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 < ∞, the Bergman space A^p(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 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:

[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 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:

[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 ℓ^p(1/(n + 1)) space. 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 L^p_1/t (0,∞) space (via the Laplace transform).

The weighted Bergman space A^p(D) is defined in an analogous way, 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] 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:

[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], 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) 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_{\varepsilon\gt 0} \int_{\overline{\mathbb{C}_+}} |f(z+\varepsilon)|^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:

[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:

[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:

[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:

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

and:

[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]

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, https://www.springer.com/mathematics/analysis/book/978-0-387-98791-0 
• Hazewinkel, Michiel, ed. (2001), "Bergman spaces", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=B/b120130 .

See also

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