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
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
This is a semi-inner product, and clearly . We may equip with an inner product given by
where is the usual inner product on The corresponding norm is given by
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
We usually have (the unit disk of the complex plane ), in that case , and if
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
- Arcozzi, Nicola; Rochberg, Richard; Sawyer, Eric T.; Wick, Brett D. (2011), "The Dirichlet space: a survey" (PDF), New York J. Math., 17a: 45–86
- 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.