Dirichlet space

In mathematics, the Dirichlet space on the domain $\Omega \subseteq ℂ,𝒟(\Omega )$ (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

$𝒟(f):=\frac{1}{\pi }{\iint }_{\Omega }|f^{\prime }(z){|}^{2}dA=\frac{1}{4\pi }{\iint }_{\Omega }|{\partial }_{x}f{|}^2+|{\partial }_{y}f{|}^{2}dxdy$

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 $𝒟(\Omega )$. It is not a norm in general, since $𝒟(f)=0$ whenever f is a constant function.

For $f,g\in 𝒟(\Omega )$, we define

$𝒟(f,g):=\frac{1}{\pi }{\iint }_{\Omega }{f}^{\prime }(z)\overline{g^{\prime }(z)}dA(z).$

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

$⟨f,g{⟩}_{𝒟(\Omega )}:=⟨f,g{⟩}_{H^2(\Omega )}+𝒟(f,g)(f,g\in 𝒟(\Omega )),$

where $⟨\cdot ,\cdot {⟩}_{H^2(\Omega )}$ is the usual inner product on $H^2(\Omega ).$ The corresponding norm $\Vert \cdot {\Vert }_{𝒟(\Omega )}$ is given by

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

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

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

$\Vert f{\Vert }_{𝒟(\Omega )\cap H^{\mathrm{\infty }}(\Omega )}:=\Vert f{\Vert }_{H^{\mathrm{\infty }}(\Omega )}+𝒟(f{)}^{1/2}(f\in 𝒟(\Omega )\cap H^{\mathrm{\infty }}(\Omega )).$

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

$f(z)=\sum _{n\ge 0}{a}_n{z}^n(f\in 𝒟),$

then

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

and

$\Vert f{\Vert }_{𝒟}^2=\sum _{n\ge 0}(n+1)|a_n{|}^2.$

Clearly, $𝒟$ contains all the polynomials and, more generally, all functions $f$, holomorphic on $𝔻$ such that $f^{\prime }$ is bounded on $𝔻$.

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

$k_w(z)=\frac{1}{z\bar{w}}\log \left(\frac{1}{1-z\bar{w}}\right)(z\in ℂ\setminus \{0\}).$

See also

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

References

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