Dirichlet space
In mathematics, the Dirichlet space on the domain [math]\displaystyle{ \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]\displaystyle{ H^2(\Omega) }[/math], for which the Dirichlet integral, defined by
- [math]\displaystyle{ \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]\displaystyle{ \mathbb{C} }[/math]). The latter is the integral occurring in Dirichlet's principle for harmonic functions. The Dirichlet integral defines a seminorm on [math]\displaystyle{ \mathcal{D}(\Omega) }[/math]. It is not a norm in general, since [math]\displaystyle{ \mathcal{D}(f) = 0 }[/math] whenever f is a constant function.
For [math]\displaystyle{ f,\, g \in \mathcal{D}(\Omega) }[/math], we define
- [math]\displaystyle{ \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]\displaystyle{ \mathcal{D}(f, \, f) = \mathcal{D}(f) }[/math]. We may equip [math]\displaystyle{ \mathcal{D}(\Omega) }[/math] with an inner product given by
- [math]\displaystyle{ \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]\displaystyle{ \langle \cdot, \, \cdot \rangle_{H^2 (\Omega)} }[/math] is the usual inner product on [math]\displaystyle{ H^2 (\Omega). }[/math] The corresponding norm [math]\displaystyle{ \| \cdot \|_{\mathcal{D}(\Omega)} }[/math] is given by
- [math]\displaystyle{ \|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]\displaystyle{ \|f\|^2 = |f(c)|^2 + \mathcal{D}(f) }[/math], for some fixed [math]\displaystyle{ c \in \Omega }[/math].
The Dirichlet space is not an algebra, but the space [math]\displaystyle{ \mathcal{D}(\Omega) \cap H^\infty(\Omega) }[/math] is a Banach algebra, with respect to the norm
- [math]\displaystyle{ \|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]\displaystyle{ \Omega = \mathbb{D} }[/math] (the unit disk of the complex plane [math]\displaystyle{ \mathbb{C} }[/math]), in that case [math]\displaystyle{ \mathcal{D}(\mathbb{D}):=\mathcal{D} }[/math], and if
- [math]\displaystyle{ f(z) = \sum_{n \ge 0} a_n z^n \; \; \; \; \; (f \in \mathcal{D}), }[/math]
then
- [math]\displaystyle{ D(f) =\sum_{n\ge 1} n |a_n|^2, }[/math]
and
- [math]\displaystyle{ \|f \|^2_\mathcal{D} = \sum_{n \ge 0} (n+1) |a_n|^2. }[/math]
Clearly, [math]\displaystyle{ \mathcal{D} }[/math] contains all the polynomials and, more generally, all functions [math]\displaystyle{ f }[/math], holomorphic on [math]\displaystyle{ \mathbb{D} }[/math] such that [math]\displaystyle{ f' }[/math] is bounded on [math]\displaystyle{ \mathbb{D} }[/math].
The reproducing kernel of [math]\displaystyle{ \mathcal{D} }[/math] at [math]\displaystyle{ w \in \mathbb{C} \setminus \{ 0 \} }[/math] is given by
- [math]\displaystyle{ k_w(z) = \frac{1}{z\overline{w}} \log \left( \frac{1}{1-z\overline{w}} \right) \; \; \; \; \; (z \in \mathbb{C} \setminus \{ 0 \}). }[/math]
See also
References
- Arcozzi, Nicola; Rochberg, Richard; Sawyer, Eric T.; Wick, Brett D. (2011), "The Dirichlet space: a survey", New York J. Math. 17a: 45–86, http://nyjm.albany.edu/j/2011/17a-4v.pdf
- El-Fallah, Omar; Kellay, Karim; Mashreghi, Javad; Ransford, Thomas (2014). A primer on the Dirichlet space. Cambridge, UK: Cambridge University Press. ISBN 978-1-107-04752-5. http://cambridge.org/9781107047525.
 |  |
