Leafwise holomorphic functions

It is a well-known and elementary fact that a holomorphic function on a compact complex manifold is necessarily constant. The purpose of the present article is to investigate whether, or to what extent, a similar property holds in the setting of holomorphically foliated spaces.

     

Proper holomorphic self-maps of smooth bounded Reinhardt domains in C~n

It is proved that every proper holomorphic self-map of a smooth bounded Reinhardt domain of D'Angelo finite type in Cn (n > 1) is an automorphism.     

Functions on discrete sets holomorphic in the sense of Isaacs, or monodiffric functions of the first kind

We study discrete analogues of holomorphic functions of one and two variables, especially those that were called monodiffric functions of the first kind by Rufus Isaacs. Discrete analogues of the Cauchy-Riemann operators, domains of holomorphy in one discrete variable, and the Hartogs phenomenon in two discrete variables are investigated.     

Restricted T-universal functions on multiply connected domains

Let G ⊂ C be a domain which is bounded by a finite number of pairwise disjoint Jordan curves. We prove the existence of a function which is holomorphic exactly on G and has universal translates with respect to a prescribed set E ⊂∂G and which in addition is continuous on G ^-\E. This revised version was published online in June 2006 with corrections to the Cover Date.     

Boundary jets of holomorphic maps between strongly pseudoconvex domains

In this paper we consider jets taken at a fixed boundary point of germs of holomorphic diffeomorphisms which send one strongly pseudoconvex domain into another. We completely describe possible first and second jets and conditions of extremality in terms of the Chern-Moser normal forms of the domains.     

Spaces of holomorphic functions on non-balanced domains

This paper studies the coincidence of the _τω${\tau }_{\omega }$ and _τδ${\tau }_{\delta }$ topologies on the space of holomorphic functions defined on an open subset U   of a Banach space. Dineen and Mujica proved that _τω=_τδ${\tau }_{\omega }={\tau }_{\delta }$ when U   is a balanced open subset of a separable Banach space with the bounded approximation property. Here, we study the _τω=_τδ${\tau }_{\omega }={\tau }_{\delta }$ problem for several types of non-balanced domains U.     

Rigidity of proper holomorphic mappings between certain unbounded non-hyperbolic domains

The Fock–Bargmann–Hartogs domain _Dn,m(μ)$D_{n,m}(\mu )$ (μ>0$\mu >0$) in C^n+m${\mathbf{C}}^{n+m}$ is defined by the inequality ‖w‖²<e^{−μ‖z‖2}${\Vert w\Vert }^2<e^{-\mu {\Vert z\Vert }^2}$, where (z,w)∈Cⁿ×C^m$(z,w)\in {\mathbf{C}}^n\times {\mathbf{C}}^m$, which is an unbounded non-hyperbolic domain in C^n+m${\mathbf{C}}^{n+m}$. Recently, Yamamori gave an explicit formula for the Bergman kernel of the Fock–Bargmann–Hartogs domains in terms of the polylogarithm functions and Kim–Ninh–Yamamori determined the automorphism group of the domain _Dn,m(μ)$D_{n,m}(\mu )$. In this article, we obtain rigidity results on proper holomorphic mappings between two equidimensional Fock–Bargmann–Hartogs domains. Our rigidity result implies that any proper holomorphic self-mapping on the Fock–Bargmann–Hartogs domain _Dn,m(μ)$D_{n,m}(\mu )$ with m≥2$m\ge 2$ must be an automorphism.     

Numerical peak holomorphic functions on Banach spaces

We introduce the notion of numerical (strong) peak function and investigate the denseness of the norm and numerical peak functions on complex Banach spaces. Let Ab(BX:X) be the Banach space of all bounded continuous functions f on the unit ball BX of a Banach space X and their restrictions to the open unit ball are holomorphic. In finite dimensional spaces, we show that the intersection of the set of all norm peak functions and the set of all numerical peak functions is a dense Gδ-subset of Ab(BX:X). We also prove that if X is a smooth Banach space with the Radon-Nikodým property, then the set of all numerical strong peak functions is dense in Ab(BX:X). In particular, when X=Lp(μ)(1<p<∞) or X=?₁, it is shown that the intersection of the set of all norm strong peak functions and the set of all numerical strong peak functions is a dense Gδ-subset of Ab(BX:X). As an application, the existence and properties of numerical boundary of Ab(BX:X) are studied. Finally, the numerical peak function in Ab(BX:X) is characterized when X=C(K) and some negative results on the denseness of numerical (strong) peak holomorphic functions are given.     

Random iterations of holomorphic maps in complex Banach spaces

Conditions guaranteeing the uniform convergence to constant maps of random iterations of holomorphic contractions on unbounded domains in complex Banach spaces are established.

     

Rigidity of proper holomorphic mappings between equidimensional bounded symmetric domains

We prove that any proper holomorphic mapping between two equidimensional irreducible bounded symmetric domains with rank is a biholomorphism. The proof of the main result in this paper will be achieved by a differential-geometric study of a special class of complex geodesic curves on the bounded symmetric domains with respect to their Bergman metrics.

     

Components exponential scator holomorphic function

The components exponential scator function, succinctly labeled “cexp,” is a function of scator variable in 1+n dimensional elliptic scator algebra. This function is introduced here, and its elementary properties are studied. In 1+1 dimensions, the cexp function becomes the usual complex exponential function of complex variable. The cexp function is shown to be scator holomorphic in the entire scator set according to the differential quotient criterion. The scator derivative of the cexp function is the cexp function itself. The relationship between the Cartesian and polar forms of the cexp function can be seen as a higher-dimensional extension of Euler formula. The mappings of grids in 1+2 dimensional space exhibit ellipses and Lissajous-like figures in addition to circles and radial lines. A cusphere surface is generated for the isometric condition in the function's image. An interesting application of the cexp function as propagators regarding the quantum measurement problem is outlined.     

Envelopes of open sets and extending holomorphic functions on dual Banach spaces

We investigate certain envelopes of open sets in dual Banach spaces which are related to extending holomorphic functions. We give a variety of examples of absolutely convex sets showing that the extension is in many cases not possible. We also establish connections to the study of iterated weak^∗ sequential closures of convex sets in the dual of separable spaces.     

Dense-lineability in spaces of continuous functions

In this paper we provide a general method to prove that certain nonlinear families of continuous functions contain dense linear manifolds. An application is furnished.

     

典型域上高阶Fréechet导数的Schwarz-Pick估计

本文给出了典型域上带正实部的全纯函数高阶Fr¶echet 导数的Schwarz-Pick 估计. 推广了单位圆盘和单位球上带正实部的全纯函数高阶偏导数的Schwarz-Pick 估计.     

The density condition and distinguishedness for weighted Fréchet spaces of holomorphic functions

We consider weighted Fréchet spaces of holomorphic functions which are defined as countable intersections of weighted Banach spaces of type H^∞. We study when these spaces have Stefan Heinrich's density condition and when they are distinguished.