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 A^p(D) is the space of all holomorphic functions $f$ in D for which the p-norm is finite:

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

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:

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

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

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

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

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

where $F(z)= \int_0^\infty f(t)e^{-tz} \, dt$, that is, A²(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.

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

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

$\|f\|_{A^p_\alpha} := \left( \frac{1}{\pi}\int_\mathbb{D} |f(z)|^p \, (1-|z|^p)^\alpha dz \right)^{1/p} < \infty,$

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

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

and this space is isometrically isomorphic, via the Laplace transform, to the space $L^2(\mathbb{R}_+, \, d\mu_\alpha)$,^[6][7] 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 $A^2_\nu$ denotes a weighted Bergman space (often called a Zen space^[3]) with respect to a translation-invariant positive regular Borel measure $\nu$ on the closed right complex half-plane $\overline{\mathbb{C}_+}$, that is

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

Reproducing kernels

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

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

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

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

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

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

In weighted case we have^[4]

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

and^[5]

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

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 (PDF) 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 264 (3), Journal of Functional Analysis, pp. 783–814
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 (2), 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 (PDF) 39 (3), 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 (PDF) 62 (1), Journal of Operator Theory, pp. 199–214

Further reading

• Bergman, Stefan (1970), The kernel function and conformal mapping, Mathematical Surveys 5 (2nd ed.), American Mathematical Society
• Hedenmalm, H.; Korenblum, B.; Zhu, K. (2000), Theory of Bergman Spaces, Springer, ISBN 978-0-387-98791-0
• Richter, Stefan (2001), "Bergman spaces", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4.

See also

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