Dirichlet space

In mathematics, the Dirichlet space on the domain $$\Omega \subseteq \mathbb{C}, \, \mathcal{D}(\Omega)$$ (named after Peter Gustav Lejeune Dirichlet), is the reproducing kernel Hilbert space of holomorphic functions, contained within the Hardy space $$H^2(\Omega)$$,  for which the Dirichlet integral, defined by


 * $$ \mathcal{D}(f) := {1\over \pi} \iint_\Omega |f^\prime(z)|^2 \, dA = {1\over 4\pi}\iint_\Omega |\partial_x f|^2 + |\partial_y f|^2 \, dx \, dy $$

is finite (here dA denotes the area Lebesgue measure on the complex plane $$\mathbb{C}$$). The latter is the integral occurring in Dirichlet's principle for harmonic functions. The Dirichlet integral defines a seminorm on $$\mathcal{D}(\Omega)$$. It is not a norm in general, since $$\mathcal{D}(f) = 0$$ whenever f is a constant function.

For $$f,\, g \in \mathcal{D}(\Omega)$$, we define


 * $$\mathcal{D}(f, \, g) : = {1\over \pi} \iint_\Omega f'(z) \overline{g'(z)} \, dA(z).$$

This is a semi-inner product, and clearly $$\mathcal{D}(f, \, f) = \mathcal{D}(f)$$. We may equip $$\mathcal{D}(\Omega)$$ with an inner product given by


 * $$ \langle f, g \rangle_{\mathcal{D}(\Omega)} := \langle f, \, g \rangle_{H^2 (\Omega)} + \mathcal{D}(f, \, g) \; \; \; \; \; (f, \, g \in \mathcal{D}(\Omega)),$$

where $$ \langle \cdot, \, \cdot \rangle_{H^2 (\Omega)}$$ is the usual inner product on $$H^2 (\Omega).$$ The corresponding norm $$ \| \cdot \|_{\mathcal{D}(\Omega)} $$ is given by


 * $$ \|f\|^2_{\mathcal{D}(\Omega)} := \|f\|^2_{H^2 (\Omega)} + \mathcal{D}(f) \; \; \; \; \; (f \in \mathcal{D} (\Omega)).$$

Note that this definition is not unique, another common choice is to take $$ \|f\|^2 = |f(c)|^2 + \mathcal{D}(f)$$, for some fixed $$ c \in \Omega $$.

The Dirichlet space is not an algebra, but the space $$\mathcal{D}(\Omega) \cap H^\infty(\Omega)$$ is a Banach algebra, with respect to the norm


 * $$ \|f\|_{\mathcal{D}(\Omega) \cap H^\infty(\Omega)} := \|f\|_{H^\infty(\Omega)} + \mathcal{D}(f)^{1/2} \; \; \; \; \; (f \in \mathcal{D}(\Omega) \cap H^\infty(\Omega)).$$

We usually have $$\Omega = \mathbb{D}$$ (the unit disk of the complex plane $$\mathbb{C}$$), in that case $$\mathcal{D}(\mathbb{D}):=\mathcal{D}$$, and if


 * $$ f(z) = \sum_{n \ge 0} a_n z^n \; \; \; \; \; (f \in \mathcal{D}), $$

then


 * $$ D(f) =\sum_{n\ge 1} n |a_n|^2,$$

and


 * $$ \|f \|^2_\mathcal{D} = \sum_{n \ge 0} (n+1) |a_n|^2. $$

Clearly, $$\mathcal{D}$$ contains all the polynomials and, more generally, all functions $$f$$, holomorphic on $$\mathbb{D}$$ such that $$f'$$ is bounded on $$\mathbb{D}$$.

The reproducing kernel of $$\mathcal{D}$$ at $$w \in \mathbb{C} \setminus \{ 0 \}$$ is given by


 * $$ k_w(z) = \frac{1}{z\overline{w}} \log \left( \frac{1}{1-z\overline{w}} \right) \; \; \; \; \; (z \in \mathbb{C} \setminus \{ 0 \}).$$