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 and also absolutely integrable. Specifically, for 0 < p < ∞, the Bergman space A^p(D) is the space of all holomorphic functions ${\displaystyle f}$ in D for which the p-norm is finite:

${\displaystyle \|f\|_{A^{p}(D)}:=\left(\int _{D}|f(x+iy)|^{p}\,\mathrm {d} x\,\mathrm {d} y\right)^{1/p}<\infty .}$

The quantity ${\displaystyle \|f\|_{A^{p}(D)}}$ is called the norm of the function f; it is a true norm if ${\displaystyle p\geq 1}$, thus A^p(D) is the subspace of holomorphic functions of the space L^p(D). The Bergman spaces are Banach spaces for ${\displaystyle 0<p<\infty }$, which is a consequence of the following estimate that is valid on compact subsets K of D:$${\displaystyle \sup _{z\in K}|f(z)|\leq C_{K}\|f\|_{L^{p}(D)}.}$$Convergence of a sequence of holomorphic functions in L^p(D) thus implies 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:

${\displaystyle \|f\|_{A^{p}(D)}:=\left(\int _{D}|f(z)|^{p}\,dA\right)^{1/p}\;\;\;\;\;(f\in A^{p}(D)),}$

where ${\displaystyle A}$ 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 ${\displaystyle \mathbb {D} }$ of the complex plane, in which case ${\displaystyle A^{p}(\mathbb {D} ):=A^{p}}$. If ${\displaystyle p=2}$, given an element ${\displaystyle f(z)=\sum _{n=0}^{\infty }a_{n}z^{n}\in A^{2}}$, we have

${\displaystyle \|f\|_{A^{2}}^{2}:={\frac {1}{\pi }}\int _{\mathbb {D} }|f(z)|^{2}\,dz=\sum _{n=0}^{\infty }{\frac {|a_{n}|^{2}}{n+1}},}$

that is, A² is isometrically isomorphic to the weighted ℓ^p(1/(n + 1)) space.^[1] In particular, not only are the polynomials dense in A², but every function ${\displaystyle f\in A^{2}}$ can be uniformly approximated by radial dilations of functions ${\displaystyle g}$ holomorphic on a disk ${\displaystyle D_{R}(0)}$, where ${\displaystyle R>1}$ and the radial dilation of a function is defined by ${\displaystyle g_{r}(z):=g(rz)}$ for ${\displaystyle 0<r<1}$.

Similarly, if D = ${\displaystyle \mathbb {C} }$₊, the right (or the upper) complex half-plane, then:

${\displaystyle \|F\|_{A^{2}(\mathbb {C} _{+})}^{2}:={\frac {1}{\pi }}\int _{\mathbb {C} _{+}}|F(z)|^{2}\,dz=\int _{0}^{\infty }|f(t)|^{2}{\frac {dt}{t}},}$

where ${\displaystyle F(z)=\int _{0}^{\infty }f(t)e^{-tz}\,dt}$, that is, A²(${\displaystyle \mathbb {C} }$₊) is isometrically isomorphic to the weighted L^p_1/t (0,∞) space (via the Laplace transform).^[2][3]

The weighted Bergman space A^p(D) is defined in an analogous way,^[1] i.e.,

${\displaystyle \|f\|_{A_{w}^{p}(D)}:=\left(\int _{D}|f(x+iy)|^{2}\,w(x+iy)\,dx\,dy\right)^{1/p},}$

provided that w : D → [0, ∞) is chosen in such way, that ${\displaystyle A_{w}^{p}(D)}$ is a Banach space (or a Hilbert space, if p = 2). In case where ${\displaystyle D=\mathbb {D} }$, by a weighted Bergman space ${\displaystyle A_{\alpha }^{p}}$^[4] we mean the space of all analytic functions f such that:

${\displaystyle \|f\|_{A_{\alpha }^{p}}:=\left((\alpha +1)\int _{\mathbb {D} }|f(z)|^{p}\,(1-|z|^{2})^{\alpha }dA(z)\right)^{1/p}<\infty ,}$

and similarly on the right half-plane (i.e., ${\displaystyle A_{\alpha }^{p}(\mathbb {C} _{+})}$) we have:^[5]

${\displaystyle \|f\|_{A_{\alpha }^{p}(\mathbb {C} _{+})}:=\left({\frac {1}{\pi }}\int _{\mathbb {C} _{+}}|f(x+iy)|^{p}x^{\alpha }\,dx\,dy\right)^{1/p},}$

and this space is isometrically isomorphic, via the Laplace transform, to the space ${\displaystyle L^{2}(\mathbb {R} _{+},\,d\mu _{\alpha })}$,^[6][7] where:

${\displaystyle d\mu _{\alpha }:={\frac {\Gamma (\alpha +1)}{2^{\alpha }t^{\alpha +1}}}\,dt.}$

Here Γ denotes the Gamma function.

Further generalisations are sometimes considered, for example ${\displaystyle A_{\nu }^{2}}$ denotes a weighted Bergman space (often called a Zen space^[3]) with respect to a translation-invariant positive regular Borel measure ${\displaystyle \nu }$ on the closed right complex half-plane ${\displaystyle {\overline {\mathbb {C} _{+}}}}$, that is:

${\displaystyle A_{\nu }^{p}:=\left\{f:\mathbb {C} _{+}\longrightarrow \mathbb {C} {\text{ analytic}}\;:\;\|f\|_{A_{\nu }^{p}}:=\left(\sup _{\varepsilon >0}\int _{\overline {\mathbb {C} _{+}}}|f(z+\varepsilon )|^{p}\,d\nu (z)\right)^{1/p}<\infty \right\}.}$

It is possible to generalise ${\displaystyle A^{2}}$ to the (weighted) Bergman space of vector-valued functions^[8], defined by$${\displaystyle A_{\alpha }^{2}(\mathbb {D}$$ ;{\mathcal {H}}):=\left\{\,f\colon \,\mathbb {D} \to {\mathcal {H}}\;|\;f{\text{ analytic and }}\|f\|_{2,\alpha }<+\infty \right\},} and the norm on this space is given as$${\displaystyle \|f\|_{2,\alpha }=\left(\int _{\mathbb {D} }\|f(z)\|_{\mathcal {H}}^{2}d\mu _{\alpha }(z)\right)^{\frac {1}{2}}.}$$The measure ${\displaystyle \mu _{\alpha }}$ is the same as the previous measure on the weighted Bergman space over the unit disk, ${\displaystyle {\mathcal {H}}}$ is a Hilbert space. In this case, the space is a Banach space for ${\displaystyle 0<p\leq \infty }$ and a (reproducing kernel) Hilbert space when ${\displaystyle p=2}$.

Reproducing kernels

The reproducing kernel ${\displaystyle k_{z}^{A^{2}}}$ of A² at point ${\displaystyle z\in \mathbb {D} }$ is given by:^[1]

${\displaystyle k_{z}^{A^{2}}(\zeta )={\frac {1}{(1-{\overline {z}}\zeta )^{2}}}\;\;\;\;\;(\zeta \in \mathbb {D} ),}$

and similarly, for ${\displaystyle A^{2}(\mathbb {C} _{+})}$ we have:^[5]

${\displaystyle k_{z}^{A^{2}(\mathbb {C} _{+})}(\zeta )={\frac {1}{({\overline {z}}+\zeta )^{2}}}\;\;\;\;\;(\zeta \in \mathbb {C} _{+}),}$

In general, if ${\displaystyle \varphi }$ maps a domain ${\displaystyle \Omega }$ conformally onto a domain ${\displaystyle D}$, then:^[1]

${\displaystyle k_{z}^{A^{2}(\Omega )}(\zeta )=k_{\varphi (z)}^{{\mathcal {A}}^{2}(D)}(\varphi (\zeta ))\,{\overline {\varphi '(z)}}\varphi '(\zeta )\;\;\;\;\;(z,\zeta \in \Omega ).}$

In weighted case we have:^[4]

${\displaystyle k_{z}^{A_{\alpha }^{2}}(\zeta )={\frac {\alpha +1}{(1-{\overline {z}}\zeta )^{\alpha +2}}}\;\;\;\;\;(z,\zeta \in \mathbb {D} ),}$

and:^[5]

${\displaystyle k_{z}^{A_{\alpha }^{2}(\mathbb {C} _{+})}(\zeta )={\frac {2^{\alpha }(\alpha +1)}{({\overline {z}}+\zeta )^{\alpha +2}}}\;\;\;\;\;(z,\zeta \in \mathbb {C} _{+}).}$

In any reproducing kernel Bergman space, functions obey a certain property. It is called the reproducing property. This is expressed as a formula as follows: For any function ${\displaystyle f\in A^{2}}$ (respectively other Bergman spaces that are RKHS), it is true that$${\displaystyle f(z)=\langle f,k_{z}^{A^{2}}\rangle _{2}=\int _{\mathbb {D} }{\frac {f(\zeta )}{(1-z{\overline {\zeta }})^{2}}}dA.}$$