Dirichlet space

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

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

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

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

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

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

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

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

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

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 $c\in \mathrm{\Omega }$.

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

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

We usually have $\mathrm{\Omega }=\mathbb{𝔻}$ (the unit disk of the complex plane $\mathbb{ℂ}$), in that case ${𝒟}(\mathbb{𝔻}):={𝒟}$, 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 $\mathbb{𝔻}$ such that $f^{\prime }$ is bounded on $\mathbb{𝔻}$.

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

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

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", New York J. Math. 17a: 45–86, http://nyjm.albany.edu/j/2011/17a-4v.pdf 
• 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. http://cambridge.org/9781107047525.