Dragan Vukotic: Around the Dirichlet space

Tuesday, 25 October 2011

Abstract: The Dirichlet space is an important conformally invariant space of analytic functions that has been studied from many points of view ever since Beurling's thesis. In this talk, we will discuss various facts about this space taken from our works with various coauthors over the last decade.

In the first part of the talk, we will discuss some arguments from the earlier works on the growth and exponential integrability of Dirichlet functions as related to the celebrated Chang-Marshall inequality and some applications of this theorem to other function spaces.

Naturally, every simply connected domain of finite area is the image of the unit disk under a conformal map that belongs to the Dirichlet space.
In the second part of the talk, we will prove the following recent result obtained jointly with J.J. Donaire and D. Girela, apparently of purely topological or geometric nature but which can be proved by using techniques from analysis: any domain of finite area contains a simply connected domain of the same area such that their difference as sets is a countable union of intervals. The construction, somewhat similar to those of P.W. Jones, uses the Whitney decomposition.