Dirichlet space

Remarks

For $f(z) = \sum_{n \ge 0} a_n z^n$, a power series on $\mathbb{D}$, one has $\int_{\mathbb{D}} \abs{z^{n-1}}^2 dA = \pi/n$, hence $\int_{\mathbb{D}} \abs{f'}^2 \, dA = \pi\sum_{n \ge 1} n \abs{a_n}^2$ and $\norm{f}_{\mathcal{D}}^2 = \abs{a_0}^2 + \sum_{n \ge 1} n\abs{a_n}^2$. This norm comes from an inner product, and $\mathcal{D}$ is a Hilbert space. Example: $f(z) = z$ has $\int_{\mathbb{D}} \abs{f'}^2 \, dA = \pi$, the area of $f(\mathbb{D}) = \mathbb{D}$. Non-example: $f(z) = \log(1 - z) = -\sum_{n \ge 1} z^n/n$ gives $\sum_{n \ge 1} n \cdot \frac{1}{n^2} = \sum 1/n = \infty$, so $f \notin \mathcal{D}$, although $f$ is in the Hardy space $H^2$. If $f$ is injective, $\int_{\mathbb{D}} \abs{f'}^2 \, dA$ is the area of $f(\mathbb{D})$, since $\abs{f'}^2$ is the Jacobian determinant of $f$. Some authors omit the term $\abs{f(0)}^2$ or replace it by $\norm{f}_{H^2}^2$; these give equivalent norms.