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

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

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

$f(z)=\sum _{n\geq 0}a_{n}z^{n}\;\;\;\;\;(f\in {\mathcal {D}}),$

then

$D(f)=\sum _{n\geq 1}n|a_{n}|^{2},$

and

$\|f\|_{\mathcal {D}}^{2}=\sum _{n\geq 0}(n+1)|a_{n}|^{2}.$

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

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

See also

• Banach space
• Bergman space
• Hardy space
• Hilbert space

References

• Arcozzi, Nicola; Rochberg, Richard; Sawyer, Eric T.; Wick, Brett D. (2011), "The Dirichlet space: a survey" (PDF), New York J. Math., 17a: 45–86
• 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.