Английская Википедия:Dirichlet space
In mathematics, the Dirichlet space on the domain <math>\Omega \subseteq \mathbb{C}, \, \mathcal{D}(\Omega)</math> (named after Peter Gustav Lejeune Dirichlet), is the reproducing kernel Hilbert space of holomorphic functions, contained within the Hardy space <math>H^2(\Omega)</math>, for which the Dirichlet integral, defined by
- <math> \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 </math>
is finite (here dA denotes the area Lebesgue measure on the complex plane <math>\mathbb{C}</math>). The latter is the integral occurring in Dirichlet's principle for harmonic functions. The Dirichlet integral defines a seminorm on <math>\mathcal{D}(\Omega)</math>. It is not a norm in general, since <math>\mathcal{D}(f) = 0</math> whenever f is a constant function.
For <math>f,\, g \in \mathcal{D}(\Omega)</math>, we define
- <math>\mathcal{D}(f, \, g) : = {1\over \pi} \iint_\Omega f'(z) \overline{g'(z)} \, dA(z).</math>
This is a semi-inner product, and clearly <math>\mathcal{D}(f, \, f) = \mathcal{D}(f)</math>. We may equip <math>\mathcal{D}(\Omega)</math> with an inner product given by
- <math> \langle f, g \rangle_{\mathcal{D}(\Omega)} := \langle f, \, g \rangle_{H^2 (\Omega)} + \mathcal{D}(f, \, g) \; \; \; \; \; (f, \, g \in \mathcal{D}(\Omega)),</math>
where <math> \langle \cdot, \, \cdot \rangle_{H^2 (\Omega)}</math> is the usual inner product on <math>H^2 (\Omega).</math> The corresponding norm <math> \| \cdot \|_{\mathcal{D}(\Omega)} </math> is given by
- <math> \|f\|^2_{\mathcal{D}(\Omega)} := \|f\|^2_{H^2 (\Omega)} + \mathcal{D}(f) \; \; \; \; \; (f \in \mathcal{D} (\Omega)).</math>
Note that this definition is not unique, another common choice is to take <math> \|f\|^2 = |f(c)|^2 + \mathcal{D}(f)</math>, for some fixed <math> c \in \Omega </math>.
The Dirichlet space is not an algebra, but the space <math>\mathcal{D}(\Omega) \cap H^\infty(\Omega)</math> is a Banach algebra, with respect to the norm
- <math> \|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)).</math>
We usually have <math>\Omega = \mathbb{D}</math> (the unit disk of the complex plane <math>\mathbb{C}</math>), in that case <math>\mathcal{D}(\mathbb{D}):=\mathcal{D}</math>, and if
- <math> f(z) = \sum_{n \ge 0} a_n z^n \; \; \; \; \; (f \in \mathcal{D}), </math>
then
- <math> D(f) =\sum_{n\ge 1} n |a_n|^2,</math>
and
- <math> \|f \|^2_\mathcal{D} = \sum_{n \ge 0} (n+1) |a_n|^2. </math>
Clearly, <math>\mathcal{D}</math> contains all the polynomials and, more generally, all functions <math>f</math>, holomorphic on <math>\mathbb{D}</math> such that <math>f'</math> is bounded on <math>\mathbb{D}</math>.
The reproducing kernel of <math>\mathcal{D}</math> at <math>w \in \mathbb{C} \setminus \{ 0 \}</math> is given by
- <math> k_w(z) = \frac{1}{z\overline{w}} \log \left( \frac{1}{1-z\overline{w}} \right) \; \; \; \; \; (z \in \mathbb{C} \setminus \{ 0 \}).</math>
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