The Levi problem for Riemann domains over Stein spaces with isolated singularities

We consider the following question: Let \({p:Y \rightarrow X}\) be an unbranched Riemann domain and assume that X is a Stein space and p is a Stein morphism. Does it follow that Y is Stein ? We show that the answer is affirmative if X has isolated singularities. This generalizes a result of Andreotti and Narasimhan.     

Riemann 映射与单连通区域分类

设G为复平面上一个单连通区域及φ为G的Riemann 映射. 本文通过φ是否属于G上多项式在不同拓扑下的闭包的情况对G进行分类. 特别地, 我们对已知的几类单连通给出了刻画.     

Bloch spaces on bounded symmetric domains in complex Banach spaces

We give a definition of Bloch space on bounded symmetric domains in arbitrary complex Banach space and prove such function space is a Banach space. The properties such as boundedness, compactness and closed range of composition operators on such Bloch space are studied. Dedicated to Professor Sheng GONG on the occasion of his 75th birthday     

Some open problems concerning q-convexity and Stein spaces

We discuss some open problems in the theory of analytic pseudoconvexity. We focus our attention especially on q-complete spaces and Stein spaces.     

Diameter preserving bijections between Grassmann spaces over Bezout domains

Let R, S be Bezout domains. Assume that n is an integer ≥ 3, 1 ≤ k ≤ n − 2. Denoted by the k-dimensional Grassmann space on . Let be a map. This paper proves the following are equivalent: (i) is an adjacency preserving bijection in both directions. (ii) is a diameter preserving bijection in both directions. Moreover, Chow’s theorem on Grassmann spaces over division rings is extended to the case of Bezout domains: If is an adjacency preserving bijection in both directions, then is induced by either a collineation or the duality of a collineation. Project 10671026 supported by National Natural Science Foundation of China.     

Weak continuity of Riemann integrable functions in Lebesgue-Bochner spaces

In general, Banach space-valued Riemann integrable functions defined on [0, 1] （equipped with the Lebesgue measure） need not be weakly continuous almost everywhere. A Banach space is said to have the weak Lebesgue property if every Riemann integrable function taking values in it is weakly continuous almost everywhere. In this paper we discuss this property for the Banach space LX^1 of all Bochner integrable functions from [0, 1] to the Banach space X. We show that LX^1 has the weak Lebesgue property whenever X has the Radon-Nikodym property and X＊ is separable. This generalizes the result by Chonghu Wang and Kang Wan [Rocky Mountain J. Math., 31（2）, 697-703 （2001）] that L^1[0, 1] has the weak Lebesgue property.     

Sobolev spaces on bounded symmetric domains

We obtain a formula for the Sobolev inner product in standard weighted Bergman spaces of holomorphic functions on a bounded symmetric domain in terms of the Peter–Weyl components in the Hua–Schmidt decomposition, and use it to clarify the relationship between the analytic continuation of these standard weighted Bergman spaces and the Sobolev spaces on bounded symmetric domains.     

Complexity spaces as quantitative domains of computation

We study domain theoretic properties of complexity spaces. Although the so-called complexity space is not a domain for the usual pointwise order, we show that, however, each pointed complexity space is an ω-continuous domain for which the complexity quasi-metric induces the Scott topology, and the supremum metric induces the Lawson topology. Hence, each pointed complexity space is both a quantifiable domain in the sense of M. Schellekens and a quantitative domain in the sense of P. Waszkiewicz, via the partial metric induced by the complexity quasi-metric.     

The approximation property for spaces of holomorphic functions on infinite dimensional spaces II

Let H(U) denote the vector space of all complex-valued holomorphic functions on an open subset U of a Banach space E. Let τω and τδ respectively denote the compact-ported topology and the bornological topology on H(U). We show that if E is a Banach space with a shrinking Schauder basis, and with the property that every continuous polynomial on E is weakly continuous on bounded sets, then (H(U),τω) and (H(U),τδ) have the approximation property for every open subset U of E. The classical space c₀, the original Tsirelson space T^∗ and the Tsirelson^∗-James space are examples of Banach spaces which satisfy the hypotheses of our main result. Our results are actually valid for Riemann domains.     

Riemann zeta函数的收敛区域

给出了Riemann zeta函数收敛区域的几种证明.     

Composition operators on Hardy spaces of Riemann surfaces

Our purpose is to define composition operators acting upon Hardy spaces of Riemann surfaces. In terms of counting functions related to analytic self-map on Riemann surfaces, the boundedness and compactness are characterized.     

Bloch spaces on bounded symmetric domains in complex Banach spaces Dedicated to Professor Sheng GONG on the occasion of his 75th birthday

We give a definition of Bloch space on bounded symmetric domains in arbitrary complex Banach space and prove such function space is a Banach space. The properties such as boundedness, compactness and closed range of composition operators on such Bloch space are studied.     

无穷维复空间的拟凸Riemann区域(英)

本文利用多极集的测度研究了凸空间上的Riemann区域的拟凸性。     

Short series over simple zeros of the Riemann zeta-function

Recently, Garaev showed that the series Σ_?∥?ζ¹(?)∥⁻¹ diverges, where the sum is taken over the simple zeros ? = β + iγ of the Riemann zeta-function ζ(s). More precisely, he proved . Using a mean-value estimate due to Ramachandra and some result on the distribution of simple zeros in short intervals on the critical line, we prove for T^0.552 ≤ H ≤ T. This leads to a slight improvement of Garaev's result in replacing his lower bound by .     

A criterion for Moishezon spaces with isolated singularities

We give a criterion for a compact complex space with isolated singularities to be Moishezon in the spirit of Siu–Demaillys solution to the Grauert–Riemenschneider conjecture. It refines a previous work by Nadel and Tsuji, and another one by Takayama, in a more specific situation.     

A Schottky-type theorem for starlike domains in Banach spaces

We show that if is a starlike domain in a Banach space and is a family of holomorphic functions on that omit two distinct values and is bounded at the origin, then is uniformly bounded on each -bounded set.

     

Solutions of the heat equation in domains with singularities

We study the asymptotics of solutions to the Dirichlet problem for the heat equation in time-dependent domains with singular points.Translated fromMatematicheskie Zametki, Vol. 64, No. 2, pp. 163–179, August, 1998.This research was supported by the Russian Foundation for Basic Research under grant No. 96-01-00504 and by INTAS under grant No. 93-351.     

Notes on ordered reproducing Hilbert spaces over the complex plane

Let f, g be entire functions. If there exist M₁,M₂>0 such that |f(z)|?M₁|g(z)| whenever |z|>M₂ we say that f?g. Let X be a reproducing Hilbert space with an orthogonal basis . We say that X is an ordered reproducing Hilbert space (or X is ordered) if f?g and g∈X imply f∈X. In this note, we show that if then X is ordered; if then X is not ordered. In the case , there are examples to show that X can be of order or opposite.