Mathematical Sciences: Bergman Spaces: Theory and Applications

这个项目是对经典函数理论的数学研究的延续，重点是伯格曼空间和最近进展的发展。当H. Hedenmalm发现Bergman空间中的函数分解类似于F. Riesz的经典分解时，根本性的突破发生了，这有助于澄清Hardy空间的结构。本研究的目标之一是描述解析Lebesgue空间的不变量子空间，理解非标准不变量子空间，确定零集上的支配关系。我们还将研究n维欧几里德空间中双调和算子的格林函数和三维空间中调和函数的代数。后者研究的是那些模量也是调和的复调和函数的定位问题。这些被认为是最接近于解析函数奇维的推广。复分析有着悠久而杰出的数学历史。它已成为数论、计算、近似理论和势理论等各个领域研究的基础。尽管经典理论已经被彻底研究了几个世纪，但新的伯格曼空间发展缓慢，尽管它们的重要性得到了公认。Hedelmalm最近的工作已经完全改变了景观，人们完全期望在这个领域发生重大的推动，类似于哈代空间理论在五六十年代蓬勃发展时发生的事情。＊＊＊

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. ***