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.
