Bergman space

In complex analysis, functional analysis and operator theory, a Bergman space, named after Stefan Bergman, 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 $0 < p < &infin;$, the Bergman space $A^{p}(D)$ is the space of all holomorphic functions $$f$$ in D for which the p-norm is finite:


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

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 Lp(D). The Bergman spaces are Banach spaces, which is a consequence of the estimate, valid on compact subsets K of D:

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


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

that is, $A^{2}$ is isometrically isomorphic to the weighted ℓp(1/(n + 1)) space. In particular the polynomials are dense in $A^{2}$. Similarly, if $D = $\mathbb{C}$_{+}$, 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, $A^{2}($\mathbb{C}$_{+})$ is isometrically isomorphic to the weighted Lp1/t (0,∞) space (via the Laplace transform).

The weighted Bergman space $A^{p}(D)$ is defined in an analogous way, i.e.,


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

provided that $w : D → [0, &infin;)$ 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 $$A^p_\alpha$$ we mean the space of all analytic functions $f$ such that:


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

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


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

and this space is isometrically isomorphic, via the Laplace transform, to the space $$L^2(\mathbb{R}_+, \, d\mu_\alpha)$$, where:


 * $$d\mu_\alpha := \frac{\Gamma(\alpha+1)}{2^\alpha t^{\alpha+1}} \, dt$$

(here $&Gamma;$ denotes the Gamma function).

Further generalisations are sometimes considered, for example $$A^2_\nu$$ 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:


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

Reproducing kernels
The reproducing kernel $$k_z^{A^2}$$ of $A^{2}$ 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:


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

In weighted case we have:


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

and:


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