Bergman 穷竭, 完备与稳定性

Bergman核是国内外研究多复变函数论的一个传统课题,本文主要介绍利用超定的非齐次Cuauchy-Riemann方程组的现代Hilbert空间理论研究Bergman核的穷竭性,稳定性与Bergman度量的完备性所取得的最新进展。     

Non-isotropic weighted Lp estimates for the $$ \bar \partial $$-equation on piecewise smooth strictly pseudoconvex domains

In this paper we obtain non-isotropic weighted L ^p estimates with the boundary distance weight function for the -equation on piecewise smooth strictly pseudoconvex domains under a hypothesis of complex transversality in ℂⁿ using the explicit formula of solutions by Berndtsson-Andersson. This work was supported by the Korea Research Foundation Grant funded by Korea Government (MOEHRD, Basic Research Promotion Fund) (Grant No. KRF-2005-070-C00007)     

The equivalence on classical metrics

In this paper we study the complete invariant metrics on Cartan-Hartogs domains which are the special types of Hua domains. Firstly, we introduce a class of new complete invariant metrics on these domains, and prove that these metrics are equivalent to the Bergman metric. Secondly, the Ricci curvatures under these new metrics are bounded from above and below by the negative constants. Thirdly, we estimate the holomorphic sectional curvatures of the new metrics, and prove that the holomorphic sectional curvatures are bounded from above and below by the negative constants. Finally, by using these new metrics and Yau's Schwarz lemma we prove that the new metrics are equivalent to the Einstein-Kahler metric. That means that the Yau's conjecture is true on Cartan-Hartogs domains.     

Behavior of the Bergman kernel and metric near convex boundary points

The boundary behavior of the Bergman metric near a convex boundary point of a pseudoconvex domain is studied. It turns out that the Bergman metric at points in the direction of a fixed vector tends to infinity, when is approaching , if and only if the boundary of does not contain any analytic disc through in the direction of .

     

Bergman kernel and metric on non-smooth pseudoconvex domains

A stability theorem of the Bergman kernel and completeness of the Bergman metric have been proved on a type of non-smooth pseudoconvex domains defined in the following way:D = {z∈U|r(z)} <whereU is a neighbourhood of andr is a continuous plurisubharmonic function onU. A continuity principle of the Bergman Kernel for pseudoconvex domains with Lipschitz boundary is also given, which answers a problem of Boas.     

Bergman type operators on a class of weakly pseudoconvex domains

A necessary and sufficient condition for the boundedness of the operator: $(T_{s,u,u} f)(\xi ) = h^{u + \tfrac{v}{a}} (\xi )\smallint _{\Omega _a } h^s (\xi ')K_{s,u,v} (\xi ,\xi ')f(\xi ')dv(\xi ') on L^p (\Omega _a ,dv_\lambda ),1< p< \infty $ , is obtained, where $\Omega _a = \left\{ {\xi = (z,w) \in \mathbb{C}^{n + m} :z \in \mathbb{C}^n ,w \in \mathbb{C}^m ,|z|^2 + |w|^{2/a}< 1} \right\},h(\xi ) = (1 - |z|^2 )^a - |w|^2 $ andK _x,u,v (ξ,ξ′).This generalizes the works in literature from the unit ball or unit disc to the weakly pseudoconvex domain ω _a . As an appli cation, it is proved thatf?L _H ^p (ω _a ,dv _λ) implies $h\tfrac{{|a|}}{a} + |\beta |(\xi )D_2^a D_z^\beta f \in L^p (\Omega _a ,dv_\lambda ),1 \leqslant p< \infty $ , for any multi-indexa=(α₁,?,α _n and ß = (ß₁, —ß). An interesting question is whether the converse holds.     

Bergman空间上复合算子的紧性刻划

研究了C^n中有界强拟凸Ω上Bergman空间A^p(Ω)上的复合算子的有界性、紧性，给出了复合算子Cψ：A^p(Ω)→A^p(Ω)紧性的一个完整刻划。     

On the Bergman metric of pseudoconvex domains in a complex projective space

We prove a localization principle of the Bergman kernel form and metric for pseudoconvex domains in the complex projective space. An estimate of the Bergman distance is also given.

     

Stein流形上的Lewy方程

本文利用Stein流形上强拟凸域的全纯支撑函数,并使用Hermite度量和陈联络定义的Koppelman-Leray核,结合边界的流形结构特点,得到了强拟凸域边界上(p,q)型Lewy方程(b-方程)解的一种整体积分表示.     

第四类Caftan-Hartogs域上Bergman度量与Einstein-Kahler度量等价

In this paper,we discuss the invariaut complete metric on the Cartan-Hartogs domain of the fourth type.Firstly,we find a new invariant complete metric,and prove the equivalence between Bergman metric and the new metric;Secondly,the Ricci curvature of the new metric has the super bound and lower bound;Thirdly,we prove that the holomorphic sectional curvature of the new metric has the negative supper bound;Finally,we obtain the equivalence between Bergman metric and Einstein-Kahler metric on the Cartan-Hartogs domain of the fourth type.     

Bergman空间上Hankel算子的本性范数的刻划

在文中,对于Ｃ＾ｍ中有界强拟凸域Ω,得到了Ｂｅｒｇｍａｎ空间Ａ＾２（Ω）上的Ｈａｎｋｅｌ算子Ｈｆ（ｆ∈Ｌ＾２（Ω,ｄｖ）的本性范数∥Ｈｆ∥ｅｓｓ（在Ｌ＾２（Ω,ｄｖ上）的等价刻划。     

关于某类Reinhardt域的Bergman核函数与解析自同构最大群

本文给出了Ｒｅｉｎｈａｒｄｔ域Ｄ＝｛ｚ＝（ｚ１，ｚ２，ｚ３）∈：｜ｚ１｜２ｋ＋｜ｚ２｜＋｜ｚ３｜２＜１，ｋ＞０｝的Ｂｅｒｇｍａｎ核函数，Ｂｅｒｇｍａｎ度量方阵及其解析自同构最大群。     

The Bergman kernels on super-Car tan domains of the first type

The Bergman kernel function and biholomorphic automorphism group for super-Cartan domain of the first type are given in explicit formulas.     

第四类超Cartan域的Bergman核函数

显式给出了第四类超Cartan域的Bergman核函数及其全纯自同构群．     

一类Hartogs域的Bergman核函数的零点问题(英文)

We obtain the Bergman kernel for a new type of Hartogs domain.The corresponding LU Qi-Keng’s problem is considered.     

第三类超Cartan域的Bergman核函数

显式给出了第三类超Cartan域的Bergman核函数及其全纯自同构群。     

一类拟凸域的Bergman度量与Kobayashi度量的比较定理

本文对一类拟凸域Ｅ（ｍ，ｎ，Ｋ）给出其不变Ｋａｈｌｅｒ度量下的全纯截曲率的显表达式，并构造了Ｅ（ｍ，ｎ，Ｋ）的一个不变的完备的Ｋａｈｌｅｒ度量，使得它大于或等于Ｂｅｒｇｍａｎ度量，而且其全纯截曲率的上界是一个负常数，从而得到Ｅ（ｍ，ｎ，Ｋ）的Ｂｅｒｇｍａｎ度量和Ｋｏｂａｙａｓｈｉ度量的比较定理。     

Singular integral on bounded strictly pseudoconvex domain

Kytmanov and Myslivets gave a special Cauchy principal value of the singular integral on the bounded strictly pseudoconvex domain with smooth boundary. By means of this Cauchy integral principal value, the corresponding singular integral and a composition formula are obtained. This composition formula is quite different from usual ones in form. As an application, the corresponding singular integral equation and the system of singular integral equations are discussed as well.