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:

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

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 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:

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

(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

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

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

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

that is, A2 is isometrically isomorphic to the weighted p(1/(n+1)) space.[1] In particular the polynomials are dense in A2. Similarly, if D = 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(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.

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

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

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

and similarly on the right half-plane (i.e. ${\displaystyle A_{\alpha }^{p}(\mathbb {C} _{+})}$) 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

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

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]

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

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]

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

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, 264 (3), Journal of Functional Analysis, pp. 783–814
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 (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, 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