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 , the Bergman space is the space of all holomorphic functions $f$ in ''D'' for which the p-norm is finite:

:$\, f\, _ := \left(\int_D , f(x+iy), ^p\,\mathrm dx\,\mathrm dy\right)^ < \infty.$

The quantity $\, f\, _$ is called the ''norm'' of the function ; it is a true norm if $p \geq 1$. Thus 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'':

Thus convergence of a sequence of holomorphic functions in implies also compact convergence, and so the limit function is also holomorphic.

If , then is a reproducing kernel Hilbert space, whose kernel is given by the Bergman kernel.

Special cases and generalisations

If the domain is bounded, then the norm is often given by

:$\, f\, _ := \left(\int_D , f(z), ^p\,dA\right)^ \; \; \; \; \; (f \in A^p(D)),$

where $A$ is a normalised Lebesgue measure of the complex plane, i.e. . Alternatively is used, regardless of the area of .
The Bergman space is usually defined on the open unit disk $\mathbb$ of the complex plane, in which case $A^p(\mathbb):=A^p$. In the Hilbert space case, given $f(z)= \sum_^\infty a_n z^n \in A^2$, we have

:$\, f\, ^2_ := \frac \int_\mathbb , f(z), ^2 \, dz = \sum_^\infty \frac,$

that is, is isometrically isomorphic to the weighted ''ℓ^p(1/(n+1))'' space. In particular the polynomials are dense in . Similarly, if , the right (or the upper) complex half-plane, then

:$\, F\, ^2_ := \frac \int_ , F(z), ^2 \, dz = \int_0^\infty , f(t), ^2\frac,$

where $F(z)= \int_0^\infty f(t)e^ \, dt$, that is, is isometrically isomorphic to the weighted ''L^p_1/t (0,∞)'' space (via the Laplace transform).

The weighted Bergman space is defined in an analogous way, i.e.

:$\, f\, _ := \left( \int_D , f(x+iy), ^2 \, w(x+iy) \, dx \, dy \right)^,$

provided that is chosen in such way, that $A^p_w(D)$ is a Banach space (or a Hilbert space, if ). In case where $D= \mathbb$, by a weighted Bergman space $A^p_\alpha$ we mean the space of all analytic functions such that

:$\, f\, _ := \left( (\alpha+1)\int_\mathbb , f(z), ^p \, (1-, z, ^2)^\alpha dA(z) \right)^ < \infty,$

and similarly on the right half-plane (i.e. $A^p_\alpha(\mathbb_+)$) we have

:$\, f\, _ := \left( \frac\int_ , f(x+iy), ^p x^\alpha \, dx \, dy \right)^,$

and this space is isometrically isomorphic, via the Laplace transform, to the space $L^2(\mathbb_+, \, d\mu_\alpha)$, where

:$d\mu_\alpha := \frac \, dt$

(here 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 $\overline$, that is

:$A^p_\nu := \left\.$

Reproducing kernels

The reproducing kernel $k_z^$ of at point $z \in \mathbb$ is given by

:$k_z^(\zeta)=\frac \; \; \; \; \; (\zeta \in \mathbb),$

and similarly for $A^2(\mathbb_+)$ we have

:$k_z^(\zeta)=\frac \; \; \; \; \; (\zeta \in \mathbb_+),$.

In general, if $\varphi$ maps a domain $\Omega$ conformally onto a domain $D$, then

:$k^_z (\zeta) = k^_(\varphi(\zeta)) \, \overline\varphi'(\zeta) \; \; \; \; \; (z, \zeta \in \Omega).$

In weighted case we have

:$k_z^ (\zeta) = \frac \; \; \; \; \; (z, \zeta \in \mathbb),$

and

:$k_z^ (\zeta) = \frac \; \; \; \; \; (z, \zeta \in \mathbb_+).$

References

Further reading

*
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See also

*Bergman kernel
*Banach space
*Hilbert space
*Reproducing kernel Hilbert space
*Hardy space
* Dirichlet space

Complex analysis
Functional analysis
Operator theory

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