关于Nevanlinna测度的结构

The Structure of Inner Multipliers on Spaces with Complete Nevanlinna Pick Kernels

We establish some multivariate generalizations of the Beurling-Lax-Halmos theorem.

Nevanlinna counting functions and pull-back measures on maximal ideal space of $H^\infty$

In this paper we give precise characterizations of the relation between the Nevanlinna counting function and pull-back measure of an analytic self-map of the unit disk near the boundary. We show that it is quite worth considering these two concepts on the maximal ideal space of the bounded analytic functions.

容许Hardy型完备Nevanlinna-Pick空间的复结构

Complex structure that admits complete Nevanlinna-Pick spaces of Hardy type

本文刻画了不可约完备Nevanlinna-Pick空间乘子范数与上确界范数相等的集合，并证明Hardy型该空间的既约复空间必双全纯同构于去掉零解析容量集的单位圆盘，从而确立Hardy空间的唯一性。

Constructing measures with identical moments

The Nevanlinna parametrization establishes a bijection between the class of all measures having a prescribed set of moments and the class of Pick functions. The fact that all measures constructed through the Nevanlinna parametrization have identical moments follows from the theory of orthogonal polynomials and continued fractions. In this paper we explore the opposite direction: we construct a set of measures and we show that they all have identical moments, and then we establish a Nevanlinna-type parametrization for this set of measures. Our construction does not require the theory of orthogo…

Ideals in some Rings of Nevanlinna-Smirnov Type

Let $N^p$ $(1<p<\infty)$ denote the algebra of holomorphic functions in the open unit disk, introduced by I.~I.~Privalov with the notation $A_q$ in [8]. Since $N^p$ becomes a ring of Nevanlinna--Smirnov type in the sense of Mortini [7], the results from [7] can be applied to the ideal structure of the ring $N^p$. In particular, we observe that $N^p$ has the Corona Property. Finally, we prove the $N^p$-analogue of the Theorem 6 in [7], which gives sufficient conditions for an ideal in $N^p$, generated by a finite number of inner functions, to be equal to the whole algebra $N^p$.

Nevanlinna理论中的组合结构

Combinatorial Structure in Nevanlinna Theory

本文推广了Weyl奇特关系，通过标准杨表揭示了Weyl--Ahlfors理论背后的组合结构，并从比较阶函数的视角给出了第二主定理的另一种证明。

Arakelov-Nevanlinna inequalities for variations of Hodge structures and applications

We prove a Second Main Theorem type inequality for any log-smooth projective pair $(X,D)$ such that $X\setminus D$ supports a complex polarized variation of Hodge structures. This can be viewed as a Nevanlinna theoretic analogue of the Arakelov inequalities for variations of Hodge structures due to Deligne, Peters and Jost-Zuo. As an application, we obtain in this context a criterion of hyperbolicity that we use to derive a vast generalization of a well-known hyperbolicity result of Nadel. The first ingredient of our proof is a Second Main Theorem type inequality for any log-smooth projective …

动量问题与谱函数

Moment Problems and Spectral Functions

本文研究了Nevanlinna-Pick插值和动量问题，证明了因果数据空间的凸性，并回顾了相关结果。

Holomorphic automorphisms of Danielewski surfaces II -- structure of the overshear group

We apply Nevanlinna theory for algebraic varieties to Danielewski surfaces and investigate their group of holomorphic automorphisms. Our main result states that the overshear group which is known to be dense in the identity component of the holomorphic automorphism group, is a free amalgamated product.

Geometric properties of measures related to holomorphic functions having positive imaginary or real part

In this paper, we study the properties of a certain class of Borel measures on $\mathbb{R}^n$ that arise in the integral representation of Herglotz-Nevanlinna functions. In particular, we find that restrictions to certain hyperplanes are of a surprisingly simple form and show that the supports of such measures can not lie within particular geometric regions, e.g. strips with positive slope. Corresponding results are derived for measures on the unit poly-torus with vanishing mixed Fourier coefficients. These measures are closely related to functions mapping the unit polydisk analytically into t…

Characterizations of the Lebesgue measure and product measures related to holomorphic functions having non-negative imaginary or real part

In this paper, we study a class of Borel measures on $\mathbb{R}^n$ that arises as the class of representing measures of Herglotz-Nevanlinna functions. In particular, we study product measures within this class where products with the Lebesgue measures play a special role. Hence, we give several characterizations of the $n$-dimensional Lebesgue measure among all such measures and characterize all product measures that appear in this class of measures. Furthermore, analogous results for the class of positive Borel measures on the unit poly-torus with vanishing mixed Fourier coefficients are als…

虚域上全纯函数的边界测度

Boundary measures of holomorphic functions on the imaginary domain

本研究围绕Pick函数的Herglotz-Nevanlinna积分表示，提出虚域上全纯函数的边界测度概念，并阐释了该测度的基本性质。

Abstract Interpolation Problem and Some Applications. II: Coefficient Matrices

The main content of this paper is Lectures 5 and 6 that continue lecture notes [20]. Content of Lectures 1-4 of [20] is reviewed for the reader's convenience in sections 1-4, respectively. It is shown in Lecture 5 how residual parts of the minimal unitary extensions, that correspond to solutions of the problem, yield some boundary properties of the coefficient matrix-function. These results generalize the classical Nevanlinna - Adamjan - Arov~- Krein theorem. Lecture 6 discusses how further properties of the coefficient matrices follow from denseness of certain sets in the associated function …

Integrals of differences of subharmonic functions. III. Hausdorff measure and content, integration over Lipschitz curves and surfaces

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…

Beurling-Carleson集、内函数与一个半线性方程

Beurling-Carleson sets, inner functions and a semi-linear equation

本文综述了Beurling-Carleson集的定义与性质，利用corona型分解揭示了奇异测度的多项性质与该集可数并集的等价关系，并证明了其与半线性方程近最大解的双射对应。

Teichmüller Structures and Dual Geometric Gibbs Type Measure Theory for Continuous Potentials

The Gibbs measure theory for smooth potentials is an old and beautiful subject and has many important applications in modern dynamical systems. For continuous potentials, it is impossible to have such a theory in general. However, we develop a dual geometric Gibbs type measure theory for certain continuous potentials in this paper following some ideas and techniques from Teichmüller theory for Riemann surfaces. Furthermore, we prove that the space of those continuous potentials has a Teichmüller structure. Moreover, this Teichmüller structure is a complete structure and is the completion of th…