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Bergman kernels of the disc, ball and polydisc, and Szegő kernels of the disc and ball

Statement

Assume the Axiom of Countable Choice ${A C}_{ω}$ (The Axiom of Countable Choice ( ${A C}_{ω}$ )), let $m \geq 1$ , and use Lebesgue measure on $C^{m} ≅ R^{2 m}$ with the first-variable-linear pairing. Write $⟨ z, w ⟩ : = \sum_{j < m} z_{j} \overline{w_{j}}$ for the standard Hermitian inner product. Then

$K_{D} (z, w) = \frac{1}{π (1 - z \overline{w})^{2}}, K_{D^{m}} (z, w) = \frac{1}{π^{m}} \prod_{j < m} \frac{1}{(1 - z_{j} \overline{w_{j}})^{2}}, K_{B^{m}} (z, w) = \frac{m!}{π^{m} (1 - ⟨ z, w ⟩)^{m + 1}} .$

Let $μ_{T}$ be normalized Haar measure on $T = \partial D$ , and let $σ_{m}$ be normalized polar surface measure on $S^{2 m - 1} = \partial B^{m}$ . The pairs $(D, μ_{T})$ and $(B^{m}, σ_{m})$ are Szegő-regular, with kernels

$S_{D} (z, w) = \frac{1}{1 - z \overline{w}}, S_{B^{m}} (z, w) = \frac{1}{(1 - ⟨ z, w ⟩)^{m}} .$

Each displayed Bergman kernel reproduces the corresponding $A^{2}$ space and is the unique such kernel. Each displayed Szegő kernel reproduces the corresponding boundary Hardy space and is its unique Riesz kernel. No smooth-boundary Szegő construction is asserted for the polydisc.

Facts & Assumptions

[A1]

The only choice axiom assumed is ${A C}_{ω}$ (The Axiom of Countable Choice ( ${A C}_{ω}$ )). It supplies the Bergman and Hardy Hilbert/Riesz constructions and the Hilbert Fourier expansion used below; no full Axiom of Choice is used.

[F1]

The normalized monomials form complete orthonormal systems in $A^{2} (D)$ , $A^{2} (B^{m})$ , and $A^{2} (D^{m})$ , with squared norms $π / (k + 1)$ , $π^{m} α! / (m + ∣ α ∣)!$ , and $π^{m} / \prod_{j < m} (α_{j} + 1)$ , respectively (Monomials form complete orthogonal systems of the Bergman spaces of the disc, the ball and the polydisc).

[F2]

For a complete orthonormal system $(e_{α})$ of a Bergman space, its kernel is $\sum_{α} e_{α} (z) \overline{e_{α} (w)}$ ; the sum converges absolutely and uniformly on compact subsets ( $A^{2} (Ω)$ is closed, and the Bergman kernel is the sum over any complete orthonormal system).

[F3]

The Cauchy product of two absolutely convergent complex series is absolutely convergent, with sum the product of the sums (The Cauchy product of two absolutely convergent complex series converges absolutely to the product of their sums).

[F4]

For a real $r$ with $0 \leq r < 1$ , $\sum_{k \geq 0} r^{k} = 1 / (1 - r)$ (For $∣ r ∣ < 1$ , $\sum_{k \geq 0} r^{k} = 1 / (1 - r)$ , and for $∣ r ∣ \geq 1$ the series diverges).

[F5]

Conjugation is multiplicative, $∣ x y ∣ = ∣ x ∣ ∣ y ∣$ , $∣ 1 ∣ = 1$ , and $∣ t ∣ = 0$ exactly when $t = 0$ (Conjugation is an involutive real-field automorphism, $z \overline{z} = ∣ z ∣^{2}$ , and modulus is definite, multiplicative, and subadditive).

[F6]

Complex series are limits of their finite partial sums, and absolute convergence is defined by the real modulus series (Complex series, absolute convergence, complex power series, and radius of convergence, Series, partial sums, convergence and the sum, divergence, and the tail series). An absolutely convergent complex series converges (Every absolutely convergent complex series converges, and rearrangements preserve its sum).

[F7]

If $0 \leq r < 1$ , then $r^{k} \to 0$ (For $∣ r ∣ < 1$ the sequence $r^{k}$ is null, and for $∣ r ∣ > 1$ the sequence $∣ r ∣^{k}$ diverges to $+ \infty$ ).

[F8]

Cauchy–Schwarz holds in complex inner-product spaces (Cauchy–Schwarz: $∣ ⟨ x, y ⟩ ∣ \leq ∥ x ∥ ∥ y ∥$ , with equality exactly for dependent pairs); the coordinate pairing $\sum_{j < m} z_{j} \overline{w_{j}}$ is the standard complex inner product (The standard formulas $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ and $\sum_{k < n} x_{k} \overline{y_{k}}$ on $C^{n}$ are inner products).

[F9]

The Euclidean norm on $C^{m}$ is $∥ z ∥ = (\sum_{j < m} ∣ z_{j} ∣^{2})^{1 / 2}$ , and its balls are the stated unit balls (Complex $m$ -space and its real coordinate dictionary, Balls, polydiscs and the distinguished boundary in $C^{m}$ ).

[F10]

The complex multinomial expansion holds for every $m, n \in N$ (The multinomial theorem for finitely many complex variables).

[F11]

For every $α \in W (n, m)$ , $ι_{C} ((\binom{n}{α})) \prod_{i < m} ι_{C} (α_{i}!) = ι_{C} (n!)$ (The multinomial theorem for finitely many complex variables).

[F12]

The disc Hardy pair is Szegő-regular and has kernel $1 / (1 - z \overline{w})$ under normalized Haar measure (The disc trace space is the Hardy boundary space and the Szegő family reproduces $H^{2}$ ).

[F13]

The ball Hardy pair is Szegő-regular; extended evaluation is bounded and has a unique Riesz representer (Polynomial traces, monomial basis and bounded evaluation for the ball Hardy space).

[F14]

The normalized ball boundary monomials form a complete orthonormal system with squared norms $w_{α} = (m - 1)! α! / (m - 1 + ∣ α ∣)!$ (Polynomial traces, monomial basis and bounded evaluation for the ball Hardy space).

[F15]

A complete orthonormal family has a norm-convergent Fourier expansion by the net of finite-subset sums (Fourier expansion in a Hilbert space).

[F16]

The binomial coefficient vanishes when its lower argument exceeds its upper argument, and $\sum_{i = 0}^{N} (\binom{i}{k}) = (\binom{N + 1}{k + 1})$ (The set $[A]^{k}$ of $k$ -element subsets and the binomial coefficient $(\binom{n}{k}) : = ∣ [n]^{k} ∣$ , Pascal's rule $(\binom{n + 1}{k + 1}) = (\binom{n}{k}) + (\binom{n}{k + 1})$ , and the hockey-stick identity $\sum_{i \leq n} (\binom{i}{k}) = (\binom{n + 1}{k + 1})$ ).

[F17]

If $k \leq n$ , then $(\binom{n}{k}) k! (n - k)! = n!$ ; every factorial is a nonzero natural ( $(\binom{n}{k}) k! (n - k)! = n!$ for $k \leq n$ ; hence $(\binom{n}{k}) k! = n^{\underline{k}}$ , the quotient $n! / (k! (n - k)!)$ is a natural number, and $(\binom{n}{k}) = (\binom{n}{n - k})$ , The factorial $n!$ and the falling factorial $n^{\underline{k}}$ , defined by recursion in $N$ ).

[F18]

On a Szegő-regular pair, each extended evaluation has a unique Riesz representer and the Szegő kernel is defined from those representers (The Hardy boundary space, Szegő projection and Szegő kernel on a smoothly bounded domain).

[F19]

The Bergman kernel reproduces evaluation on $A^{2} (Ω)$ (Reproducing property, Bergman projection and the extremal characterization).

[F20]

Each point evaluation on $A^{2} (Ω)$ has a unique Riesz representer, whose holomorphic representative defines the Bergman kernel (The Bergman space $A^{2} (Ω)$ and the Bergman kernel).

[F21]

The canonical natural in $C$ is the image of the real canonical natural under the embedded copy $R ↪ C$ ; the natural map into $R$ preserves finite sums and products, and positive natural numbers map to positive reals (The canonical natural $ι (n) = n \cdot 1_{F}$ of a field, $C = R [x] / (x^{2} + 1)$ is a field, every element is uniquely $a + b i$ , and every nonzero element has inverse $(a - b i) / (a^{2} + b^{2})$ , Laws of finite sums and products in $N$ , and $ι (\sum_{k < n} a_{k}) = \sum_{k < n} ι (a_{k})$ , Canonical naturals are positive and strictly increasing).

[F22]

The degree shells ${α \in N^{m} : ∣ α ∣ = k}$ are finite, every finite subset of $N^{m}$ is contained in a finite union of initial degree shells, and finite sums may be reindexed along bijections (The multinomial coefficient $(\binom{n}{k_{0}, \dots, k_{m - 1}})$ as the number of ordered partitions of an $n$ -set into blocks of prescribed sizes, A finite sum in a commutative monoid indexed by an arbitrary finite set, Finite commutative-monoid sums are invariant under bijective reindexing, split over disjoint unions, and satisfy the finite Fubini rule).

[F23]

Induction on the natural numbers is valid (The principle of mathematical induction).

[F24]

Natural powers are recursively defined; $(a b)^{k} = a^{k} b^{k}$ , and conjugation of a natural power is the corresponding power of the conjugate (Integer powers in the complex field, Laws of integer exponents, with the last identity following by induction from the recursion and [F5]).

Proof

technique · direct, using complete monomial systems and the binomial series

Given: ${A C}_{ω}$ , the unit disc, unit ball and unit polydisc, and the normalized boundary measures.

1.1F5F8F9given

The series parameters lie in the unit disc. If $z, w \in D$ , or if $z, w \in D^{m}$ , then $∣ z \overline{w} ∣ < 1$ or $∣ z_{j} \overline{w_{j}} ∣ < 1$ coordinatewise. If $z, w \in B^{m}$ , [F8] gives $∣ ⟨ z, w ⟩ ∣ \leq ∥ z ∥ ∥ w ∥ < 1$ by [F9]. Thus every denominator below is nonzero.

1.2A1F12F18

The disc pair with normalized Haar measure is Szegő-regular and has kernel $1 / (1 - z \overline{w})$ by [F12]. The Szegő definition [F18] makes each kernel section the unique Riesz representer of its extended evaluation, so this is the unique disc Szegő kernel.

2.1F4F5F6F7F21F24step 1.1given

Fix a complex $t$ with $∣ t ∣ < 1$ . By induction from the power recursion [F24] and modulus multiplicativity [F5], $∣ t^{k} ∣ = ∣ t ∣^{k}$ for every $k$ . The real geometric series for $∣ t ∣$ converges by [F4], so $\sum_{k \geq 0} t^{k}$ is absolutely convergent and converges by [F6], say to $G (t)$ . The finite identity $(1 - t) \sum_{k = 0}^{N} t^{k} = 1 - t^{N + 1}$ follows by telescoping in the field [F21]; [F7] gives $∣ t ∣^{N + 1} \to 0$ , hence $t^{N + 1} \to 0$ . Since $∣ t ∣ < 1 = ∣ 1 ∣$ , $t \neq 1$ ; taking limits gives $G (t) = 1 / (1 - t)$ .

3.1F3F16F21F23step 2.1given

For every integer $q \geq 1$ , define $B_{q} (t) : = \sum_{k \geq 0} (\binom{q + k - 1}{q - 1}) t^{k}$ . Step 2.1 is the base $q = 1$ . If $B_{q} (t)$ is absolutely convergent with sum $(1 - t)^{- q}$ , its Cauchy product with $\sum_{j \geq 0} t^{j}$ is absolutely convergent and has sum $(1 - t)^{- (q + 1)}$ by [F3]. The coefficient of $t^{k}$ in this product is $\sum_{j = 0}^{k} (\binom{q + j - 1}{q - 1})$ . Set $i = q + j - 1$ ; the hockey-stick identity [F16] sums $(\binom{i}{q - 1})$ from $i = 0$ to $q + k - 1$ , and [F16] makes each omitted term with $i < q - 1$ zero. Thus the coefficient is $(\binom{q + k}{q})$ . By [F21] this natural identity also holds for the coefficients embedded in $C$ . Thus the product is exactly $B_{q + 1} (t)$ , proving the identity for every $q \geq 1$ by induction [F23].

4.1A1F1F2F3F6F22F23F24step 3.1

On the disc, the normalized monomials from [F1] give $K_{D} (z, w) = π^{- 1} \sum_{k \geq 0} (k + 1) (z \overline{w})^{k} = π^{- 1} B_{2} (z \overline{w})$ , which is the stated formula by step 3.1. On the polydisc, [F2] gives the monomial expansion. Its finite degree shells are cofinal among finite subsets by [F22], so the shell sums have the same limit as the kernel expansion. Repeated Cauchy products [F3] show that the degree- $k$ shell sum is the degree- $k$ coefficient in the product of the $m$ absolutely convergent series $π^{- 1} B_{2} (z_{j} \overline{w_{j}})$ ; hence summing the shells gives their product.

4.2A1F1F2F10F11F17F21F22F24step 3.1

For $z, w \in B^{m}$ , put $t = ⟨ z, w ⟩$ . The Bergman expansion [F2] and ball monomial norms [F1] give $K_{B^{m}} (z, w) = π^{- m} \sum_{α \in N^{m}} \frac{(m + ∣ α ∣)!}{α!} z^{α} \overline{w^{α}}$ . The degree shells are finite and cofinal by [F22], so their partial sums converge to this kernel. In the shell $∣ α ∣ = k$ , the multinomial expansion [F10] applied to $x_{j} = z_{j} \overline{w_{j}}$ gives $\sum_{∣ α ∣ = k} ι_{C} ((\binom{k}{α})) z^{α} \overline{w^{α}} = t^{k}$ , using [F24] for powers and conjugation. Also [F11] gives $ι_{C} ((\binom{k}{α})) \prod_{j < m} ι_{C} (α_{j}!) = ι_{C} (k!)$ , while [F17] gives $ι_{C} ((\binom{m + k}{m})) ι_{C} (m!) ι_{C} (k!) = ι_{C} ((m + k)!)$ after [F21] identifies the real and complex canonical naturals. The factorial denominator is positive by [F17, F21], so division gives $(m + k)! / α! = m! (\binom{m + k}{m}) (\binom{k}{α})$ in the corresponding real scalars. The degree- $k$ block is therefore $\frac{m!}{π^{m}} (\binom{m + k}{m}) t^{k}$ . Step 3.1 sums these blocks to the stated formula.

4.3A1F10F11F13F14F15F17F18F21F22F24step 3.1

Let $S_{w}$ be the ball Szegő Riesz representer from [F13], and let $e_{α} (ζ) = ζ^{α} / \sqrt{w_{α}}$ be its complete orthonormal boundary monomials by [F14]. By [F15], $S_{w} = \sum_{α} ⟨ S_{w}, e_{α} ⟩ e_{α}$ . For any finite subset of indices, the first-variable-linear reproducing identity gives $⟨ e_{α}, S_{w} ⟩ = e_{α} (w)$ and hence $⟨ S_{w}, e_{α} ⟩ = \overline{e_{α} (w)}$ . The finite degree shells are cofinal by [F22]; applying the bounded evaluation at $z$ from [F13] to the Fourier sums over those shells gives $S_{B^{m}} (z, w) = \sum_{k \geq 0} \sum_{∣ α ∣ = k} z^{α} \overline{w^{α}} / w_{α}$ . For each shell, [F10] applied to $x_{j} = z_{j} \overline{w_{j}}$ gives $\sum_{∣ α ∣ = k} ι_{C} ((\binom{k}{α})) z^{α} \overline{w^{α}} = ⟨ z, w ⟩^{k}$ , using [F24] for powers and conjugation. Since $w_{α} = (m - 1)! α! / (m - 1 + k)!$ , [F11] gives $ι_{C} ((\binom{k}{α})) \prod_{j < m} ι_{C} (α_{j}!) = ι_{C} (k!)$ , and [F17] gives $ι_{C} ((\binom{m + k - 1}{m - 1})) ι_{C} ((m - 1)!) ι_{C} (k!) = ι_{C} ((m - 1 + k)!)$ after [F21] identifies canonical natural scalars. The factorial denominator is positive by [F17, F21], so division gives the degree- $k$ block $(\binom{m + k - 1}{m - 1}) ⟨ z, w ⟩^{k}$ . Step 3.1 with $q = m$ sums these blocks to $(1 - ⟨ z, w ⟩)^{- m}$ . The Szegő definition [F18] gives uniqueness of this Riesz kernel.

5.1F2F19F20step 1.2step 4.1step 4.2step 4.3∎

The basis expansions in steps 4.1 and 4.2 are the Bergman Riesz kernels by [F2], so [F19] supplies their reproducing identities; uniqueness follows from the Bergman-space Riesz definition [F20]. Steps 1.2 and 4.3 establish the two Szegő reproducing kernels and uniqueness. Setting either kernel variable to $0$ in the displayed formulas (equivalently, retaining only the degree-zero monomial term) gives $K_{D} (0, w) = 1 / π$ , $K_{D^{m}} (0, w) = 1 / π^{m}$ , $K_{B^{m}} (0, w) = m! / π^{m}$ , and both stated Szegő values $1$ . When $m = 1$ , $B^{1} = D$ and $D^{1} = D$ , and the corresponding formulas agree. The polydisc appears only in the Bergman product formula, so no boundary regularity is claimed for it.

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Sources

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