Referenced prerequisite statements

Airy functions and fold model operators

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The phase, density and normalization are those of [Fold amplitudes and critical densities](fold-amplitudes-and-critical-densities.md). The uniform even/odd coefficient construction is in [Folds, reflections and uniform smooth descent](folds-reflections-and-uniform-descent.md). We use the one-dimensional stationary-phase theorem of [Stationary phase and critical manifolds](stationary-phase-and-critical-manifolds.md), the principal-symbol isomorphism of [Gaussian lines, densities and invariant symbols](gaussian-lines-and-invariant-symbols.md), and the every-order regularity criterion of [Recognizing a Lagrangian distribution intrinsically](intrinsic-lagrangian-regularity.md).

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Boundary flux and weak identities

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We use [Order, positivity and distributional limits](order-positivity-and-limits.md) for measures and finite-order test extensions, [Local data and compatible products](local-data-and-compatible-products.md) for localization, and [Weak equations and classical functions](weak-equations-and-classical-functions.md) for the distinction between weak and pointwise equations. Entry prerequisites are the \(C^1\) implicit function theorem, ordinary change of variables, compact cutoffs and partitions of unity. [Dyatlov 2026], §10.1.6, discusses the surface-measure distribution for a regular level set. An integrable weak divergence suffices for the flux identity proved below.

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Boundary flux and weak identities

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We use [Order, positivity and distributional limits](order-positivity-and-limits.md) for measures and finite-order test extensions, [Local data and compatible products](local-data-and-compatible-products.md) for localization, and [Weak equations and classical functions](weak-equations-and-classical-functions.md) for the distinction between weak and pointwise equations. Entry prerequisites are the \(C^1\) implicit function theorem, ordinary change of variables, compact cutoffs and partitions of unity. [Dyatlov 2026], §10.1.6, discusses the surface-measure distribution for a regular level set. An integrable weak divergence suffices for the flux identity proved below.

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Cauchy kernels and distributional boundary limits

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[Avi Zeff’s lecture on Pompeiu’s formula](https://math.berkeley.edu/~avizeff/complex_analysis_S26/lecture_12.html), Section 2, is a primary human reference for the complex Green identity and punctured-circle argument. [Dyatlov 2026], Proposition 9.8, states the fundamental solution of the Cauchy–Riemann operator. We give the kernel and finite-order boundary proofs here. We use [Boundary flux and weak identities](boundary-flux-and-weak-identities.md) for Green’s formula and pointwise-to-weak equations, and [Order, positivity and distributional limits](order-positivity-and-limits.md) for finite-regularity tests. We use the elliptic regularity theorem in [Elliptic tests and intrinsic wavefront](https://kokunoyumeto.github.io/open-mathematics-courses/courses/AN-03/AN03-U012.html#AN03-GEO-009); Section 2 checks its hypotheses for this operator. Entry prerequisites are multivariable calculus, dominated convergence and elementary geometric series.

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Compatible jets on closed sets

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The prerequisites are [Jets, supported distributions and local operators](jets-supported-distributions-and-local-operators.md), [When a kernel is smooth](when-a-kernel-is-smooth.md), Taylor's formula, smooth mollifiers and elementary Euclidean distance geometry. We prove the extension construction and its bounds, including the partition used in the gluing. References are the original paper [Whitney 1934], the jet perspective of [Kolář–Michor–Slovák 1993], and the distribution background in [Dyatlov 2026].

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Convolution as addition of supports

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We use [Tensor products and parameter-dependent distributions](tensor-products-and-parameters.md) for independent pairings and their limits, [Local data and compatible products](local-data-and-compatible-products.md) for tests compact only on a distribution’s support, and [Order, positivity and distributional limits](order-positivity-and-limits.md) for finite-regularity tests. The last section uses [Distributions as kernels of continuous operators](distributions-as-kernels.md), [Weak equations and classical functions](weak-equations-and-classical-functions.md), and the compact approximation result in [When a kernel is smooth](when-a-kernel-is-smooth.md). Entry requirements are ordinary integration, Taylor’s formula, compact cutoffs and partitions of unity. Basic references are [Dyatlov 2026], for convolution, and [Melrose 2016] and [Schwartz 1952], for the kernel perspective.

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Convolution as addition of supports

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We use [Tensor products and parameter-dependent distributions](tensor-products-and-parameters.md) for independent pairings and their limits, [Local data and compatible products](local-data-and-compatible-products.md) for tests compact only on a distribution’s support, and [Order, positivity and distributional limits](order-positivity-and-limits.md) for finite-regularity tests. The last section uses [Distributions as kernels of continuous operators](distributions-as-kernels.md), [Weak equations and classical functions](weak-equations-and-classical-functions.md), and the compact approximation result in [When a kernel is smooth](when-a-kernel-is-smooth.md). Entry requirements are ordinary integration, Taylor’s formula, compact cutoffs and partitions of unity. Basic references are [Dyatlov 2026], for convolution, and [Melrose 2016] and [Schwartz 1952], for the kernel perspective.

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Convolution as addition of supports

This exact referenced proof is not bundled or certified here.

We use [Tensor products and parameter-dependent distributions](tensor-products-and-parameters.md) for independent pairings and their limits, [Local data and compatible products](local-data-and-compatible-products.md) for tests compact only on a distribution’s support, and [Order, positivity and distributional limits](order-positivity-and-limits.md) for finite-regularity tests. The last section uses [Distributions as kernels of continuous operators](distributions-as-kernels.md), [Weak equations and classical functions](weak-equations-and-classical-functions.md), and the compact approximation result in [When a kernel is smooth](when-a-kernel-is-smooth.md). Entry requirements are ordinary integration, Taylor’s formula, compact cutoffs and partitions of unity. Basic references are [Dyatlov 2026], for convolution, and [Melrose 2016] and [Schwartz 1952], for the kernel perspective.

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Convolution as addition of supports

This exact referenced proof is not bundled or certified here.

We use [Tensor products and parameter-dependent distributions](tensor-products-and-parameters.md) for independent pairings and their limits, [Local data and compatible products](local-data-and-compatible-products.md) for tests compact only on a distribution’s support, and [Order, positivity and distributional limits](order-positivity-and-limits.md) for finite-regularity tests. The last section uses [Distributions as kernels of continuous operators](distributions-as-kernels.md), [Weak equations and classical functions](weak-equations-and-classical-functions.md), and the compact approximation result in [When a kernel is smooth](when-a-kernel-is-smooth.md). Entry requirements are ordinary integration, Taylor’s formula, compact cutoffs and partitions of unity. Basic references are [Dyatlov 2026], for convolution, and [Melrose 2016] and [Schwartz 1952], for the kernel perspective.

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Cubic scaling and necessary continuity

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We use the signed cotangent forms, ordinary symbol classes and half-density normalization from [Oscillatory distributions and their order](oscillatory-distributions-and-order.md). [Corank geometry and sufficient continuity](corank-geometry-and-sufficient-continuity.md) proves the pointwise tangent normalization and partial Fourier form used here, including its zero two-jet. [Graph operators, continuity and Egorov](graph-operators-continuity-and-egorov.md) proves proper elliptic graph quantizations and their microlocal inverses. [Homogeneous submanifold normal forms](homogeneous-submanifold-normal-forms.md) gives the sharper constant-rank result and explains why flattening the entire relation requires more than tangent normalization.

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Folds, reflections and uniform smooth descent

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[Phase space and generating families](phase-space-and-generating-families.md) supplies the manifold, tangent and cotangent conventions. [Homogeneous submanifold normal forms](homogeneous-submanifold-normal-forms.md) explains characteristic directions and function-preserving symplectic pairs; we will need the fold facts here before constructing simultaneous symplectic fold coordinates. [Cubic scaling and necessary continuity](cubic-scaling-and-necessary-continuity.md) shows why the order \(-1/6\) is a possible necessary threshold when corank is two. The present lesson supplies supporting smooth geometry and amplitude realization, not the later Airy continuity theorem.

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Holomorphic boundaries in convex cones

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We use [Cauchy kernels and distributional boundary limits](cauchy-kernels-and-boundary-limits.md) for the one-variable finite-test calculation and circle formula, and [Gluing holomorphic sides](gluing-holomorphic-sides.md) for one-variable zero-boundary uniqueness. [Chakrabarti and Shafikov 2017], Introduction and §2.6, provides primary human context for distributional holomorphic boundaries and their relation to weak Cauchy–Riemann derivatives. We prove the convex-cone boundary theorem directly below. Entry prerequisites are ordinary convex geometry, multivariable calculus and dominated convergence.

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- [Debraj Chakrabarti and Rasul Shafikov, *Distributional boundary values of holomorphic functions on product domains*, Mathematische Zeitschrift 286 (2017), 1145–1171](https://math.sci.uwo.ca/~shafikov/papers/MathZ2017.pdf). The introduction and §2.6 discuss the boundary-current setting and its Cauchy–Riemann derivative relation. That theorem concerns generic corners and currents; the finite-test cone limit and uniqueness above have their own proofs.
- [*Cauchy kernels and distributional boundary limits*](cauchy-kernels-and-boundary-limits.md), Theorem 3.1 and Corollary 4.2. The one-variable cancellation and normalized pole limits used here.
- [*Gluing holomorphic sides*](gluing-holomorphic-sides.md), Corollary 3.3. The exact one-variable zero-boundary result used for \(Q\) in (3.2).
- [Avi Zeff, *Lecture 12: Pompeiu’s formula*, March 6, 2026](https://math.berkeley.edu/~avizeff/complex_analysis_S26/lecture_12.html). The complex integral identity underlying the one-variable prerequisites.

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Jets, supported distributions and local operators

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The prerequisites are [Distributions as kernels of continuous operators](distributions-as-kernels.md), [Tensor products and parameter-dependent distributions](tensor-products-and-parameters.md), Taylor's formula, smooth cutoffs and convolution with a smooth compactly supported function. We use ordinary derivatives \(\partial\), with the bilinear distribution convention; no powers of \(i\) are hidden in the coefficients. Basic references are [Dyatlov 2026], [Melrose 2016] and [Whitney 1934]. The special extension needed here is proved directly by smoothing at a scale set by the distance to the plane.

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Tensor products and parameter-dependent distributions

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The prerequisites are [Distributions as kernels of continuous operators](distributions-as-kernels.md), especially density of product tests, and [When a kernel is smooth](when-a-kernel-is-smooth.md), for compact distributions and bounded test families. We assume Taylor's formula and differentiation on compact subsets. Basic references are [Dyatlov 2026], [Melrose 2016] and [Schwartz 1952].

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Tensor products and parameter-dependent distributions

This exact referenced proof is not bundled or certified here.

The prerequisites are [Distributions as kernels of continuous operators](distributions-as-kernels.md), especially density of product tests, and [When a kernel is smooth](when-a-kernel-is-smooth.md), for compact distributions and bounded test families. We assume Taylor's formula and differentiation on compact subsets. Basic references are [Dyatlov 2026], [Melrose 2016] and [Schwartz 1952].

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Weak equations and classical functions

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We use [Local data and compatible products](local-data-and-compatible-products.md) for restriction, derivative signs and the product rule, and [Tensor products and parameter-dependent distributions](tensor-products-and-parameters.md) for separated variables. Integration, smooth cutoffs and elementary finite-dimensional linear algebra are prerequisites. The distribution perspective is [Dyatlov 2026], Chapter 3. For systems we use the proved noncommuting matrix transport in [Building a local inverse from radial singularities, “Matrix transport and the order of multiplication”](https://kokunoyumeto.github.io/open-mathematics-courses/courses/AN-03/AN03-U020.html#AN03-EHP-006), with an explicit specialization below.

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