# Bergman space

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 A$^p$(D) is the space of all holomorphic functions $f$ in D for which the p-norm is finite:
    $\|f\|_{A^{p}(D)}:=\left(\int _{D}|f(x+iy)|^{p}\,\mathrm {d} x\,\mathrm {d} y\right)^{1/p}<\infty .$
The quantity $\|f\|_{A^{p}(D)}$ is called the norm of the function f; it is a true norm if $p\geq 1$. Thus A$^p$(D) is the subspace of holomorphic functions that are in the space L$^p$(D). The Bergman spaces are Banach spaces, which is a consequence of the estimate, valid on compact subsets K of D:
| $\sup _{z\in K}|f(z)|\leq C_{K}\|f\|_{L^{p}(D)}.$ | (1) |
Thus convergence of a sequence of holomorphic functions in L$^p$(D) implies also compact convergence, and so the limit function is also holomorphic.
If p = 2, then A$^p$(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:
    $\|f\|_{A^{p}(D)}:=\left(\int _{D}|f(z)|^{p}\,dA\right)^{1/p}\;\;\;\;\;(f\in A^{p}(D)),$
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 $\mathbb {D}$ of the complex plane, in which case $A^{p}(\mathbb {D} ):=A^{p}$. In the Hilbert space case, given:$f(z)=\sum _{n=0}^{\infty }a_{n}z^{n}\in A^{2}$, we have:
    $\|f\|_{A^{2}}^{2}:={\frac {1}{\pi }}\int _{\mathbb {D} }|f(z)|^{2}\,dz=\sum _{n=0}^{\infty }{\frac {|a_{n}|^{2}}{n+1}},$
that is, A$^2$ is isometrically isomorphic to the weighted ℓ$^p$(1/(n + 1)) space. In particular the polynomials are dense in A$^2$. Similarly, if D = $\mathbb {C}$+, the right (or the upper) complex half-plane, then:
    $\|F\|_{A^{2}(\mathbb {C} _{+})}^{2}:={\frac {1}{\pi }}\int _{\mathbb {C} _{+}}|F(z)|^{2}\,dz=\int _{0}^{\infty }|f(t)|^{2}{\frac {dt}{t}},$
where $F(z)=\int _{0}^{\infty }f(t)e^{-tz}\,dt$, that is, A$^2$($\mathbb {C}$+) is isometrically isomorphic to the weighted L$^p$$_{1/t}$ (0,∞) space (via the Laplace transform).
The weighted Bergman space A$^p$(D) is defined in an analogous way, i.e.,
    $\|f\|_{A_{w}^{p}(D)}:=\left(\int _{D}|f(x+iy)|^{2}\,w(x+iy)\,dx\,dy\right)^{1/p},$
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 $D=\mathbb {D}$, by a weighted Bergman space $A_{\alpha }^{p}$ we mean the space of all analytic functions f such that:
    $\|f\|_{A_{\alpha }^{p}}:=\left((\alpha +1)\int _{\mathbb {D} }|f(z)|^{p}\,(1-|z|^{2})^{\alpha }dA(z)\right)^{1/p}<\infty ,$
and similarly on the right half-plane (i.e., $A_{\alpha }^{p}(\mathbb {C} _{+})$) we have:
    $\|f\|_{A_{\alpha }^{p}(\mathbb {C} _{+})}:=\left({\frac {1}{\pi }}\int _{\mathbb {C} _{+}}|f(x+iy)|^{p}x^{\alpha }\,dx\,dy\right)^{1/p},$
and this space is isometrically isomorphic, via the Laplace transform, to the space $L^{2}(\mathbb {R} _{+},\,d\mu _{\alpha })$, where:
    $d\mu _{\alpha }:={\frac {\Gamma (\alpha +1)}{2^{\alpha }t^{\alpha +1}}}\,dt$
(here Γ denotes the Gamma function).
Further generalisations are sometimes considered, for example $A_{\nu }^{2}$ 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 ${\overline {\mathbb {C} _{+}}}$, that is:
    $A_{\nu }^{p}:=\left\{f:\mathbb {C} _{+}\longrightarrow \mathbb {C} {\text{ analytic}}\;:\;\|f\|_{A_{\nu }^{p}}:=\left(\sup _{\varepsilon >0}\int _{\overline {\mathbb {C} _{+}}}|f(z+\varepsilon )|^{p}\,d\nu (z)\right)^{1/p}<\infty \right\}.$

## Reproducing kernels

The reproducing kernel $k_{z}^{A^{2}}$ of A$^2$ at point $z\in \mathbb {D}$ is given by:
    $k_{z}^{A^{2}}(\zeta )={\frac {1}{(1-{\overline {z}}\zeta )^{2}}}\;\;\;\;\;(\zeta \in \mathbb {D} ),$
and similarly, for $A^{2}(\mathbb {C} _{+})$ we have:
    $k_{z}^{A^{2}(\mathbb {C} _{+})}(\zeta )={\frac {1}{({\overline {z}}+\zeta )^{2}}}\;\;\;\;\;(\zeta \in \mathbb {C} _{+}),$
In general, if $\varphi$ maps a domain $\Omega$ conformally onto a domain $D$, then:
    $k_{z}^{A^{2}(\Omega )}(\zeta )=k_{\varphi (z)}^{{\mathcal {A}}^{2}(D)}(\varphi (\zeta ))\,{\overline {\varphi '(z)}}\varphi '(\zeta )\;\;\;\;\;(z,\zeta \in \Omega ).$
In weighted case we have:
    $k_{z}^{A_{\alpha }^{2}}(\zeta )={\frac {\alpha +1}{(1-{\overline {z}}\zeta )^{\alpha +2}}}\;\;\;\;\;(z,\zeta \in \mathbb {D} ),$
and:
    $k_{z}^{A_{\alpha }^{2}(\mathbb {C} _{+})}(\zeta )={\frac {2^{\alpha }(\alpha +1)}{({\overline {z}}+\zeta )^{\alpha +2}}}\;\;\;\;\;(z,\zeta \in \mathbb {C} _{+}).$