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 and also absolutely integrable. Specifically, for $0 < p < \infty$ , 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)} := {(\int_{D} | f (x + i y) |^{p} d x d y)}^{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 of the space L^p(D). The Bergman spaces are Banach spaces for $0 < p < \infty$ , which is a consequence of the following estimate that is valid on compact subsets K of D: $sup_{z \in K} | f (z) | \leq C_{K} ‖ f ‖_{L^{p} (D)} .$ Convergence of a sequence of holomorphic functions in $L p (D)$ thus implies 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)} := {(\int_{D} | f (z) |^{p} d A)}^{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 $𝔻$ of the complex plane, in which case $A^{p} (𝔻) := A^{p}$ . If $p = 2$ , given an element $f (z) = \sum_{n = 0}^{\infty} a_{n} z^{n} \in A^{2}$ , we have

‖ f ‖_{A^{2}}^{2} := \frac{1}{π} \int_{𝔻} | f (z) |^{2} d z = \sum_{n = 0}^{\infty} \frac{| a_{n} |^{2}}{n + 1},

that is, $A 2$ is isometrically isomorphic to the weighted ℓ^p(1/(n + 1)) space. In particular, not only are the polynomials dense in $A 2$ , but every function $f \in A^{2}$ can be uniformly approximated by radial dilations of functions $g$ holomorphic on a disk $D_{R} (0)$ , where $R > 1$ and the radial dilation of a function is defined by $g_{r} (z) := g (r z)$ for $0 < r < 1$ .

Similarly, if $ℂ$ , the right (or the upper) complex half-plane, then:

‖ F ‖_{A^{2} (ℂ_{+})}^{2} := \frac{1}{π} \int_{ℂ_{+}} | F (z) |^{2} d z = \int_{0}^{\infty} | f (t) |^{2} \frac{d t}{t},

where $F (z) = \int_{0}^{\infty} f (t) e^{- t z} d t$ , that is, $A^{2} (ℂ_{+})$ 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.,

‖ f ‖_{A_{w}^{p} (D)} := {(\int_{D} | f (x + i y) |^{2} w (x + i y) d x d y)}^{1 / p},

provided that $w : D \to [0, \infty)$ is chosen in such way, that $A_{w}^{p} (D)$ is a Banach space (or a Hilbert space, if $p = 2$ ). In case where $D = 𝔻$ , by a weighted Bergman space $A_{α}^{p}$ we mean the space of all analytic functions $f$ such that:

‖ f ‖_{A_{α}^{p}} := {((α + 1) \int_{𝔻} | f (z) |^{p} (1 - | z |^{2})^{α} d A (z))}^{1 / p} < \infty,

and similarly on the right half-plane (i.e., $A_{α}^{p} (ℂ_{+})$ ) we have:

‖ f ‖_{A_{α}^{p} (ℂ_{+})} := {(\frac{1}{π} \int_{ℂ_{+}} | f (x + i y) |^{p} x^{α} d x d y)}^{1 / p},

and this space is isometrically isomorphic, via the Laplace transform, to the space $L^{2} (ℝ_{+}, d μ_{α})$ , where:

d μ_{α} := \frac{Γ (α + 1)}{2^{α} t^{α + 1}} d t .

Here $Γ$ denotes the Gamma function.

Further generalisations are sometimes considered, for example $A_{ν}^{2}$ denotes a weighted Bergman space (often called a Zen space) with respect to a translation-invariant positive regular Borel measure $ν$ on the closed right complex half-plane $\overline{ℂ_{+}}$ , that is:

A_{ν}^{p} := {f : ℂ_{+} ⟶ ℂ analytic : ‖ f ‖_{A_{ν}^{p}} := {(sup_{ε > 0} \int_{\overline{ℂ_{+}}} | f (z + ε) |^{p} d ν (z))}^{1 / p} < \infty} .

It is possible to generalise $A^{2}$ to the (weighted) Bergman space of vector-valued functions, defined by $A_{α}^{2} (𝔻; ℋ) := {f : 𝔻 \to ℋ | f analytic and ‖ f ‖_{2, α} < + \infty},$ and the norm on this space is given as $‖ f ‖_{2, α} = {(\int_{𝔻} ‖ f (z) ‖_{ℋ}^{2} d μ_{α} (z))}^{\frac{1}{2}} .$ The measure $μ_{α}$ is the same as the previous measure on the weighted Bergman space over the unit disk, $ℋ$ is a Hilbert space. In this case, the space is a Banach space for $0 < p \leq \infty$ and a (reproducing kernel) Hilbert space when $p = 2$ .

Reproducing kernels

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

k_{z}^{A^{2}} (ζ) = \frac{1}{(1 - \overline{z} ζ)^{2}} (ζ \in 𝔻),

and similarly, for $A^{2} (ℂ_{+})$ we have:

k_{z}^{A^{2} (ℂ_{+})} (ζ) = \frac{1}{(\overline{z} + ζ)^{2}} (ζ \in ℂ_{+}),

In general, if $φ$ maps a domain $Ω$ conformally onto a domain $D$ , then:

k_{z}^{A^{2} (Ω)} (ζ) = k_{φ (z)}^{𝒜^{2} (D)} (φ (ζ)) \overline{φ^{'} (z)} φ^{'} (ζ) (z, ζ \in Ω) .

In weighted case we have:

k_{z}^{A_{α}^{2}} (ζ) = \frac{α + 1}{(1 - \overline{z} ζ)^{α + 2}} (z, ζ \in 𝔻),

and:

k_{z}^{A_{α}^{2} (ℂ_{+})} (ζ) = \frac{2^{α} (α + 1)}{(\overline{z} + ζ)^{α + 2}} (z, ζ \in ℂ_{+}) .

In any reproducing kernel Bergman space, functions obey a certain property. It is called the reproducing property. This is expressed as a formula as follows: For any function $f \in A^{2}$ (respectively other Bergman spaces that are RKHS), it is true that $f (z) = ⟨ f, k_{z}^{A^{2}} ⟩_{2} = \int_{𝔻} \frac{f (ζ)}{(1 - z \overline{ζ})^{2}} d A .$

References

1 2 3 4 Duren, Peter L.; Schuster, Alexander (2004), Bergman spaces, Mathematical Series and Monographs, American Mathematical Society, ISBN 978-0-8218-0810-8
↑ Duren, Peter L. (1969), Extension of a theorem of Carleson (PDF), vol. 75, Bulletin of the American Mathematical Society, pp. 143–146
1 2 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.
1 2 Cowen, Carl; MacCluer, Barbara (1995-04-27), Composition Operators on Spaces of Analytic Functions, Studies in Advanced Mathematics, CRC Press, p. 27, ISBN 9780849384929
1 2 3 Elliott, Sam J.; Wynn, Andrew (2011), "Composition Operators on the Weighted Bergman Spaces of the Half-Plane", Proceedings of the Edinburgh Mathematical Society, 54 (2): 374–379, arXiv:0910.0408, doi:10.1017/S0013091509001412, S2CID 18811195
↑ 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, vol. 39, Bulletin of the London Mathematical Society, pp. 459–466, archived from the original on 2015-12-24
↑ Gallrado-Gutiérez, Eva A.; Partington, Jonathan R.; Segura, Dolores (2009), Cyclic vectors and invariant subspaces for Bergman and Dirichlet shifts (PDF), vol. 62, Journal of Operator Theory, pp. 199–214
↑ Aleman, Alexandru; Constantin, Olivia (2004). "Hankel operators on Bergman spaces and similarity to contractions". International Mathematics Research Notices. 2004 (35): 1785–1801. doi:10.1155/S1073792804140105. ISSN 1687-0247.{{cite journal}}: CS1 maint: unflagged free DOI (link)

Bergman space

Special cases and generalisations

Reproducing kernels

References

Further reading

See also