In complex analysis, functional analysis and operator theory, a Bergman space, named after Stefan Bergman, 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 0 < p < ∞ , the Bergman space Ap(D) is the space of all holomorphic functions f in D for which the p-norm is finite: The quantity is called the norm of the function f Ap(D) is the subspace of holomorphic functions that are in the space Lp(D). 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 Lp(D) implies also compact convergence, and so the limit function is also holomorphic. If , then Ap(D) is a reproducing kernel Hilbert space, whose kernel is given by the Bergman kernel.

Special cases and generalisations

If the domain D is bounded, then the norm is often given by: where A is a normalised Lebesgue measure of the complex plane, i.e. . Alternatively is used, regardless of the area of D . The Bergman space is usually defined on the open unit disk \mathbb{D} of the complex plane, in which case. In the Hilbert space case, given:, we have: that is, A2 is isometrically isomorphic to the weighted ℓp(1/(n + 1)) space. In particular the polynomials are dense in A2 . Similarly, if , the right (or the upper) complex half-plane, then: where, that is, A2(\mathbb{C}+) is isometrically isomorphic to the weighted Lp1/t (0,∞) space (via the Laplace transform). The weighted Bergman space Ap(D) is defined in an analogous way, i.e., provided that w : D → [0, ∞) is chosen in such way, that A^p_w(D) is a Banach space (or a Hilbert space, if ). In case where, by a weighted Bergman space A^p_\alpha we mean the space of all analytic functions f such that: and similarly on the right half-plane (i.e., ) we have: and this space is isometrically isomorphic, via the Laplace transform, to the space, where: (here Γ denotes the Gamma function). Further generalisations are sometimes considered, for example A^2_\nu denotes a weighted Bergman space (often called a Zen space ) with respect to a translation-invariant positive regular Borel measure \nu on the closed right complex half-plane, that is:

Reproducing kernels

The reproducing kernel k_z^{A^2} of A2 at point is given by: and similarly, for we have: In general, if \varphi maps a domain \Omega conformally onto a domain D, then: In weighted case we have: and: