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

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.

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