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
in D for which the p-norm is finite: The quantity
is called the norm of the function f; it is a true norm if
Thus 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 p = 2, then Ap(D) is a reproducing kernel Hilbert space, whose kernel is given by the Bergman kernel.
If the domain D is bounded, then the norm is often given by: where
is a normalised Lebesgue measure of the complex plane, i.e. dA = dz/Area(D).
Alternatively dA = dz/π is used, regardless of the area of D. 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, A2 is isometrically isomorphic to the weighted ℓp(1/(n + 1)) space.
+, the right (or the upper) complex half-plane, then: where
+) is isometrically isomorphic to the weighted Lp1/t (0,∞) space (via the Laplace transform).
[2][3] The weighted Bergman space Ap(D) is defined in an analogous way,[1] i.e., provided that w : D → [0, ∞) is chosen in such way, that
, by a weighted Bergman space
[4] we mean the space of all analytic functions f such that: and similarly on the right half-plane (i.e.,
) we have:[5] and this space is isometrically isomorphic, via the Laplace transform, to the space
,[6][7] where: (here Γ denotes the Gamma function).
denotes a weighted Bergman space (often called a Zen space[3]) with respect to a translation-invariant positive regular Borel measure
on the closed right complex half-plane
maps a domain
conformally onto a domain