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The Bergman space $A^{2} (Ω)$ and the Bergman kernel

Definition

Assume the Axiom of Countable Choice ${A C}_{ω}$ (The Axiom of Countable Choice ( ${A C}_{ω}$ )), let $m \geq 1$ , and let $Ω \subseteq C^{m}$ be a nonempty open set. Read $C^{m}$ as $R^{2 m}$ through Complex $m$ -space and its real coordinate dictionary. Give $Ω$ the trace of the Lebesgue sigma-algebra and the restricted measure $λ_{Ω}$ induced by $λ_{2 m}$ , namely $λ_{Ω} (E) : = λ_{2 m} (E)$ for $E \in L (R^{2 m}) ∣_{Ω}$ ; write $L^{2} (Ω)$ for the resulting complex $L^{2}$ space.

Let $A^{2} (Ω)$ be the holomorphic functions $f$ on $Ω$ with $\int_{Ω} ∣ f ∣^{2} d λ_{Ω} < \infty$ , and let $A^{2} (Ω) : = {[f]_{L^{2} (Ω)} : f \in A^{2} (Ω)} .$ The argument below shows that each such class has a unique holomorphic representative and that $A^{2} (Ω)$ is a closed complex linear subspace of $L^{2} (Ω)$ . Give it the inherited inner product $⟨ [f], [g] ⟩ : = \int_{Ω} f \overline{g} d λ_{Ω},$ which is linear in the first variable. Thus $A^{2} (Ω)$ is a Hilbert space.

For each $w \in Ω$ , evaluation $E_{w} ([f]) = f (w)$ is a bounded linear functional on $A^{2} (Ω)$ . Riesz representation gives a unique $k_{w} \in A^{2} (Ω)$ such that $f (w) = ⟨ [f], k_{w} ⟩ (f \in A^{2} (Ω)) .$ The Bergman kernel is $K_{Ω} (z, w) : = k_{w} (z), z, w \in Ω,$ where $k_{w} (z)$ means evaluation of the unique holomorphic representative of $k_{w}$ . In particular $k_{z} = K_{Ω} (\cdot, z)$ and the displayed Riesz identity is the reproducing property. For $Ω = C^{m}$ , $A^{2} (Ω) = {0}$ and $K_{Ω} \equiv 0$ ; no positivity of $K_{Ω}$ is asserted for general unbounded $Ω$ .

Facts & Assumptions

[A1]

The only choice principle is ${A C}_{ω}$ ; it is used by the Lebesgue, complex $L^{2}$ , Bergman mean/evaluation, and Riesz suppliers, and to select an approximating sequence in the closedness argument. No full Axiom of Choice is used (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

[F1]

$λ_{2 m}$ is a complete measure on the Lebesgue sigma-algebra of $R^{2 m}$ . Its trace on the open, hence Lebesgue-measurable, set $Ω$ is a measure on the trace sigma-algebra $L (R^{2 m}) ∣_{Ω}$ (Lebesgue measurable sets, the family $L (R^{n})$ , and the restricted set function $λ_{n}$ , Assuming countable choice, $L (R^{n})$ is a sigma-algebra containing every elementary set and $λ_{n}$ is a complete measure extending elementary volume, The trace of a sigma-algebra on a subset, The trace of a sigma-algebra is a sigma-algebra on the traced subset, Assuming countable choice, every Borel subset of $R^{n}$ is Lebesgue measurable).

[F2]

Holomorphic functions on $Ω$ are continuous; their restrictions are Borel measurable on the subspace $Ω$ , and the subspace Borel sigma-algebra is the trace of the ambient Borel sigma-algebra, hence is contained in the trace Lebesgue sigma-algebra (Holomorphic functions on an open subset of $C^{m}$ , Holomorphic functions of several variables are smooth and their complex derivatives are holomorphic, The Borel sigma-algebra of a subspace is the trace of the ambient Borel sigma-algebra, Assuming countable choice, every Borel subset of $R^{n}$ is Lebesgue measurable).

[F3]

Complex linear combinations of holomorphic functions are holomorphic (Sums, products and nonvanishing quotients of holomorphic functions are holomorphic).

[F4]

With the pairing $\int f \overline{g} d λ_{Ω}$ , complex $L^{2} (Ω)$ is a Hilbert space under ${A C}_{ω}$ ; the pairing is linear in its first variable and induces the quotient $L^{2}$ norm (The complex $L^{2}$ pairing on equivalence classes, Complex Lp classes and Euclidean test-function conventions, $L^{2}$ with the integral pairing is a Hilbert space).

[F5]

On each nonempty compact $K \subseteq Ω$ , point evaluation is bounded by $\sup_{K} ∣ f ∣ \leq C_{K} ∥ f ∥_{L^{2} (Ω)}$ (Sup-norm and first-derivative bounds by the $L^{2}$ norm on compact subsets).

[F6]

An $L^{2}$ limit class of holomorphic $L^{2}$ functions has a holomorphic representative (Sup-norm and first-derivative bounds by the $L^{2}$ norm on compact subsets).

[F7]

For a holomorphic $f$ on a polydisc with closure in $Ω$ , $∣ f (a) ∣^{2} \leq \frac{1}{π^{m} \prod_{j < m} r_{j}^{2}} \int_{Δ_{r} (a)} ∣ f ∣^{2} d λ_{2 m},$ where $Δ_{r} (a)$ is the polydisc of positive radii (Balls, polydiscs and the distinguished boundary in $C^{m}$ , The mean-value $L^{2}$ bound for holomorphic functions on a polydisc).

[F8]

Every bounded linear functional on a complex Hilbert space has a unique Riesz representer $y$ with $F (x) = ⟨ x, y ⟩$ (Riesz representation for Hilbert spaces).

Proof

technique · direct, using the preceding compact evaluation and mean-value estimates

Given: ${A C}_{ω}$ , $m \geq 1$ , a nonempty open $Ω \subseteq C^{m}$ , and the Lebesgue measure and function-space conventions above.

1.1F2F3F4F5given

By [F2], every holomorphic function is measurable for $λ_{Ω}$ , so the definition of $A^{2} (Ω)$ is meaningful. If $f, g \in A^{2} (Ω)$ determine the same $L^{2}$ class, then $h = f - g$ is holomorphic by [F3] and has $∥ h ∥_{L^{2} (Ω)} = 0$ by [F4]. For each $w \in Ω$ , the singleton ${w}$ is compact, so [F5] gives $∣ h (w) ∣ \leq C_{{w}} ∥ h ∥_{L^{2} (Ω)} = 0$ . Thus $f = g$ on $Ω$ , proving uniqueness of the holomorphic representative and well-definedness of evaluation.

1.2A1F1F4F7given

If $Ω = C^{m}$ , then for every $a \in C^{m}$ and every $R > 0$ , [F7] gives $∣ f (a) ∣^{2} \leq (π R^{2})^{- m} \int_{Δ_{R} (a)} ∣ f ∣^{2} d λ_{2 m} \leq (π R^{2})^{- m} ∥ f ∥_{L^{2} (C^{m})}^{2}$ . Letting $R \to \infty$ and using $m \geq 1$ gives $f (a) = 0$ for every $a$ , so $A^{2} (C^{m}) = {0}$ .

2.1A1F3F4F6step 1.1given

The image $A^{2} (Ω)$ is a complex linear subspace by [F3] and [F4]. To prove it is closed, let $g$ lie in its $L^{2}$ closure. For each $n \geq 1$ , the set of $A^{2}$ classes within distance $1 / n$ of $g$ is nonempty; [A1] selects a sequence of such classes, and step 1.1 gives each a unique holomorphic representative $f_{n}$ . Then $f_{n} \to g$ in $L^{2} (Ω)$ , so [F6] supplies a holomorphic representative $F$ of $g$ . Since $g \in L^{2} (Ω)$ , this representative belongs to $A^{2} (Ω)$ , and $g \in A^{2} (Ω)$ . Hence the image is closed.

3.1A1F3F4F5F8step 1.1step 2.1given

The closed subspace $A^{2} (Ω)$ of the Hilbert space in [F4] is complete: each Cauchy sequence in it converges in $L^{2} (Ω)$ and its limit lies in $A^{2} (Ω)$ by step 2.1. For fixed $w \in Ω$ , evaluation is well-defined by step 1.1, complex-linear by [F3], and bounded by [F5] with $K = {w}$ . Applying [F8] gives a unique $k_{w} \in A^{2} (Ω)$ such that $f (w) = ⟨ [f], k_{w} ⟩$ for every $f \in A^{2} (Ω)$ .

4.1step 1.1step 1.2step 3.1∎

Define $K_{Ω} (z, w) = k_{w} (z)$ using the unique holomorphic representative from step 1.1. Then $k_{z} = K_{Ω} (\cdot, z)$ and the identity in step 3.1 is exactly the reproducing property. If $Ω = C^{m}$ , step 1.2 gives $A^{2} (C^{m}) = {0}$ , so every evaluation functional and its unique Riesz representer vanish; hence $K_{C^{m}} \equiv 0$ .

Depends on

Used by

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128 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Jiří Lebl, Tasty Bits of Several Complex Variables (standard reference, not scraped)
Zbigniew Błocki, The Bergman Kernel and Metric (standard reference, not scraped)

The Bergman space A2(Ω)A2(Ω) and the Bergman kernel