Mathematical Sciences: Bergman Spaces: Theory and Applications

数学科学：伯格曼空间：理论与应用

• 批准号: 9302985
• 负责人: Boris Korenblum
• 金额: $ 3.5万
• 依托单位: SUNY at Albany
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1993
• 资助国家: 美国
• 起止时间: 1993-07-15 至 1995-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9302985&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Bergman; Spaces; Theory

Korenblum 9302985 This project is a continuation of mathematical research into classical function theory focusing on Bergman spaces and developments which have issued from recent advances. A fundamental breakthrough occurred when H. Hedenmalm discovered a factorization of functions in Bergman space analogous to the classical one of F. Riesz which helped clarify the structure of the Hardy spaces. Among the goals of this research, work will be done to describe the invariant subspaces of the analytic Lebesgue spaces, on understanding the nonstandard invariant subspaces and determining the domination relations on zero sets. Work will also be done on the Green's function for the biharmonic operator in n-dimensional Euclidean space and algebras of harmonic functions in three-dimensional space. The latter is concerned with locating those complex harmonic functions whose modulus is also harmonic. These are thought to be the closest generalization to odd dimensions of analytic functions. Complex analysis has a long, distinguished mathematical history. It has been a basis for studies in diverse areas as number theory, computation, approximation theory and potential theory. Although the classical theory has been thoroughly investigated for centuries, the newer Bergman spaces have been slow to develop, despite their recognized importance. The recent work of Hedelmalm has completely changed the landscape to a point where one fully expects major thrusts to take place in this area similar to those which occurred when the theory of Hardy spaces flourished in the 1950's and 60's. ***

科伦布卢姆9302985 这个项目是对经典函数论的数学研究的延续，重点是伯格曼空间， 从最近的进展中产生的发展。 当H. Hedenmalm在Bergman空间中发现了一种类似于经典F. Riesz这有助于澄清结构的哈代空间。 在本研究的目标中，工作将完成描述解析Lebesgue空间的不变子空间，理解非标准不变子空间和确定零集上的控制关系。 工作也将完成的绿色的功能，双调和算子在n维欧氏空间和代数的调和函数在三维空间。 后者是关于定位那些复调和函数，其模也是调和的。 这些被认为是最接近的推广奇数维的解析函数。 复分析有着悠久而杰出的数学历史。 它是数论、计算、近似理论和势理论等不同领域研究的基础。 虽然经典理论已经被彻底研究了几个世纪，但较新的伯格曼空间发展缓慢，尽管它们的重要性得到了公认。 最近的工作Hedelmalm已经完全改变了景观到一个点，人们完全预计主要推力发生在这一领域类似于那些发生时，理论的哈代空间蓬勃发展，在20世纪50年代和60年代。 ***