Internet Analysis Seminar

The topic for the Internet Analysis Seminar for Fall 2010 will be the Dirichlet Space of Analytic Functions. The classical Dirichlet space $\mathcal{D}$ of holomorphic functions $f$ on the unit disk D satisfying

$$
\left\Vert f \right\Vert _{\mathcal{D}}=\left\{ \left\vert f(0)\right\vert^{2}+\int_{D}\left\vert
f^{\prime }\left( z \right) \right\vert ^{2}dxdy\right\} ^{\frac{1}{2}}<\infty
$$

occupies a pivotal endpoint niche in the theory of Hilbert spaces of holomorphic functions satisfying Sobolev type conditions. As such, $\mathcal{D}$ inherits much of the character of the space BMO of functions of bounded mean oscillation on the real line, which in turn occupies a pivotal endpoint niche among the somewhat different scale of Lebesgue spaces on the line.

One of the primary themes of this seminar is an exposition of how the Carleson program has unfolded for the Dirichlet space. The Carleson program examines the questions of interpolating sequences for $\mathcal{D}$ and the existence/absence of coronas in Banach algebras associated with $\mathcal{D}$. Each of these questions was first considered and answered by Lennart Carleson in the context of the classical Hardy space $H^{2}$ in the late 1950's and early 1960's. The analogous questions for the Dirichlet space were obtained much more recently and have since been an active area of mathematical research.

We first will review some of the necessary details about the classical Hardy space and then compare and contrast this with the corresponding results for the Dirichlet space.