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 0 < p < ∞, the Bergman space Ap(D) is the space of all holomorphic functions f in D for which the p-norm is finite:
∥ f ∥ A p ( D ) := ( ∫ D | f ( x + i y ) | p d x d y ) 1 / p < ∞ . The quantity ∥ f ∥ A p ( D ) is called the norm of the function f; it is a true norm if p ≥ 1 . 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.
Special cases and generalisations
If the domain D is bounded, then the norm is often given by
∥ f ∥ A p ( D ) := ( ∫ D | f ( z ) | p d A ) 1 / p ( f ∈ A p ( D ) ) , where A 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 D of the complex plane, in which case A p ( C ) := A p . In the Hilbert space case, given f ( z ) = ∑ n = 0 ∞ a n z n ∈ A 2 , we have
∥ f ∥ A 2 2 := 1 π ∫ D | f ( z ) | 2 d z = ∑ n = 0 ∞ | a n | 2 n + 1 , that is, A2 is isometrically isomorphic to the weighted ℓp(1/(n+1)) space. In particular the polynomials are dense in A2. Similarly, if D = ℂ+, the right (or the upper) complex half-plane, then
∥ F ∥ A 2 ( C + ) 2 := 1 π ∫ C + | F ( z ) | 2 d z = ∫ 0 ∞ | f ( t ) | 2 d t t , where F ( z ) = ∫ 0 ∞ f ( t ) e − t z d t , that is, A2(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.
∥ f ∥ A w p ( D ) := ( ∫ D | f ( x + i y ) | 2 w ( x + i y ) d x d y ) 1 / p , provided that w : D → [0, ∞) is chosen in such way, that A w p ( D ) is a Banach space (or a Hilbert space, if p = 2). In case where D = D , by a weighted Bergman space A α p we mean the space of all analytic functions f such that
∥ f ∥ A α p := ( 1 π ∫ D | f ( z ) | p ( 1 − | z | p ) α d z ) 1 / p < ∞ , and similarly on the right half-plane (i.e. A α p ( C + ) ) we have
∥ f ∥ A α p ( C + ) := ( 1 π ∫ C + | f ( x + i y ) | p x α d x d y ) 1 / p , and this space is isometrically isomorphic, via the Laplace transform, to the space L 2 ( R + , d μ α ) , where
d μ α := Γ ( α + 1 ) 2 α t α + 1 d t (here Γ denotes the Gamma function).
Further generalisations are sometimes considered, for example A ν 2 denotes a weighted Bergman space (often called a Zen space) with respect to a translation-invariant positive regular Borel measure ν on the closed right complex half-plane C + ¯ , that is
A ν p := { f : C + ⟶ C analytic : ∥ f ∥ A ν p := ( sup ϵ > 0 ∫ C + ¯ | f ( z + ϵ ) | p d ν ( z ) ) 1 / p < ∞ } . The reproducing kernel k z A 2 of A2 at point z ∈ D is given by
k z A 2 ( ζ ) = 1 ( 1 − z ¯ ζ ) 2 ( ζ ∈ D ) , and similarly for A 2 ( C + ) we have
k z A 2 ( C + ) ( ζ ) = 1 ( z ¯ + ζ ) 2 ( ζ ∈ C + ) , .
In general, if φ maps a domain Ω conformally onto a domain D , then
k z A 2 ( Ω ) ( ζ ) = k φ ( z ) A 2 ( D ) ( φ ( ζ ) ) φ ′ ( z ) ¯ φ ′ ( ζ ) ( z , ζ ∈ Ω ) . In weighted case we have
k z A α 2 ( ζ ) = α + 1 ( 1 − z ¯ ζ ) α + 2 ( z , ζ ∈ D ) , and
k z A α 2 ( C + ) ( ζ ) = 2 α ( α + 1 ) ( z ¯ + ζ ) α + 2 ( z , ζ ∈ C + ) . 