Dirichlet space explained

In mathematics, the Dirichlet space on the domain

\Omega \subseteq C , l{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 byl{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

C ). The latter is the integral occurring in Dirichlet's principle for harmonic functions. The Dirichlet integral defines a seminorm on l{D}(\Omega) . It is not a norm in general, since l{D}(f) = 0 whenever f is a constant function.

For

f, g \in l{D}(\Omega) , we definel{D}(f, g) : = { 1 \over \pi} \iint _{\Omega} f'(z) \overline{g'(z)} dA(z).

This is a semi-inner product, and clearly

l{D}(f, f) = l{D}(f) . We may equip l{D}(\Omega) with an inner product given by\langle f, g \rangle _{l{D} (\Omega)} : = \langle f, g \rangle _{H ^{2} (\Omega)} + l{D}(f, g) (f, g \in l{D}(\Omega)),

where

\langle ⋅ , ⋅ \rangle _{H ^{2} (\Omega)} is the usual inner product on H ^{2} (\Omega). The corresponding norm \| ⋅ \| _{l{D} (\Omega)} is given by\|f\| ^{2} _{l{D} (\Omega)} : = \|f\| ^{2} _{H ^{2} (\Omega)} + l{D}(f) (f \in l{D} (\Omega)).

Note that this definition is not unique, another common choice is to take

\|f\| ^{2} = |f(c)| ^{2} + l{D}(f) , for some fixed c \in \Omega .

The Dirichlet space is not an algebra, but the space

l{D}(\Omega) \cap H ^{infty(\Omega)} is a Banach algebra, with respect to the norm\|f\| _{l{D} (\Omega) \cap H ^{infty(\Omega)}} : = \|f\| _{H ^{infty(\Omega)}} + l{D}(f) ^{1/2} (f \in l{D}(\Omega) \cap H ^{infty(\Omega)).}

We usually have

\Omega = D (the unit disk of the complex plane C ), in that case l{D}( D ):= l{D} , and if f(z) = \sum _{n} a _{n} z ^{n} (f \in l{D}),

then

D(f) = \sum _{n\ge} n |a ^{2 ,} _{n|}

and

\|f \| ^{2} _{l{D} =} \sum _{n} (n +1 ) |a ^{2 .} _{n|}

Clearly,

l{D} contains all the polynomials and, more generally, all functions f , holomorphic on D such that f' is bounded on D .

The reproducing kernel of

l{D} at w \in C \setminus \{ 0 \} is given byk _{w(z)} = (1)/(z\overline{w) } \log \left(\frac \right) \; \; \; \; \; (z \in \mathbb \setminus \).

See also

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

References

• • Book: Omar. El-Fallah. Karim. Kellay. Javad. Mashreghi. Thomas. Ransford. A primer on the Dirichlet space. 2014. Cambridge University Press. Cambridge, UK. 978-1-107-04752-5.