BREAKING NEWS

Summary

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 $f$ 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 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:

$\sup _{z\in K}|f(z)|\leq C_{K}\|f\|_{L^{p}(D)}.$ (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

$\|f\|_{A^{p}(D)}:=\left(\int _{D}|f(z)|^{p}\,dA\right)^{1/p}\;\;\;\;\;(f\in A^{p}(D)),$ 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, A2 is isometrically isomorphic to the weighted p(1/(n+1)) space. In particular the polynomials are dense in A2. 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, A2($\mathbb {C}$ +) is isometrically isomorphic to the weighted Lp1/t (0,∞) space (via the Laplace transform).

The weighted Bergman space Ap(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

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

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

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

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

1. ^ a b c d 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. ^ a b Jacob, Brigit; Partington, Jonathan R.; Pott, Sandra (2013-02-01). "On Laplace-Carleson embedding theorems". Journal of Functional Analysis. 264 (3): 783–814. arXiv:1201.1021. doi:10.1016/j.jfa.2012.11.016. S2CID 7770226.
4. ^ a b 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. ^ a b c 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, archived from the original on 2015-12-24
7. ^ Gallrado-Gutiérez, Eva A.; Partington, Jonathan R.; Segura, Dolores (2009), Cyclic vectors and invariant subspaces for Bergman and Dirichlet shifts (PDF), 62, Journal of Operator Theory, pp. 199–214