Bergman space


In complex analysis, functional analysis and operator theory, a Bergman space is a function space of holomorphic functions in a domain D of the complex plane that are sufficiently well-behaved at the boundary that they are absolutely integrable. Specifically, for, the Bergman space is the space of all holomorphic functions in D for which the p-norm is finite:
The quantity is called the norm of the function ; it is a true norm if. Thus is the subspace of holomorphic functions that are in the space Lp. The Bergman spaces are Banach spaces, which is a consequence of the estimate, valid on compact subsets K of D:
Thus convergence of a sequence of holomorphic functions in implies also compact convergence, and so the limit function is also holomorphic.
If, then is a reproducing kernel Hilbert space, whose kernel is given by the Bergman kernel.

Special cases and generalisations

If the domain is bounded, then the norm is often given by
where is a normalised Lebesgue measure of the complex plane, i.e.. Alternatively is used, regardless of the area of.
The Bergman space is usually defined on the open unit disk of the complex plane, in which case. In the Hilbert space case, given, we have
that is, is isometrically isomorphic to the weighted ℓp space. In particular the polynomials are dense in. Similarly, if, the right complex half-plane, then
where, that is, is isometrically isomorphic to the weighted Lp1/t space.
The weighted Bergman space is defined in an analogous way, i.e.
provided that is chosen in such way, that is a Banach space. In case where, by a weighted Bergman space we mean the space of all analytic functions such that
and similarly on the right half-plane we have
and this space is isometrically isomorphic, via the Laplace transform, to the space, where
.
Further generalisations are sometimes considered, for example denotes a weighted Bergman space with respect to a translation-invariant positive regular Borel measure on the closed right complex half-plane, that is

Reproducing kernels

The reproducing kernel of at point is given by
and similarly for we have
In general, if maps a domain conformally onto a domain, then
In weighted case we have
and