On the Dirichlet space for finitely connected regions

Conformal Welding for Finitely Connected Regions

We discuss a numerical implementation of conformal welding for finitely connected regions using the geodesic zipper algorithm and Koebe’s iterative method for computing conformal maps to regions bounded by circles. We also show that a conformal map from a finitely connected region to a region bounded by circles can be written as a composition of finitely many conformal maps of simply connected ...

Harmonic Coordinates on Finitely Connected Fractafolds

We define finitely connected fractafolds, which are generalizations of p.c.f. self-similar sets introduced by Kigami and of fractafolds introduced by Strichartz. Any self-similarity is not assumed, and countably infinite ramification is allowed. We prove that if a fractafold has a resistance form in the sense of Kigami that satisfies certain assumptions, then there exists a weak Riemannian metr...

The Dirichlet Space on the bi-disc

Bilinear Forms on the Dirichlet Space

Hankel operators on the Hardy space of the disk, H (D) , can be studied as linear operators from H (D) to its dual space, as conjugate linear operators from H (D) to itself, or, in the viewpoint we will take here, as bilinear functionals on H (D) × H (D) . In that formulation, given a holomorphic symbol function b we consider the bilinear Hankel form, defined initially for f, g in P (D) , the s...

The classical Dirichlet space

D |f ′|2 dx dy. In particular, we will cover the basic structure of these functions their boundary values and their zeros along with two important operators that act on this space of functions the forward and backward shifts. This survey is by no means complete. For example, we will not cover the Toeplitz or Hankel operators on these functions, nor will we cover the important topic of interpola...