Dirichlet space

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

${\displaystyle {\mathcal {D}}(f):={1 \over \pi }\iint _{\Omega }|f^{\prime }(z)|^{2}\,dA={1 \over 4\pi }\iint _{\Omega }|\partial _{x}f|^{2}+|\partial _{y}f|^{2}\,dx\,dy}$

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

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

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

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

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

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

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

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

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

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

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

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

then

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

and

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

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

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

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