Approximation of subharmonic functions in the unit disk
Igor Chyzhykov
设u是D={ |z| <1}中的下调和函数.在D中存在一个绝对常数C和一个解析函数f，使得\int_D| u（z）-log| f（z）||其中m表示平面勒贝格测度。我们还考虑了一致逼近。
Let u be a subharmonic function in D={|z|<1}. There exist an absolute constant C and an analytic function f in D such that \int_D |u(z)-log|f(z)|| dm(z)<C where m denotes the plane Lebesgue measure. We also consider uniform approximation.
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Answering a question of A.Zygmund in \cite{MR} G.MacLane and L.Rubel described boundedness of $L_2$-norm w.r.t. the argument of $\log |B|$, where $B$ is a Blaschke product. We generalize their results in several directions. We describe growth of $p$th means, $p\in(1, \infty)$, of subharmonic functions bounded from above in the unit disc. Necessary and sufficient conditions are formulated in terms of the complete measure (of a subharmonic function) in the sense of A.Grishin. We also prove sharp estimates of the growth of $p$th means of analytic and subharmonic functions of finite order in the u…
General estimates from below of holomorphic and subharmonic functions play one of the key roles in the theory of growth of holomorphic and subharmonic functions and in general in the theory of potential. At the same time, the most diverse technical apparatus was often used for this. In one of our recent papers, we showed in fairly general cases that in fact such a wide set of tools is not required. The classic Harnack distance is almost always enough. In this article, we will illustrate how this works in the cases of the unit disc in the complex plane.
Contraction property of certain classes of log-$\mathcal{M}-$subharmonic functions in the unit ball in $\mathbb{R}^n$
本文证明单位球中绝对值对数超调和的光滑函数类的压缩性质，推广Kulikov结果至高维，并应用于复平面调和映射获得新结果。
Accuracy of approximation of subharmonic functions by logarithms of moduli of analytic ones in Chebyshev metrics
已知有穷级$ρ$的下调和函数可以通过在直到$C\log|z|$的例外集之外的点$z$处的整函数的模的对数来近似。在这篇文章中，我们证明，如果这样的近似是更精确的，i。e.常数$C$减小，然后，从$C=ρ/4$开始，异常集的大小显著增大。对无穷级的下调和函数和单位圆内的次调和函数也证明了类似的结果。这些定理改进和补充了Yulmukhametov的一个结果。
Let $u\not\equiv -\infty$ be a subharmonic function on the complex plane $\mathbb C$. In 2016, we obtained a result on the existence of an entire function $f\neq 0$ satisfying the estimate $\log|f|\leq {\sf B}_u$ on $\mathbb C$, where functions ${\sf B}_u$ are integral averages of $u$ for rapidly shrinking disks as it approaches infinity. We give another equivalent version of this result with $\log |f|\leq u$ outside a very small exceptional set if $u$ is of finite order.
Let $M$ be a subharmonic function on a domain $D$ in the complex plane $\mathbb C$ with the Riesz measure $ν_M$. Let $f$ be a non-zero holomorphic function on $D$ such that $\log |f|\leq M$ on $D$ and the function $f$ vanish on a sequence ${\tt Z}=\{{\tt z}_k\}_{k=1,2, \dots}\subset D$ {\large(}$u\not\equiv -\infty$ be a subharmonic function on $D$ with the Riesz measure or the mass distribution $ν_u$, and $u\leq M$ on $D$ resp.{\large)}. Then restrictions on the growth of the Riesz measure $ν_M$ of the function $M$ near the boundary of the domain $D$ entail certain restrictions on the distrib…
Given two compact sets, $E$ and $F$, on the unit circle, we study the class of subharmonic functions on the unit disk which can grow at the direction of $E$ and $F$ (sets of singularities) at different rate. The main result concerns the Blaschke-type condition for the Riesz measure of such functions. The optimal character of such condition is demonstrated.
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Minimal m-subharmonic functions with nonmaximal Hessian measures
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Stability of Wehrl-type Functionals and Concentration Estimates on Bergman Spaces of Log-Subharmonic Functions on the Unit Sphere
本文研究了Wehrl型函数在Bergman空间中的稳定性，通过等周不等式证明了对数亚调和函数的极值问题和集中估计的稳定性。
Additional integral inequalities are obtained for integrals of the differences of subharmonic functions by Borel measures on balls in a multidimensional Euclidean space. These integrals are still estimated from above through the Nevanlinna characteristic and various characteristics of the Borel measure and its support. All the results are also new for logarithms of modules of meromorphic functions on discs in the complex plane. Integration by Borel measures with a support on fractal sets is allowed, and estimates in these cases are given by the Hausdorff measure and the Hausdorff content for t…
Let $M$\/ be a subharmonic function with Riesz measure $μ_M$ on the unit disk $\mathbb D$ in the complex plane $\mathbb C$. Let $f$ be a nonzero holomorphic function on $\mathbb D$ such that $f$ vanishes on ${\sf Z}\subset \mathbb D$, and satisfies $|f| \leq \exp M$ on $\mathbb D$. Then restrictions on the growth of $μ_M$ near the boundary of $D$ imply certain restrictions on the distribution of $\sf Z$. We give a quantitative study of this phenomenon in terms of special non-radial test functions constructed using $ρ$-trigonometrically convex functions.
Complete harmonic metrics and subharmonic functions on the unit disc
本文将 Li-Mochizuki 关于循环希格斯丛完全调和度量唯一性的定理推广到任意次调和权函数，并证明单位圆盘上完全度量总存在且可由光滑权函数逼近的度量族单调收敛得到。
The Hessian equation in quaternionic space
本文在四元数空间$\mathbb{H}^{n}$中引入$m$-次调和函数并定义四元数Hessian算子，解决了单位球上四元数Hessian方程的齐次Dirichlet问题。
Let $AL^{2}_ϕ(\mathbb{D})$ denote the closed subspace of $L^{2}(\mathbb{D},e^{-2ϕ}dλ)$ consisting of holomorphic functions in the unit disc ${\mathbb D}$. For certain class of subharmonic funcions $ϕ: {\mathbb D}\rightarrow{\mathbb D}$, we prove upper pointwise estimate for the Bergman kernel for $AL^{2}_ϕ(\mathbb{D})$
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本文借助Jack引理建立解析函数的微分不等式，并据此提出单位圆盘内新的单叶性判定准则。
We provide some new sharp embeddings for p-Carleson and related measures in the unit disk of the complex plane
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Quasiconformal Extensions of Harmonic Univalent Mappings of the Unit Disk
本文研究单位圆盘内调和单叶映射的拟共形延拓条件，证明调和强星形映射可延拓至全平面并给出显式延拓函数，同时探讨圆盘外调和映射的可延拓性。
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Quaternionic metamonogenic functions in the unit disk
本研究构造了单位圆盘内一组除低阶小子空间外正交的四元数亚单演函数，可在四元数Hilbert模L²中逼近所有亚单演函数，并用于表示极坐标下虚时间波动方程的含时解。
Topological Characteristics of Harmonic Quasiconformal Unit Disk Automorphisms in the Uniform Topology
本文研究赋予一致收敛拓扑的单位圆盘调和拟共形自同胚空间 $HQ(\mathbb{D})$，探讨并揭示了其可分性、紧性、道路连通性与完备性等重要拓扑性质。
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本文研究单位圆盘上Toeplitz算子从Triebel和Besov空间到BMOA型空间的尖锐作用结果，将已有s<1情形推广到s≥1，并适当修改了原有证明。
Weighted contractivity for derivatives of functions in the Bergman space on the unit disk
本文推广了Ramos和Tilli关于单位圆盘解析函数的锐利不等式，将其扩展到Bergman空间中函数导数关于两种不同测度的情形，并讨论与Fock空间类似结果的联系。
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Nonlinear resolvents in the unit disk: geometry and dynamics
本文提出统一方法研究全纯生成子非线性预解式的几何与动力学性质，通过应用偏差定理确定其螺旋相似阶数与强星形阶数，消除拟共形延拓限制，并证明全单位圆盘一致收敛性及半群特征。
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\|_…
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本文证明了复平面单位圆盘上的真全纯映射在Moebius变换的后合成下由它的临界点唯一确定。
In this paper we study an algebraic and topological structure inside the following sets of special functions: Bloch functions defined on the open unit disk that are unbounded and analytic functions of bounded type defined a Banach algebra E into E, which are not Lorch-analytic.
Indefinite structure of the Bergman kernel on the open unit disk
本文研究了开单位圆上伯格曼核的不定结构，通过构造不变距离、双射映射和不定核来探讨其相互作用。
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