Problems in Function Theory

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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.