On Valiron's Theorem

Generalization of proximate order and applications

We introduce a concept of a quasi proximate order which is a generalization of a proximate order and allows us to study efficiently analytic functions whose order and lower order of growth are different. We prove an existence theorem of a quasi proximate order, i.e. a counterpart of Valiron's theorem for a proximate order. As applications, we generalize and complement some results of M. Cartwright and C.~N.~Lin\-den on asymptotic behavior of analytic functions in the unit disc.

随机解析函数的增长率，并应用于线性动力学

Rate of growth of random analytic functions, with an application to linear dynamics

本文建立随机整函数与单位圆盘随机解析函数的 Wiman–Valiron 型不等式，改进已有结果，并用于刻画任意混沌加权后移算子频繁超循环函数在例外集外的增长率。

某些有界多解析函数与约化多解析函数的Landau型定理

Landau-type theorems for certain bounded poly-analytic and reduced poly-analytic functions

本文针对有界多解析函数与约化多解析函数建立若干Landau型定理，推广了已有结果。

Cross theorem with singularities - pluripolar vs. analytic case

We prove that in the extension theorem for separately holomorphic functions on an $N$-fold cross with singularities the case of analytic singularities follows from the case of pluripolar singularities.

An Analytic Proof of the Matrix Spectral Factorization Theorem

An analytic proof is proposed of Wiener's theorem on factorization of positive definite matrix-functions.

关于多项解析函数的Landau型定理

On Landau-Type Theorems for Poly-Analytic Functions

本文研究了多项解析函数的Landau型定理及其双Lipschitz定理，推广了双解析函数的相关结果。

Tauberian theorems for $k$--summability with respect to an analytic germ

The goal of this article is to establish tauberian theorems for the $k$--summability processes defined by germs of analytic functions in several complex variables. The proofs are based on the tauberian theorems for $k$--summability in one variable and in monomials, and a method of monomialization of germs of analytic functions.

锥内解析函数的Cartright型和Bernstein型定理

Cartwright-type and Bernstein-type theorems for functions analytic in a cone

将以前只在$\C^n$中对指数型函数已知的Cartright型和Bernstein型定理推广到锥中任意阶函数的情形。

The theorems of Schottky and Landau for analytic functions omitting the n-th roots of unity

We prove sharp Landau- and Schottky-type theorems for analytic functions which omit the $n$-th roots of unity. The proofs are based on a sharp lower bound for the Poincaré metric of the complex plane punctured at the roots of unity.

Macdonald's Theorem for Analytic Functions

A proof is reconstructed for a useful theorem on the zeros of derivatives of analytic functions due to H. M. Macdonald, which appears to be now little known. The Theorem states that, if a function $f(z)$ is analytic inside a bounded region bounded by a contour on which the modulus of $f(z)$ is constant, then the number of zeros (counted according to multiplicity) of $f(z)$ and of its derivative in the region differ by unity.

Addendum to Theorem 10.4 in "Boundaries of Analytic Varieties"

The main result of [HL] (Annals of Math. 102 (1975), 223--290) gives a characterization of the boundaries of complex subvarieties in C^n. An application of this result concerning pseudoconvexity, namely Theorem 10.4 in [HL], is amplified.

Gonchar-Stahl's $ρ^2$-theorem and associated directions in the theory of rational approximation of analytic functions

Gonchar-Stahl's $ρ^2$-theorem characterizes the rate of convergence of best uniform (Chebyshev) rational approximations (with free poles) for one basic class of analytic functions. The theorem itself, its modifications and generalizations, methods involved in the proof and other related details constitute an important subfield in the theory of rational approximations of analytic functions and complex analysis. The paper briefly outlines essentials of the subfield. Fundamental contributions by A. A. Gonchar and H. Stahl are in the center of the exposition.

A Peak Point Theorem for Uniform Algebras on Real-Analytic Varieties

It was once conjectured that if $A$ is a uniform algebra on its maximal ideal space $X$, and if each point of $X$ is a peak point for $A$, then $A = C(X)$. This peak-point conjecture was disproved by Brian Cole in 1968. Here we establish a peak-point theorem for uniform algebras generated by real-analytic functions on real-analytic varieties, generalizing previous results of the authors and John Wermer.

Banach Analytic Sets and a Non-Linear Version of the Levi Extension Theorem

We prove a certain non-linear version of the Levi extension theorem for meromorphic functions. This means that the meromorphic function in question is supposed to be extendable along a sequence of complex curves, which are arbitrary, not necessarily straight lines. Moreover, these curves are not supposed to belong to any finite dimensional analytic family. The conclusion of our theorem is that nevertheless the function in question meromorphically extends along an (infinite dimensional) analytic family of complex curves and its domain of existence is a pinched domain filled in by this analytic …

Log-p-解析函数的进展：Landau 型定理及其改进

Advancements in Log-P-Analytic Functions: Landau-Type Theorems and Their Refinements

该研究引入 log-p-解析函数概念，建立并改进多解析函数的 Landau 型定理，还针对有界情形给出专门版本与精确结论。

A sharp inclusion theorem for a certain class of analytic functions defined by the Salagean derivative

In this short note we employ the Briot-Bouquet differential subordination to determine the best possible inclusion relation within a certain family of analytic functions defined by the Salagean derivative.

Landau-type theorems for certain bounded bi-analytic functions and biharmonic mappings

In this paper, we establish three new versions of Landau-type theorems for bounded bi-analytic functions of the form $F(z)=\bar{z}G(z)+H(z)$, where $G$ and $H$ are analytic in the unit disk $|z|<1$ with $G(0)=H(0)=0$ and $H'(0)=1$. In particular, two of them are sharp while the other one either generalizes or improves the corresponding result of Abdulhadi and Hajj. As consequences, several new sharp versions of Landau-type theorems for certain subclasses of bounded biharmonic mappings are proved.

关于幂级数解析延拓的 Le Roy-Lindelöf 定理的多变量类比

Multivariate analog of the Le Roy-Lindelöf theorem about power series analytic continuation

本文研究了多重幂级数在扇形域中的可延拓性问题，通过证明 Le Roy-Lindelöf 定理的多变量类比，揭示了插值级数系数的全纯函数增长与解析延拓扇形域之间的联系。

Extension of analytic covers by extension of their ramification divisor

This paper deals with extension of analytic covers. We prove topological extension theorems for analytic covers. The main result is an extension theorem which only uses the extension of the ramification divisor. We give also a Thullen-type and a Hartogs-type extensions theorems.

关于自由解析sheaves在最大和齐性空间上的相对Riemann-Hilbert和Newlander-Nirenberg定理

Relative Riemann-Hilbert and Newlander-Nirenberg Theorems for torsion-free analytic sheaves on maximal and homogeneous spaces

本文基于自由解析sheaves在最大和齐性空间上，证明了相对Riemann-Hilbert定理在考虑torsion时仍成立，并推导出Newlander-Nirenberg型定理，揭示了tame flat连接与局部系统及基本群表示的对应关系。

多解析函数与对数-α解析函数的Landau型定理

Landau-type Theorems for Polyanalytic and Log-$α$-analytic functions

该研究针对多解析函数与log-α解析函数的单叶性问题，通过新思路证明改进引理并给出单位圆盘上有界多解析函数的系数估计，提出三种Landau型定理并确定单叶域与单叶圆盘半径，同时推导出log-α解析函数的相应Landau型定理。

Uniform estimates in the Poincare-Aronszajn Theorem on the separation of singularities of analytic functions

We study the possibility of splitting any bounded analytic function with singularities in a closed set E union F as a sum of two bounded analytic functions with singularities in E and F respectively. We obtain some results under geometric restrictions on the sets E and F and we provide some examples showing the sharpness of the positive results.

On the duality theorem on an analytic variety

The duality theorem for Coleff-Herrera products on a complex manifold says that if $f = (f_1,\dots,f_p)$ defines a complete intersection, then the annihilator of the Coleff-Herrera product $μ^f$ equals (locally) the ideal generated by $f$. This does not hold unrestrictedly on an analytic variety $Z$. We give necessary, and in many cases sufficient conditions for when the duality theorem holds. These conditions are related to how the zero set of $f$ intersects certain singularity subvarieties of the sheaf $\mathcal{O}_Z$.

带非齐次解析伸缩的调和映射的Bloch型与Landau型定理

Bloch and Landau Type Theorems for Harmonic Mappings with Inhomogeneous Analytic Dilatation

该研究针对单位圆盘内满足非齐次解析伸缩方程的保向调和映射，在相应约束条件下建立了Bloch型定理，并在附加假设下得到两种版本的Landau定理，丰富了调和映射相关理论。

Analytic inversion of adjunction: L^2 extension theorems with gain

We establish new results on weighted $L^2$ extension of holomorphic top forms with values in a holomorphic line bundle, from a smooth hypersurface cut out by a holomorphic function. The weights we use are determined by certain functions that we call denominators. We give a collection of examples of these denominators related to the divisor defined by the submanifold.

Propagation of boundary CR foliations and Morera type theorems for manifolds with attached analytic discs

We prove that generic homologically nontrivial $(2n-1)$-parameter family of analytic discs attached by their boundaries to a CR manifold $Ω$ in $\mathbb C^n, n \le 2$ tests CR functions: if a smooth function on $Ω$ extends analytically inside each analytic disc then it satisfies the tangential CR equations. In particular, we answer, in real analytic category, two open questions: on characterization of analytic functions in planar domains (the strip-problem), and on characterization of boundary values of holomorphic functions in domains in $\mathbb C^n$ (a conjecture of Globevnik and Stout). We…

A uniqueness theorem for bounded analytic functions on the polydisc

For each n,N>0 we construct a set of points x_1,...,x_M in D^n with the following property: if f is a rational inner function on D^n of degree strictly less than N and g is an analytic function mapping D^n to D that satisfies g(x_i)=f(x_i) for each i=1,...,M, then g=f on D^n. In terms of the Pick problem on D^n, our result implies that for any rational inner f of degree less than N, the Pick problem with data x_1,...,x_M and f(x_1),...,f(x_M) has a unique solution.

Rigidity theorem of the Bergman kernel by analytic capacity

In [7], Dong and I proved that the domains $D \subset \mathbb{C}$ of finite volume whose on-diagonal Bergman kernels $K(\cdot, \cdot)$ satisfy $K(z_0, z_0) = Volume(D)^{-1}$ are disks minus closed polar sets. We utilized the solution of the Suita conjecture, a deep theorem of several complex variables. In this note, I present a significantly more elementary proof of this theorem that does not use several complex variables. As a corollary, a new lower bound for the on-diagonal Bergman kernel is given. Finally, I show that the only real ellipsoid in Webster normal form which satisfies $K(0, 0) =…

A Characterization of regular points by $L^2$ Extension Theorem

In this article, we present that the germ of a complex analytic set at the origin in $\mathbb{C}^n$ is regular if and only if the related $L^2$ extension theorem holds. We also obtain a necessary condition of the $L^2$ extension of bounded holomorphic sections from singular analytic sets.

A generalization of Takegoshi's relative vanishing theorem

We present a generalization of Takegoshi's relative version of the Grauert-Riemenschneider vanishing theorem. Under some natural assumptions, we extend Takegoshi's vanishing theorem to the case of Nakano semi-positive coherent analytic sheaves on singular complex spaces. We also obtain some new results about proper modifications of torsion-free coherent analytic sheaves.

Extension theorems for analytic objects associated to foliations

In this paper we will establish a structure theorem concerning the extension of analytic objects associated to germs of dimension one foliations on surfaces, through one-dimensional barriers. As an application, an extension theorem for projective transverse structures is obtained.

A probabilistic proof of the Open Mapping Theorem for analytic functions

The conformal invariance of Brownian motion is used to give a short proof of the Open Mapping Theorem for analytic functions.

The Lefschetz theorem for CR manifolds and the nonexistence of real analytic Levi flat submanifolds

We prove the Lefchetz theorem for CR submanifolds in Hermitian symmetric spaces. As an application we prove the nonexistence of real analytic Levi flat submanifolds in such manifolds.

Bergman kernel asymptotics and a pure analytic proof of Kodaira embedding theorem

In this paper, we survey some recent results about the asymptotic expansion of Bergman kernel and we give a Bergman kernel proof of Kodaira embedding theorem.

关于解析空间消失定理的注记

On vanishing theorems for analytic spaces

本文报告了复解析空间间射影态射的新消失定理，推广了Kollár的挠自由性与消失定理至解析简单正规交叉对，并在复解析极小模型研究中起关键作用。

实解析集的 Cartan 定理 B 的逆定理

A converse of Cartan's Theorem B for Real Analytic Set

本文证明了由 Fernando 和 Ghiloni 提出的关于实解析集的 Cartan 定理 B 的逆定理。

Local ABC theorems for analytic functions

The classical $abc$ theorem for polynomials (often called Mason's theorem) deals with nontrivial polynomial solutions to the equation $a+b=c$. It provides a lower bound for the number of distinct zeros of the polynomial $abc$ in terms of $°{a}$, $°{b}$, and $°{c}$. We prove some "local" $abc$-type theorems for general analytic functions living on a reasonable bounded domain $Ω\subset\mathbb C$, rather than on the whole of $\mathbb C$. The estimates obtained are sharp, for any $Ω$, and they imply (a generalization of) the original "global" $abc$ theorem by a limiting argument.

新曼泰贝定理、里曼-勒贝格引理与抽象解析数论

Newman's Tauberian theorem, the Riemann-Lebesgue Lemma, and abstract analytic number theory

本文改进了新曼泰贝定理，并应用其研究抽象解析半群，证明了素数定理及梅特恩函数的估计。

The Bernstein-Walsh-Siciak Theorem for analytic hypersurfaces

As a first step towards a general set-theoretic counterpart of the remarkable Bernstein-Walsh-Siciak Theorem concerning the rapidity of polynomial approximation of a holomorphic function on polynomially convex compact sets in $\mathbb{C}^n$, we prove a version of this theorem for analytic hypersurfaces.

复解析空间之间射影态射的锥与收缩定理

Cone and contraction theorem for projective morphisms between complex analytic spaces

本文在复解析空间中建立正规对的锥与收缩定理，作为复解析对数典范对极小模型纲领的起点，并重点处理奇点比川又对数终端更差的情形。

Analytic Bertini theorem

We prove an analytic Bertini theorem, generalizing a previous result of Fujino and Matsumura.

复解析空间之间射影态射的消失定理

Vanishing theorems for projective morphisms between complex analytic spaces

本文讨论复解析空间射影态射的消失定理，从最小模型理论观点建立了Kollár纲领的终极推广，对最小模型理论起关键作用。

解析贝蒂尼定理II——局部情形

Analytic Bertini theorem II --- The local case

研究证明局部解析贝蒂尼定理，证实布克森猜想

Representability theorem in derived analytic geometry

We prove the representability theorem in derived analytic geometry. The theorem asserts that an analytic moduli functor is a derived analytic stack if and only if it is compatible with Postnikov towers, has a global analytic cotangent complex, and its truncation is an analytic stack. Our result applies to both derived complex analytic geometry and derived non-archimedean analytic geometry (rigid analytic geometry). The representability theorem is of both philosophical and practical importance in derived geometry. The conditions of representability are natural expectations for a moduli functor.…

The Theorem of Jentzsch--Szegő on an analytic curve. Application to the irreducibility of truncations of power series

The theorem of Jentzsch--Szegő describes the limit measure of a sequence of discrete measures associated to the zeroes of a sequence of polynomials in one variable. Following the presentation of this result by Andrievskii and Blatt in their book, we extend this theorem to compact Riemann surfaces, then to analytic curves over an ultrametric field. The particular case of the projective line over an ultrametric field gives as corollaries information about the irreducibility of the truncations of a power series in one variable.

依赖参数的多重次调和函数的解析性定理

Analyticity theorems for parameter-dependent plurisubharmonic functions

研究依赖参数的（拟）多重次调和函数的水平集解析性，证明 Lelong 数上水平集之并通常多重极，并给出复奇点指数次水平集及相对 Bergman 核对数极点的解析性条件与非解析反例。

Operator theory and the Oka extension theorem

For $δ$ an $m$-tuple of analytic functions, we define an algebra $\hidg$, contained in the bounded analytic functions on the analytic polyhedron $ {|δ^l(z)| < 1, \ 1 \leq l \leq m}$, and prove a representation formula for it. We give conditions whereby every function that is analytic on a neighborhood of $ {|δ^l(z)| \leq 1, \ 1 \leq l \leq m}$ is actually in $\hidg$. We use this to give a proof of the Oka extension theorem with bounds. We define an $\hidg$ functional calculus for operators.

Chow's theorem for real analytic Levi-flat hypersurfaces

In this article we provide a version of Chow's theorem for real analytic Levi-flat hypersurfaces in the complex projective space $\mathbb{P}^{n}$, $n \geq 2$. More specifically, we prove that a real analytic Levi-flat hypersurface $M \subset \mathbb{P}^{n}$, with singular set of real dimension at most $2n-4$ and whose Levi leaves are contained in algebraic hypersurfaces, is tangent to the levels of a rational function in $\mathbb{P}^{n}$. As a consequence, $M$ is a semialgebraic set. We also prove that a Levi foliation on $\mathbb{P}^{n}$ - a singular real analytic foliation whose leaves are i…

Rectilinearization of sub analytic sets as a consequence of local monomialization

We give a new proof of the rectilinearization theorem of Hironaka. We deduce rectilinearization as a consequence of our theorem on local monomialization for complex and real analytic morphisms.

An "Analytic" Version of Menshov's Representation Theorem

Every measurable function f on the circle can be represented as a sum of harmonics with positive spectrum, converging in measure. For convergence almost everywhere this is not true. We discuss several other subsets of Z for which one might get a Menshov type representation converging almost everywhere or in measure.

解析 Lipschitz 函数的近似 Taylor 定理

Approximate Taylor theorem for analytic Lipschitz functions

本文研究复平面有界开集上解析小 Lipschitz 函数空间，证明有界点导子的存在性蕴含相应点处的近似 Taylor 定理，从而推广已有有界点导子结果。

实分析和双Lipschitz版本的Nobile定理

Real and bi-Lipschitz versions of the Theorem of Nobile

本文证明了Nobile定理的实分析和双Lipschitz版本，展示了实解析集和复解析集的光滑性与Nash变换的微分同胚性质之间的等价关系。

A General Non-Vanishing Theorem and an Analytic Proof of the Finite Generation of the Canonical Ring

On August 5, 2005 in the American Mathematical Society Summer Institute on Algebraic Geometry in Seattle and later in several conferences I gave lectures on my analytic proof of the finite generation of the canonical ring for the case of general type. After my lectures many people asked me for a copy of the slides which I used for my lectures. Since my slides were quite sketchy because of the time limitation for the lectures, I promised to post later on a preprint server my detailed notes from which my slides were extracted. Here are my detailed notes giving the techniques and the proof.

The Analytic Expression for Riemann's Prime Counting Function via the Residue Theorem

In his paper "On the Number of Primes Less Than a Given Magnitude", Bernhard Riemann introduced a prime counting function F(x) which counts the number of primes under x. Riemann obtained an analytic expression for F(x) by evaluating an inverse Laplace Transform. His method involved advanced techniques of analysis. However, this transform can be evaluated using the Residue Theorem when an appropriate branch of log(zeta) is defined. In this paper, a method for constructing a holomorphic branch of log(zeta) extending to the left half-plane is described along with it's geometry surrounding the log…