Mathematical Sciences: Bergman Spaces: Theory and Applications

Korenblum 9302985这个项目是对经典函数论的数学研究的继续，重点是Bergman空间和最近的进展所产生的发展。H·海登马尔姆发现了Bergman空间中的函数分解，类似于F.Riesz的经典分解，这有助于阐明Hardy空间的结构，这是一个根本性的突破。本研究的目的之一是刻画解析勒贝格空间的不变子空间，理解非标准不变子空间，确定零集上的控制关系。还将研究n维欧氏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。后者主要研究模为调和的复调和函数的定位问题。这些被认为是解析函数最接近奇数维的推广。复数分析有着悠久而杰出的数学历史。它一直是数论、计算、逼近理论和位势理论等多个领域研究的基础。尽管经典理论已经被彻底研究了几个世纪，但较新的Bergman空间发展缓慢，尽管它们的重要性得到了公认。赫德马尔姆最近的工作已经完全改变了地貌，以至于人们完全可以预见这一地区将发生重大冲断，类似于哈代空间理论在1950年代的S和60年代的S蓬勃发展时所发生的情况。

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. ***