ExampleConstruction: AI-adaptedVerification: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-6.1-sol)audited 2026-10-08

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The polydisc boundary is not a smooth hypersurface, so the Szegő definition does not apply

Example

Assume ${A C}_{ω}$ (The Axiom of Countable Choice ( ${A C}_{ω}$ )) and let $m \geq 2$ . The topological boundary $\partial D^{m}$ is not a $C^{1}$ hypersurface. Hence the bounded- $C^{1}$ -domain hypothesis in The Hardy boundary space, Szegő projection and Szegő kernel on a smoothly bounded domain fails, so that definition does not supply a surface-measure Hardy space or a Szegő kernel for $D^{m}$ . The distinguished torus $T^{m}$ is a proper subset of $\partial D^{m}$ : $(1, 0, 1, \dots, 1)$ is a boundary point but is not in $T^{m}$ .

Facts & Assumptions

[A1]

The only choice principle is ${A C}_{ω}$ (The Axiom of Countable Choice ( ${A C}_{ω}$ )), inherited through the bounded- $C^{1}$ -domain, surface-integration and Szegő-definition interfaces; no full Axiom of Choice is used.

[F1]

With zero-based coordinates, $D^{m} = {z \in C^{m} : ∣ z_{j} ∣ < 1 for every j < m}$ and its closure is ${z : ∣ z_{j} ∣ \leq 1 for every j < m}$ . Its topological boundary is the points in this closure with at least one coordinate of modulus one, while its distinguished torus is $T^{m} : = {z : ∣ z_{j} ∣ = 1 for every j < m}$ (Balls, polydiscs and the distinguished boundary in $C^{m}$ ).

[F2]

A $C^{1}$ boundary chart is locally, after a rigid coordinate change, a graph of a $C^{1}$ real function over an open subset of $R^{2 m - 1}$ ; the graph tangent at a point has dimension $2 m - 1$ (Bounded C1 domains and their outward normals, The total (Fréchet) derivative $D f (a)$ as the linear first-order approximation with $o (∥ h ∥_{2})$ remainder).

[F3]

Under the real coordinate dictionary, $C^{m}$ is $R^{2 m}$ and the vectors $e_{0}, i e_{0}, \dots, e_{m - 1}, i e_{m - 1}$ form a real basis (Complex $m$ -space and its real coordinate dictionary).

[F4]

For real $t$ , $∣ e^{i t} ∣ = 1$ and $\frac{d}{d t} e^{i t} = i e^{i t}$ ; the derivative follows from the complex exponential derivative and its addition law ( $\exp (x + i y) = e^{x} (\cos y + i \sin y)$ , $∣ \exp (x + i y) ∣ = e^{x}$ , and $e^{i π} + 1 = 0$ , The complex exponential is entire and its complex derivative is itself, $\exp (z + w) = \exp z \exp w$ , and the complex exponential extends the real exponential).

[F5]

The library's Szegő definition takes a bounded connected open set with $C^{1}$ boundary and its surface measure as input; that measure is defined on a compact embedded $C^{1}$ hypersurface (The Hardy boundary space, Szegő projection and Szegő kernel on a smoothly bounded domain, Surface integration on compact C1 hypersurfaces).

Verification

technique · direct, using boundary curves and the tangent space of a $C^1$ graph

Given: ${A C}_{ω}$ , $m \geq 2$ , the unit polydisc and the definitions of a $C^{1}$ boundary and Szegő regularity.

1.1F1F3F4given

Let $p = (1, \dots, 1)$ . For each $j < m$ , the circle curve $γ_{j} (t) = (1, \dots, e^{i t}, \dots, 1)$ lies in $\partial D^{m}$ and has velocity $i e_{j}$ at $t = 0$ by [F1, F4]. For each $j < m$ , the one-sided radial curve $δ_{j} (t) = (1, \dots, 1 - t, \dots, 1)$ , $0 \leq t < 1$ , also lies in $\partial D^{m}$ : for $t > 0$ its $j$ th coordinate has modulus below one, while another coordinate remains of modulus one since $m \geq 2$ . Its right velocity at $0$ is $- e_{j}$ . The $2 m$ velocities $i e_{j}, - e_{j}$ form a real basis by [F3].

1.2F1given

Let $q_{0} = 1$ , $q_{1} = 0$ , and $q_{j} = 1$ for $2 \leq j < m$ . All coordinate moduli are at most one and $∣ q_{0} ∣ = 1$ , so $q \in \partial D^{m}$ by [F1]. Since $∣ q_{1} ∣ = 0$ , not every coordinate has modulus one, so $q \notin T^{m}$ by [F1]. The torus is contained in $\partial D^{m}$ by [F1], so it is a proper subset.

2.1F2step 1.1given

Suppose $\partial D^{m}$ had a $C^{1}$ boundary chart at $p$ . After a rigid coordinate change it would locally be a graph $x_{2 m} = h (y)$ with $h$ of class $C^{1}$ , whose tangent space at $p$ has dimension $2 m - 1$ by [F2]. By continuity, each curve from step 1.1 remains in this chart neighborhood for sufficiently small parameter. Write a curve in the chart as $(y (t), x_{2 m} (t))$ with $y (t) = y_{0} + t v + o (t)$ ; differentiability of $h$ gives $x_{2 m} (t) = h (y_{0}) + t D h (y_{0}) v + o (t)$ . Thus each velocity lies in the graph tangent space ${(v, D h (y_{0}) v) : v \in R^{2 m - 1}}$ . The $2 m$ independent velocities from step 1.1 cannot all lie in this $(2 m - 1)$ -dimensional space; rigid coordinate changes preserve independence. Therefore $\partial D^{m}$ is not a $C^{1}$ hypersurface.

3.1A1F5step 2.1step 1.2given∎

By [F5], the library's Szegő definition requires a bounded connected open set with $C^{1}$ boundary and its surface measure on the associated compact $C^{1}$ hypersurface. Step 2.1 proves that $D^{m}$ fails the boundary hypothesis. Hence this definition does not apply to the polydisc, and the distinguished-torus construction in step 1.2 uses a different boundary set.

Depends on

Used by

The polydisc Bergman product and the different distinguished-torus Hardy kernel Example

Dependency tree · two levels

61 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)