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 interpolation. These topics are surveyed in a nice paper of Wu [64]. In order to make this survey more manageable, we will also restrict ourselves to this space of functions with finite Dirichlet integral and will not try to cover the many related Dirichlet-type spaces. We refer the reader to the papers [11, 41, 47, 54] for more on this.

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