Problems in Function Theory

函数论问题

• 批准号: 0401720
• 负责人: John Garnett
• 金额: $ 12.51万
• 依托单位: University of California-Los Angeles
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2004
• 资助国家: 美国
• 起止时间: 2004-07-01 至 2008-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0401720&HistoricalAwards=false
• 关键词: Problems; Function; Theory

DMS 0401720J B GarnettUCLAGarnett and his students will work on several problems in classical one dimensional complex analysis. The first problem is to approximate any Blaschke product uniformly on the open disc by Blaschke products whose zeros aresufficiently spread apart and thin that the corresponding Riesz mass is bounded in all holomorphic coordinate systems (i. e. is a Carleson measure). The approximation should be effected using explicit constructions. The second problem is to give a direct proof of the equivalence of two weight conditions, the Muckenhoupt $A_2$ condition and the Helson-Szeg\"o condition, that are necessary and sufficient for the Hilbert transform to be bounded on $L^2$(weight).The third problem is to construct non-constant bounded analytic functions on the complement of a positive length subset of a Lipschitz graph. The fourth problemis a corona problem for infinitely connected plane domains whose boundaries lie on certain regular Cantor sets. It too requires some new explicit constructions. The fifth problem is to prove the $n$-dimensional Lipschitzharmonic capacity is a bilipschitz invariant.The methods to be used on these problems will be constructive so that they can be give explicit computer aided constructions of analytic functions and conformal mappings. Analytic functions and conformal mappings havebroad applications in fluid dynamics, acoustics, and electrical engineering, and in these applications constructions are more useful than general existence theorems.

B . GarnettUCLAGarnett和他的学生将研究经典一维复分析中的几个问题。第一个问题是用Blaschke积均匀地近似开盘上的任何Blaschke积，这些Blaschke积的零足够分散和薄，使得相应的Riesz质量在所有全纯坐标系（即Carleson测度）中有界。应该使用显式结构来实现近似。第二个问题是直接证明两个权条件的等价性，即Muckenhoupt $A_2$和Helson-Szeg $ o条件，它们是Hilbert变换在$L^2$（权）上有界的充分必要条件。第三个问题是在Lipschitz图的正长度子集的补上构造非常有界解析函数。第四个问题是边界位于正则康托集上的无限连通平面域的电晕问题。它也需要一些新的明确的结构。第5个问题是证明n维lipschitz调和容量是一个bilipschitz不变量。用于这些问题的方法将是建设性的，以便它们可以给出解析函数和保角映射的显式计算机辅助构造。解析函数和保角映射在流体动力学、声学和电气工程中有广泛的应用，在这些应用中，构造比一般的存在定理更有用。