p$ estimate for maximal Riesz transform

In this paper, we prove that the $L^p(\mathbb{R}^d)$ norm of the maximal truncated Riesz transform in terms of the $L^p(\mathbb{R}^d)$ norm of Riesz transform is dimension-free for any $2\leq p<\infty$, using integration by parts formula for radial Fourier multipliers. Moreover, we show that $$\|R_j^*f\|_{L^p}\leq \left({2+\frac{1}{\sqrt{2}}}\right)^{\frac{2}{p}}\|R_jf\|_{L^p},\ \ \mbox{for}\ \ p\geq2,\ \ d\geq2.$$ As by products of our calculations, we infer the $L^p$ norm contractivity of the truncated Riesz transforms $R^t_j$ in terms of $R_j$, and their accurate $L^p$ norms. More precisely, we prove: $$\|R^t_jf\|_{L^p}\leq\|R_jf\|_{L^p}$$ and $$\|R^t_j\|_{L^p}=\|R_j\|_{L^p},$$ for all $1 0.$

math.CA↗
arXiv2022

Composition operators on Bloch and Hardy type spaces

The main purpose of this paper is to discuss Hardy type spaces, Bloch type spaces and the composition operators of complex-valued harmonic functions. We first establish a sharp estimate of the Lipschitz continuity of complex-valued harmonic functions in Bloch type spaces with respect to the pseudo-hyperbolic metric, which gives an answer to an open problem. Then some classes of composition operators on Bloch and Hardy type spaces will be investigated. The obtained results improve and extend some corresponding known results.

math.CV↗
arXiv2021

Lipschitz property of harmonic mappings with respect to the pseudo-hyperbolic metric

In this paper, we show that harmonic Bloch mappings are Lipschitz continuous with respect to the pseudo-hyperbolic metric. This result improves the corresponding result of Theorem 1 of [P. Ghatage, J. Yan, and D. Zheng, Composition operators with closed range on the Bloch space, Proc. Amer. Math. Soc. 129 (2000), 2039-2044]. Furthermore, we prove the similar property for harmonic quasiregular Bloch-type mappings.

math.CV↗
arXiv2020

$L^p\to L^q$ norm estimates of Cauchy transforms on the Dirichlet problem and their applications

Denote by $C^α(\mathbb{D})$ the space of the functions $f$ on t}he unit disk $\mathbb{D}$ which are Hölder continuous with the exponent $α$, and denote by $C^{1, α}(\mathbb{D})$ the space which consists of differentiable functions $f$ such that their derivatives are in the space $C^α(\mathbb{D})$. Let $\mathcal{C}$ be the Cauchy transform of Dirichlet problem. In this paper, we obtain the norm estimates of $\|\mathcal{C}\|_{L^p\to L^q}$, where $3/2<p<2$ and $q=p/(p-1)$. As an application, we show that if $3/2<p<2$, then $u\in C^μ(\mathbb{D})$, where $μ=2/p-1$. We also show that if $2<p<\infty$, then $u\in C^{1, ν}(\mathbb{D})$, where $ν=1-2/p$. Finally, for the case $p=\infty$, we show that $u$ is not necessarily in $C^{1, 1}(\mathbb{D})$, but its gradient, i.e., $|\nabla u|$ is Lipschitz continuous with respect to the pseudo-hyperbolic metric. This paper is inspired by Chapter 4 of [Astala, Iwaniec, Martin: Elliptic partial differential equations and quasiconformal mappings in the plane, Princeton Mathematical Series, Vol. 48, Princeton University Press, Princeton, NJ, 2009, p. xviii+677] and [Kalaj, Cauchy transform and Poisson's equation, Adv. Math. \textbf{231} (2012), 213--242]

math.FA↗
arXiv2020

$L^p$-theory for Cauchy-transform on the unit disk

Let $\mathbb{D}$ be the unit disk and $φ\in L^p(\mathbb{D}, \mathrm{d}A)$, where $1\leq p\leq\infty$. For $z\in\mathbb{D}$, the Cauchy-transform on $\mathbb{D}$, denote by $\mathcal{P}$, is defined as follows: $$\mathcal{P}[φ](z)=-\int_{\mathbb{D}}\left(\frac{φ(w)}{w-z}+\frac{z\overline{φ(w)}}{1-\bar{w}z}\right)\mathrm{d}A(w).$$ The Beurling transform on $\mathbb{D}$, denote by $\mathcal{H}$, is now defined as the $z$-derivative of $\mathcal{P}$. In this paper, by using Hardy's type inequalities and Bessel functions, we show that $\|\mathcal{P}\|_{L^2\to L^2}=α\approx1.086$, where $α$ is a solution to the equation: $2J_0(2/α)-αJ_1(2/α)=0$, and $J_0$, $J_1$ are Bessel functions. Moreover, for $p>2$, by using Taylor expansion, Parseval's formula and hypergeometric functions, we also prove that $\|\mathcal{P}\|_{L^p\to L^{\infty}}=2(Γ(2-q)/Γ^2(2-\frac{q}{2}))^{1/q}$, where $q=p/(p-1)$ is the conjugate exponent of $p$, and $Γ$ is the Gamma function. Finally, applying the same techniques developed in this paper, we show that the Beurling transform $\mathcal{H}$ acts as an isometry of $L^2(\mathbb{D}, \mathrm{d}A)$.

math.CV↗
arXiv2020

Norm estimates of the Cauchy transform and related operators

Suppose $f\in L^p(\mathbb{D})$, where $p\geq1$ and $\mathbb{D}$ is the unit disk. Let $\mathfrak{J}_0$ be the integral operator defined as follows: $\mathfrak{J}_0[f](z)=\int_{\mathbb{D}}\frac{z}{1-\bar{w}z}f(w)\mathrm{d}A(w)$, where $z$, $w\in\mathbb{D}$ and $\mathrm{d}A(w)=\frac{1}π\mathrm{d}x\mathrm{d}y$ is the normalized area measure on $\mathbb{D}$. Suppose $\mathfrak{J}_0^*$ is the adjoint operator of $\mathfrak{J}_0$. Then $\mathfrak{J}^*_0=\mathfrak{B}\mathfrak{C}$, where $\mathfrak{B}$ and $\mathfrak{C}$ are the operators induced by the Bergman projection and Cauchy transform, respectively. In this paper, we obtain the $L^1$, $L^2$ and $L^{\infty}$ norm of the operator $\mathfrak{J}_0^*$. Moreover, we obtain the $L^p(\mathbb{D})\rightarrow L^\infty(\mathbb{D})$ norm of the operators $\mathfrak{C}$ and $\mathfrak{J}_0^*$, provided that $p>2$. This study is a continuation of the investigations carried out in [4] and [11].

math.CV↗
arXiv2020

Norm estimates of the partial derivatives for harmonic mappings and harmonic quasiregular mappings

Suppose $p\geq1$, $w=P[F]$ is a harmonic mapping of the unit disk $\mathbb{D}$ satisfying $F$ is absolutely continuous and $\dot{F}\in L^p(0, 2π)$, where $\dot{F}(e^{it})=\frac{\mathrm{d}}{\mathrm{d}t}F(e^{it})$. In this paper, we obtain Bergman norm estimates of the partial derivatives for $w$, i.e., $\|w_z\|_{L^p}$ and $\|\overline{w_{\bar{z}}}\|_{L^p}$, where $1\leq p<2$. Furthermore, if $w$ is a harmonic quasiregular mapping of $\mathbb{D}$, then we show that $w_z$ and $\overline{w_{\bar{z}}}$ are in the Hardy space $H^p$, where $1\leq p\leq\infty$. The corresponding Hardy norm estimates, $\|w_z\|_{p}$ and $\|\overline{w_{\bar{z}}}\|_{p}$, are also obtained.

math.CV↗
arXiv2020

Boundary Schwarz lemma for harmonic mappings having zero of order $p$

Suppose $w$ is a sense-preserving harmonic mapping of the unit disk $\mathbb{D}$ such that $w(\mathbb{D})\subseteq\mathbb{D}$ and $w$ has a zero of order $p\geq1$ at $z=0$. In this paper, we first improve the Schwarz lemma for $w$, and then, we establish its boundary Schwarz lemma. Moreover, by using the automorphism of $\mathbb{D}$, we further generalize this result.

math.CV↗
arXiv2019

Neohookean deformations of annuli in the higher dimensional Euclidean space

Let $n\ge 2$ be an integer and assume that $\mathbb{A}=\{x\in\mathbf{R}^n:1<|x|<R\}$ and $\A_\ast = \{y \in \mathbf{R}^n: 1 < |y| < R_\ast\}$ be two annuli in Euclidean space $\mathbf{R}^n$. Assume that $\mathcal{F}(\A, \A_\ast)$ (resp. $\mathcal{R}(\A, \A_\ast)$) be the class of all orientation preserving (resp. radial) homeomorphisms $h : \A \mapsto \A_\ast$ in the Sobolev space $\mathcal{W} ^{1,n}(\A, \A_\ast)$ which keep the boundary circles in the same order. In this paper, we extended the corresponding results of Iwaniec and Onninen which was published in {\it Math. Ann.} Vol. 348, 2010.

math.CV↗