Bergman space

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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 [math]\displaystyle{ f }[/math] in D for which the p-norm is finite:

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

The quantity [math]\displaystyle{ \|f\|_{A^p(D)} }[/math] is called the norm of the function f; it is a true norm if [math]\displaystyle{ p \geq 1 }[/math]. 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:

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

 

 

 

 

(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

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

where [math]\displaystyle{ A }[/math] 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 [math]\displaystyle{ \mathbb{D} }[/math] of the complex plane, in which case [math]\displaystyle{ A^p(\mathbb{D}):=A^p }[/math]. In the Hilbert space case, given [math]\displaystyle{ f(z)= \sum_{n=0}^\infty a_n z^n \in A^2 }[/math], we have

[math]\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}, }[/math]

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 = [math]\displaystyle{ \mathbb{C} }[/math]+, the right (or the upper) complex half-plane, then

[math]\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}, }[/math]

where [math]\displaystyle{ F(z)= \int_0^\infty f(t)e^{-tz} \, dt }[/math], that is, A2([math]\displaystyle{ \mathbb{C} }[/math]+) 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.

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

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

[math]\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, }[/math]

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

[math]\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}, }[/math]

and this space is isometrically isomorphic, via the Laplace transform, to the space [math]\displaystyle{ L^2(\mathbb{R}_+, \, d\mu_\alpha) }[/math],[6][7] where

[math]\displaystyle{ d\mu_\alpha := \frac{\Gamma(\alpha+1)}{2^\alpha t^{\alpha+1}} \, dt }[/math]

(here Γ denotes the Gamma function).

Further generalisations are sometimes considered, for example [math]\displaystyle{ A^2_\nu }[/math] denotes a weighted Bergman space (often called a Zen space[3]) with respect to a translation-invariant positive regular Borel measure [math]\displaystyle{ \nu }[/math] on the closed right complex half-plane [math]\displaystyle{ \overline{\mathbb{C}_+} }[/math], that is

[math]\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\}. }[/math]

Reproducing kernels

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

[math]\displaystyle{ k_z^{A^2}(\zeta)=\frac{1}{(1-\overline{z}\zeta)^2} \; \; \; \; \; (\zeta \in \mathbb{D}), }[/math]

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

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

In general, if [math]\displaystyle{ \varphi }[/math] maps a domain [math]\displaystyle{ \Omega }[/math] conformally onto a domain [math]\displaystyle{ D }[/math], then[1]

[math]\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). }[/math]

In weighted case we have[4]

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

and[5]

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

References

  1. 1.0 1.1 1.2 1.3 Duren, Peter L.; Schuster, Alexander (2004), Bergman spaces, Mathematical Series and Monographs, American Mathematical Society, ISBN 978-0-8218-0810-8, https://www.ams.org/bookpages/surv-100 
  2. Duren, Peter L. (1969), Extension of a theorem of Carleson, 75, Bulletin of the American Mathematical Society, pp. 143–146, https://www.ams.org/journals/bull/1969-75-01/S0002-9904-1969-12181-6/S0002-9904-1969-12181-6.pdf 
  3. 3.0 3.1 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. 4.0 4.1 Cowen, Carl; MacCluer, Barbara (1995-04-27), Composition Operators on Spaces of Analytic Functions, Studies in Advanced Mathematics, CRC Press, p. 27, ISBN 9780849384929, https://www.crcpress.com/Composition-Operators-on-Spaces-of-Analytic-Functions/Jr-MacCluer/9780849384929 
  5. 5.0 5.1 5.2 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, http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=8267653&fileId=S0013091509001412 
  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, http://blms.oxfordjournals.org/content/39/3/459.full.pdf+html 
  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, http://www.theta.ro/jot/archive/2009-062-001/2009-062-001-010.pdf 

Further reading

See also