Problems in Function Theory

首席研究员和他的学生将致力于复杂分析的三个领域的问题。第一组问题涉及用Blaschke积以一致范数或BMO范数逼近任意Blaschke积，这些Blaschke积的零点充分分离，使得相应的Riesz质量在所有全纯坐标系中是有界的(即，它是Carleson测度)。这一领域的一个特殊问题是直接证明两个条件的等价性，即Muckenoupt条件和Helson-Szego条件，这两个条件是希尔伯特变换在加权希尔伯特空间上有界的充要条件。另一种是解决Nikolski猜想，即对任何模型子空间(即Hilbert空间上单边移位的不变子空间的正交补)，都存在赋值泛函的Riesz基。解决这些问题需要对希尔伯特变换和双曲度规有更深的理解。第二组问题证明了几类无限连通平面域的冠状定理，包括Cantor中间三分集与其自身的乘积的补集和Lipschitz图的正长子集的补集。第三个问题集涉及将David，Tolsa，Volberg等人最近关于解析容量的漂亮工作推广到高维欧氏空间中的Lipschitz调和容量，特别是证明了n维Lipschitz调和容量是bilipschitz不变量.这些问题的方法将是建设性的，以便在某些情况下它们可以得到解析函数和共形映射的显式计算机辅助构造.解析函数和共形映射在流体力学、声学和电学中有着广泛的应用，在这些应用中，显式构造比一般存在定理更有用。研究这些问题所需要的数学方法包括调和分析、偏微分方程、双曲几何和几何测度论的许多思想和方法。因此，研究这些问题的学生将获得广泛的数学培训。

The principal investigator and his students will work on problems in three areas of complex analysis. The first set of problems concerns approximating an arbitrary Blaschke product, in uniform norm or in BMO norm, by Blaschke products whose zeros are sufficiently separated that the corresponding Riesz mass is bounded in all holomorphic coordinate systems (i.e., it is a Carleson measure). A particular problem in this area is to find a direct proof of the equivalence of two conditions, the Muckenhoupt condition and the Helson-Szego condition, that are necessary and sufficient for the Hilbert transform to be bounded on a weighted Hilbert space. Another is to resolve the Nikolski conjecture that there is a Riesz basis of evaluation functionals for any model subspace (i.e., the orthogonal complement of an invariant subspace for the unilateral shift on Hilbert space). Solving these problems should require deeper understandings of the Hilbert transform and the hyperbolic metric. The second set of problems it to prove the corona theorem for some classes of infinitely connected plane domains, including the complement of the product of the Cantor middle-third set with itself and the complements of positive length subsets of Lipschitz graphs. The third problem set involves extending beautiful recent work of David, Tolsa, Volberg, and others on analytic capacity to Lipschitz harmonic capacity in higher dimensional Euclidean space, in particular, to prove that n-dimensional Lipschitz harmonic capacity is a bilipschitz invariant.The methods to be used on these problems will be constructive, so that in some cases they can yield explicit computer-aided constructions of analytic functions and conformal mappings. Analytic functions and conformal mappings have broad applications in fluid dynamics, acoustics, and electrical engineering, and in these applications explicit constructions are more useful than general existence theorems. The mathematics needed to study these problems includes many ideas and methods from harmonic analysis, partial differential equations, hyperbolic geometry, and geometric measure theory. As a result, the students who work on these problems will obtain a broad mathematical training.