# Bergman space

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 $\displaystyle{ f }$ in D for which the p-norm is finite:

$\displaystyle{ \|f\|_{A^p(D)} := \left(\int_D |f(x+iy)|^p\,\mathrm dx\,\mathrm dy\right)^{1/p} \lt \infty. }$

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 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:

$\displaystyle{ \sup_{z\in K} |f(z)| \le C_K\|f\|_{L^p(D)}. }$

(1)

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

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

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 $\displaystyle{ A^p(\mathbb{D}):=A^p }$. In the Hilbert space case, given $\displaystyle{ f(z)= \sum_{n=0}^\infty a_n z^n \in A^2 }$, we have

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

that is, A2 is isometrically isomorphic to the weighted p(1/(n+1)) space.[1] In particular the polynomials are dense in A2. Similarly, if D = $\displaystyle{ \mathbb{C} }$+, the right (or the upper) complex half-plane, then

$\displaystyle{ \|F\|^2_{A^2(\mathbb{C}_+)} := \frac{1}{\pi} \int_{\mathbb{C}_+} |F(z)|^2 \, dz = \int_0^\infty |f(t)|^2\frac{dt}{t}, }$

where $\displaystyle{ F(z)= \int_0^\infty f(t)e^{-tz} \, dt }$, that is, A2($\displaystyle{ \mathbb{C} }$+) is isometrically isomorphic to the weighted 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.

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

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

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

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

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

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 $\displaystyle{ A^2_\nu }$ 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

$\displaystyle{ A^p_\nu := \left\{ f : \mathbb{C}_+ \longrightarrow \mathbb{C} \; \text{analytic} \; : \; \|f\|_{A^p_\nu} := \left( \sup_{\epsilon\gt 0} \int_{\overline{\mathbb{C}_+}} |f(z+\epsilon)|^p \, d\nu(z) \right)^{1/p} \lt \infty \right\}. }$

## 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]

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

In weighted case we have[4]

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

and[5]

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

## References

1. Duren, Peter L.; Schuster, Alexander (2004), Bergman spaces, Mathematical Series and Monographs, American Mathematical Society, ISBN 978-0-8218-0810-8
2. Duren, Peter L. (1969), Extension of a theorem of Carleson, 75, Bulletin of the American Mathematical Society, pp. 143–146
3. Jacob, Brigit; Partington, Jonathan R.; Pott, Sandra (2013-02-01). "On Laplace-Carleson embedding theorems". Journal of Functional Analysis 264 (3): 783–814. doi:10.1016/j.jfa.2012.11.016.
4. Cowen, Carl; MacCluer, Barbara (1995-04-27), Composition Operators on Spaces of Analytic Functions, Studies in Advanced Mathematics, CRC Press, p. 27, ISBN 9780849384929
5. Elliott, Sam J.; Wynn, Andrew (2011), Composition Operators on the Weighted Bergman Spaces of the Half-Plane, 54, Proceedings of the Edinburgh Mathematical Society, pp. 374–379
6. Duren, Peter L.; Gallardo-Gutiérez, Eva A.; Montes-Rodríguez, Alfonso (2007-06-03), A Paley-Wiener theorem for Bergman spaces with application to invariant subspaces, 39, Bulletin of the London Mathematical Society, pp. 459–466
7. Gallrado-Gutiérez, Eva A.; Partington, Jonathan R.; Segura, Dolores (2009), Cyclic vectors and invariant subspaces for Bergman and Dirichlet shifts, 62, Journal of Operator Theory, pp. 199–214