In mathematics, the Dirichlet space on the domain
(named after Peter Gustav Lejeune Dirichlet), is the reproducing kernel Hilbert space of holomorphic functions, contained within the Hardy space
, 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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/faae9e3b90d982a601148cdb8ab943a6db26b4b0)
is finite (here dA denotes the area Lebesgue measure on the complex plane
). The latter is the integral occurring in Dirichlet's principle for harmonic functions. The Dirichlet integral defines a seminorm on
. It is not a norm in general, since
whenever f is a constant function.
For
, we define
![{\displaystyle {\mathcal {D}}(f,\,g):={1 \over \pi }\iint _{\Omega }f'(z){\overline {g'(z)}}\,dA(z).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/640feab898d15ea2c5f166be7667fa79f2c0bc8f)
This is a semi-inner product, and clearly
. We may equip
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 )),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a0bbf7ad53aa5e1de5fad830c88d803189989d00)
where
is the usual inner product on
The corresponding norm
is given by
![{\displaystyle \|f\|_{{\mathcal {D}}(\Omega )}^{2}:=\|f\|_{H^{2}(\Omega )}^{2}+{\mathcal {D}}(f)\;\;\;\;\;(f\in {\mathcal {D}}(\Omega )).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4aafcce5dc9fb8428d00f4d283098a0e4b4c3937)
Note that this definition is not unique, another common choice is to take
, for some fixed
.
The Dirichlet space is not an algebra, but the space
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 )).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2e092d7a843a5ae22c4850b0732c91624f47ed54)
We usually have
(the unit disk of the complex plane
), in that case
, and if
![{\displaystyle f(z)=\sum _{n\geq 0}a_{n}z^{n}\;\;\;\;\;(f\in {\mathcal {D}}),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/21a3b88bae99d339b1ccc2b3b1d9a414d385c2a7)
then
![{\displaystyle D(f)=\sum _{n\geq 1}n|a_{n}|^{2},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0c980210241edbe1f00ded83894e942cdc632863)
and
![{\displaystyle \|f\|_{\mathcal {D}}^{2}=\sum _{n\geq 0}(n+1)|a_{n}|^{2}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7b57ee51e6a4f141c16ccec8f9d80348c309874b)
Clearly,
contains all the polynomials and, more generally, all functions
, holomorphic on
such that
is bounded on
.
The reproducing kernel of
at
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\}).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2c67f6ba7ab1fc0d94d3adadd214988ba6501989)