Problems in Function Theory

DMS 0401720 J 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乘积在开圆盘上一致逼近任何Blaschke乘积，这些Blaschke乘积的零点足够分散和薄，使得相应的Riesz质量在所有全纯坐标系中是有界的。e.是Carleson测度）。近似值应使用显式构造来实现。 第二个问题是直接证明Hilbert变换在L^2 $（权）上有界的两个权条件Muckenhoupt A_2 $条件和Helson-Szego条件的等价性。第三个问题是构造Lipschitz图的正长子集的补集上的非常数有界解析函数。第四个问题是无穷连通平面区域的冠状问题，其边界位于某些正则Cantor集上。 它也需要一些新的明确的结构。 第五个问题是证明$n$维Lipschitz调和容量是bilipschitz不变量，所用的方法是构造性的，这样就可以给出解析函数和共形映射的明确的计算机辅助构造。 解析函数和共形映射在流体力学、声学和电气工程中有着广泛的应用，在这些应用中，构造比一般的存在定理更有用。

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.