Sources, authorship and component terms

The 148 course lessons are original exposition by GPT-6.1 Sol (OpenAI), Ultra, September–October 2026, dedicated under CC0 1.0. They are self-checked drafts. Its two declared general analytic prerequisite proofs remain planned. Its construction covers nonzero complex constant-coefficient operators with all lower order terms and unrestricted growth. A separate bounded revision review accepts the annotation repair and the general total-curvature proof. The construction imposes no growth condition at infinity and asserts no global temperedness. Every specialized argument is written in the lesson; the Fourier and Banach providers and ordinary entry calculus retain their exact declared scope. Its scalar theorem retains the nowherezero real-frequency highest-time coefficient and the spatial growth class; the characteristic heat plane is included. The explicit delay inverse does not itself prove the general compact-kernel theorem. The added support-cone lesson proves the general forward support/reciprocal implication; the qualified punctured-cone converse is proved in L117, while the original cone convention and unqualified equivalence remain unresolved. The original 1938 Petrovskii article is historical attribution only and supplies no required proof. The support-cone lesson, with its seven solutions and two original diagrams, is self-checked by the writing AI. Its conditional CD9 implication retains D7/C7/C6/C8, rational-form completeness, tube/duality, orientation/multiplicity, source-coefficient and full component-constancy obligations. Its cone is nonempty, ordinarily convex, open, positive-dilation invariant and excludes zero. Uniform closed-angular reciprocal bounds beyond logarithmic height yield one common inverse with every directional support inequality; temperedness is not assumed or concluded. The source H2 convention and unrestricted equivalence remain unresolved. Gerd Grubb’s freely readable Fourier chapter joins Semyon Dyatlov’s notes as credited comparison background; all analytic steps needed by this construction are written in the learner. They supply fixed-polynomial local imaginary tubes and power bounds, the open permitted graph, closed positive-polar graph and W_F, compact permitted families and C7/AH3 field adapters, relative to the stated C1/D1 and D4 inputs. Integral prism, small-chain, Mayer–Vietoris, sphere-generator and mapping-torus arguments prove the actual outward and canonical spheres are the nonzero order-two class. Characteristic-zero disappearance, the nonzero mod-2 class and the two-dimensional reduced-degree-zero exception remain explicit. The source coefficient convention remains unresolved; no source-error, novelty-priority, general integral torsion or general component-constancy claim is made. Atiyah, Bott, Gårding and Hatcher receive precise free-primary method credits; the fixed-polynomial arguments and needed integral chain facts are written in full. The fixed maximum-degree construction supplies smooth nonnegative coefficient-scalar-invariant entire-function averaging, common compact punctured support for n≥1, a uniform denominator and every real coefficient differential; constants, actual degree drops and the separate n=0 unit-mass endpoint remain explicit. Scalar Taylor, finite-dimensional compactness/Cauchy–Schwarz, cutoffs, polynomial root finiteness/uniform continuity and Lebesgue/Fubini inputs remain declared. Its ordinary complex top-degree comparison is relative to CD034; its integral basis, primitive normal-circle-first tube, coefficient-one affine/projective identity, d=1 ordinary point convention and separate hemisphere factor are retained. General rational completeness, tube injection, affine identity, component constancy and original source coefficients remain open. It proves compact-convex growth, uniform entire polynomial division with the same K and N, the unique compact inverse and ordinary hull equality, and all-center full-multiplicity exponential-polynomial annihilation with divisor P(-z). The new learner includes the whole formal proof and a coordinate rectangle calculation; original source and renderer downloads are unchanged. The Fourier and Cauchy entries retain their lower provider scope. Its exact original files and component terms remain downloadable. L011 links that precise local input; L013 receives only the L111 halfspace link, retaining its wavefront and other lower entries. The complete TP041 reader and MH043 reader retain their proof bodies, examples, all six solutions each and five full-caption native diagrams. Their complete programme sources and notices and exact original/corrected alternatives remain readable and downloadable under their actual component terms. Current H and real comparisons are qualified outside the original historical notices. L125 adds reversible delimiters for original inline indices and powers. L123 incorporates an inline-typesetting repair, retaining all 142 original expressions, 42 added inline expressions, eleven exact original downloads and all other source/reader bytes. That earlier reader edition preserved the other 122 lesson records. This source revision updates the references in L001, L002, L003, L018, L020, L042 and L043, the provider and context notices in L124–L125, and the required hypoellipticity premise in L031. All mathematical expressions, worked examples and complete solutions are retained. A subsequent bounded source revision updates only the references in L092, L093, L095 and L096. Their written proofs, mathematical expressions, examples, solutions and figures retain their exact bytes; other source and reader files retain their previous bytes. The next bounded source revision updates references in L049, L050, L053, L094 and L097. Their written proofs, mathematical expressions, examples, solutions and figures retain their exact bytes; other source and reader files retain their previous bytes. The next source revision gives L018 direct links to its existing continuation and bounded-extension proofs, and clarifies the internal proof routes and background references in L006, L007, L098 and L099. All five mathematical bodies, examples, solutions and figures retain their exact bytes.

Human mathematical sources credited within the lessons include Lars Hörmander’s The Analysis of Linear Partial Differential Operators I–II, Gerd Grubb’s author-hosted lecture chapters, Michael Taylor’s Partial Differential Equations, Michel Coste’s notes on semialgebraic geometry, Arne Enqvist’s work on cone-supported fundamental solutions, Romain Crétier’s notes on analytic Cauchy problems, Laurent Schwartz’s work on evolution and convolution, Jan Kisyński’s Petrovskii-correctness manuscript and I. G. Petrovskii’s historical root-growth criterion, Semyon Dyatlov’s distributions notes, Georges de Rham’s classical period theorem and Allen Hatcher’s prism and subdivision constructions. These citations identify the mathematical sources. The weighted Fourier, local-regularity, continuous-functional and Lebesgue-duality lessons now identify their written internal functional-analysis proofs. Three Fourier-background references link the exact freely readable Grubb lecture chapter §5. Each cited passage retains its stated scope. The published payload contains original lessons and separately licensed open proof readings.

The CC0 dedication covers independently written programme expression. Cited books and papers retain their own rights; included proof readings retain their listed component licences and notices. Scholarly credit and a source's reading availability do not grant permission to reproduce its protected expression or replace a required proof. Each lesson and provider keeps its stated mathematical dependency scope.

Supporting proof readings

The 42 listed supporting readings below retain their individual availability status, original author notices and component licence. The tangent-zoom entry remains unavailable with its source absent; the Green entry is restored with both complete original sources and readers; the four added CC0 readings include their complete original sources and readers. The 15 elliptic-analysis readings use CC0 1.0; their CC0 dedication, component credits and source records accompany them.

• Airy functions and fold model operators — CC0-1.0.
• Banach estimates, quotient spaces and compact parameter arguments — CC0 1.0.
• Boundary flux and weak identities — CC0-1.0.
• Building a local inverse from radial singularities — CC0 1.0.
• Cauchy kernels and distributional boundary limits — CC0-1.0.
• Causal kernels, initial data, and short-time geometry — CC0 1.0.
• Compatible jets on closed sets — CC0-1.0.
• Convex supports and convolution cancellation — CC0-1.0.
• Convolution as addition of supports — CC0-1.0.
• Corank geometry and sufficient continuity — CC0-1.0.
• Cubic scaling and necessary continuity — CC0-1.0.
• Detecting regularity without choosing coordinates — CC0 1.0.
• Folds, reflections and uniform smooth descent — CC0-1.0.
• Fourier transforms, finite spectra and convex separation — CC0 1.0.
• From Weyl symbols to operators and changes of coordinates — CC0 1.0.
• From symbol estimates to operators on every Sobolev scale — CC0 1.0.
• Gaussian lines, densities and invariant symbols — CC0-1.0.
• Graph operators, continuity and Egorov — CC0-1.0.
• Hamilton fields and subprincipal transport — CC0-1.0.
• Holomorphic boundaries in convex cones — CC0-1.0.
• Jets, supported distributions and local operators — CC0-1.0.
• Kernels, adjoints and clean composition — CC0-1.0.
• Local inverses and distance-weighted elliptic estimates — CC0 1.0.
• Localizing symbols when the measuring scale moves — CC0 1.0.
• Metric and topological foundations — CC0 1.0.
• Oscillatory distributions and their order — CC0-1.0.
• Phase space and generating families — CC0-1.0.
• Polynomial and contour interfaces for stable boundary models — CC0 1.0.
• Quadratic Fourier multipliers at a moving scale — CC0 1.0.
• Recognizing a Lagrangian distribution intrinsically — CC0-1.0.
• Reducing Gaussian symbols and composing linear relations — CC0-1.0.
• Singularities along a submanifold and smooth boundary passage — CC0 1.0.
• Stable modes and the algebra of boundary data — CC0 1.0.
• Stationary phase and critical manifolds — CC0-1.0.
• Tangent zooms and quadratic models — CC0-1.0.
• Tensor products and parameter-dependent distributions — CC0-1.0.
• Two measuring scales, one Weyl product — CC0 1.0.
• Weak equations and classical functions — CC0-1.0.
• Local data and compatible products — CC0-1.0; complete original notice and source retained.
• Order, positivity and distributional limits — CC0-1.0; complete original notice and source retained.
• Distributions as kernels of continuous operators — CC0-1.0; complete original notice and source retained.
• When a kernel is smooth — CC0-1.0; complete original notice and source retained.

Planned foundation proofs records each prerequisite’s status. The characteristic-halfspace entry links to its complete written construction; other entries retain their individual statuses. The Schwartz evolution lesson states its Fourier, Fréchet and finite-algebra providers exactly; the general compact-kernel forward interface is proved in L115, and the qualified punctured-cone converse is proved in L117. The original H2 cone convention, unqualified equivalence and full Notes variant remain incomplete. Four added complete distribution readings resolve their cached-text reader absence. The protected statement gateway and inherited provider-reader links remain edition records; the added navigation leads to complete texts. The full smooth-period detection proof closes its bounded comparison implication; all stated projective/component premises remain open. L118 supplies its bounded homogeneous-polynomial local-cone receiver relative to C1/D1 and D4. L119 proves its actual integral wave-sphere result, with all coefficient and orientation qualifications. The inherited foundation and provider statuses remain explicit; these results do not admit the general AN-01 interfaces, full C6–C8, rational-form completeness, tube injectivity or full component constancy. L120 supplies the exact written averaging prerequisite and Lemma 2.1 denominator receiver in L004. L122 supplies the internal compact-division and entire exponential-polynomial annihilator input in L010 entry(3) and the corresponding L011 Fourier statements, plus an additional ordinary-hull proof for L010 entry(2). The valid AN-01 Corollary4.3 alternative remains available. The local polynomial-annihilator germ equivalence, its convergence issue and compact singular-support hull remain separate and open. The complete Green reader and its earlier alternative now supply their reviewed boundary, finite-corner, inside weak-divergence and everywhere pointwise-to-weak arguments at declared lower entries. This does not newly admit every lower foundation or the entire Cauchy/weak-elliptic providers; the L122 finite rectangle proof remains unchanged. The general AN-01 theory, general topology, component constancy and the full prerequisites of whole lessons are not proved here. Recursive prerequisite closure remains incomplete.

Boundary flux and weak identities now has both complete Green proofs and their six solutions each, with the full selected scalar, algebra and measure prerequisites and their distinct CC0 terms. Versions, exact downloads and terms. This restores the bounded Green provider at its declared entries; lower foundations, entire Cauchy/weak providers and course closure remain separate.

The three additional complete selected Green foundations remain CC0 1.0, with their component notices, selection history and distinct reader history. Their complete proof bodies include the added angular inverse and scalar convexity proof. The existing complete foundation alternatives remain unchanged. The two complete Green main proofs remain CC0. Original human-source credits remain in both full proofs.

Reader software and illustrations

MathJax retains Apache 2.0. Font notices are included with the reader and the supporting illustrations. Original course figures retain their stated CC0 attribution; third-party font components retain their own notices.

Editable Markdown, figures and reproducible figure sources accompany the course download.

L126–L130 add the compact closed–open sequence, form/right-cap comparison, rational coefficient and canonical connecting maps, and the smooth-wavefront convolution estimate. Their original exposition, examples, solutions and figures use the accompanying CC0 terms. Proofs, native figures and reproducible sources. External human references retain their own rights. The coefficient, orientation, right-cap, excision, compact-distribution, Fourier and test-family hypotheses are stated in the lessons.

Fourier limits and Global solvability link Grubb’s freely readable Fourier chapter and the written Banach, compactness and Fourier-duality proofs. Their analytic and hyperbolic application hypotheses remain explicit.

Local Newtonian potentials and subharmonic regularity proves the local integrability and convolution continuity result associated with Hörmander’s Proposition 16.1.1. The complete original argument, examples, solutions and diagrams use CC0 1.0; the figure font notice retains its terms. Editable sources and reproduction files.

Poisson extension and subharmonic comparison gives the normalized ball Poisson operator, four equivalent comparison criteria and the usc finite-above supremum theorem associated with Hörmander’s Lemma 16.1.3, Proposition 16.1.4 and Corollary 16.1.5. The complete original argument, examples, solutions and diagrams use CC0 1.0. Editable source and reproducible figures.

Horizontal envelopes and their limiting slope proves convexity of slice suprema and the finite limiting-slope estimate for subharmonic functions under a linear height bound on the upper half-space. Classical source: Hörmander, The Analysis of Linear Partial Differential Operators II, §16.1, Lemma 16.1.6 and the following discussion, printed page 308 (PDF page 321); 1983 edition, second revised printing 1990, reprint 2005. Original exposition and diagrams: CC0 1.0. Figure font notice. Editable proof and reproduction files.

Distributional limits of subharmonic functions proves strong local norm convergence and the two upper-limit conclusions associated with Hörmander’s Proposition 16.1.2. The complete original argument, examples, solutions and diagrams use CC0 1.0; the figure font notice retains its terms. Editable sources and reproduction files.

Boundary measures and Green potentials in a half-space proves the representation of a subharmonic function under a linear upper height bound, the two exact weighted measure conditions and the weak boundary trace. Classical source: Hörmander, The Analysis of Linear Partial Differential Operators II, §16.1, Theorem 16.1.7, printed pages 310–312 (PDF pages 323–325); 1983 edition, second revised printing 1990, reprint 2005. Original proof, examples, solutions and diagrams: CC0 1.0. Figure font notice. Editable sources and reproduction files.

A linear profile under dilation gives the integrated Green and Poisson kernel proof of the dilation conclusion associated with Hörmander’s Theorem 16.1.8, with its exact representation input and a direct one-dimensional supplement. The original proof, examples, solutions and diagrams use CC0 1.0; the figure font notice retains its terms. Editable sources and reproduction files.

Fourier endpoints and the asymptotic density of zeros gives exact endpoint growth, the logarithmic multiplicity measure and the disk zero-count limit associated with Hörmander’s Theorem 16.1.9. The original proof exposition, examples, solutions and diagrams use CC0 1.0. Editable source and reproducible figures.

Compact support and the Carleman condition proves the reciprocal derivative-growth necessity condition associated with Hörmander’s Theorem 16.1.10. The complete original argument, examples, solutions and diagrams use CC0 1.0; the figure font notice retains its terms. Editable sources and reproduction files.

Plurisubharmonic envelopes and support functions proves the finite horizontal envelope, its compact convex set of slopes, and global bounded-above PSH constancy associated with Hörmander’s Lemmas 16.2.1–16.2.2. The original exposition, proofs, examples, solutions and diagrams use CC0 1.0. Editable source and reproducible figures.

Constant upper envelopes and scaled plurisubharmonic averages proves almost-everywhere constancy of a globally bounded pointwise upper limit through a direct PSH upper-envelope construction, including every real dilation parameter. Classical source: Hörmander, The Analysis of Linear Partial Differential Operators II, §16.2, Lemma 16.2.3 and its following real-parameter remark, printed page 316 (PDF page 329); 1983 edition, second revised printing 1990, reprint 2005. Original proof, examples, solutions and diagrams: CC0 1.0. Figure font notice. Editable sources and reproduction files.

Directional averages and additivity of growth indicators proves the directional and ball-average statements and indicator additivity associated with Hörmander’s Theorem 16.2.4, Corollaries 16.2.5–16.2.6 and Theorem 16.2.7. The original arguments, examples, solutions and figures use CC0 1.0; the figure font notice retains its terms. Editable proof and reproducible figures.

Fourier indicators and the convex hull of a measure's support. The Fourier transform of a compact complex measure has a PSH logarithm whose horizontal indicator equals its convex support function, including cancellation and nonatomic endpoints. Classical source: Hörmander II, §16.3, Lemma 16.3.1, printed p. 319; 1983 edition, second revised printing 1990, reprint 2005. Original material: CC0 1.0. Editable sources and original figures.

Local compactness and Hartogs bounds. A locally upper-bounded PSH sequence on a connected domain either collapses uniformly on compacts or has a proper local L1 limit along a subsequence. Explicit selection and moving maxima prove the compact Hartogs bound. Classical source: Hörmander I, Theorem 4.1.9, printed pp. 94–96; first edition 1983, second edition 1990, reprint 2003. The additional PSH closure is proved explicitly. Original material: CC0 1.0. Editable sources and original figures.

Solving the Cauchy–Riemann equations with a weight. Positive Levi curvature yields a weighted solution of closed Cauchy–Riemann data. Full graph-domain and weak-limit arguments prove the strict estimate and the general PSH estimate with exact factor 2 and square weight. Classical source: Hörmander II, §15.1, Theorems 15.1.1–15.1.2, printed pp. 271–274; 1983 edition, second revised printing 1990, reprint 2005. Original material: CC0 1.0. Editable sources and original figures.

Weighted holomorphic extensions and their growth. An entire function on a complex linear subspace extends with the exact codimension-dependent weighted norm bound. Full restriction and iteration arguments give the precise polynomial growth transfer. Classical source: Hörmander II, §15.1, Theorem 15.1.3 and Corollary 15.1.4, printed pp. 274–276; 1983 edition, second revised printing 1990, reprint 2005. Original material: CC0 1.0. Editable sources and original figures.

Fourier transforms of analytic functionals on a real convex carrier. The Fourier transform of an analytic functional has arbitrarily small exponential losses. A diagonal holomorphic extension constructs the functional and proves compatibility, uniqueness and local holomorphic test bounds. Classical source: Hörmander II, §15.1, Theorem 15.1.5, printed p. 276; I, Definition 9.1.1 and the local-test discussion, printed pp. 326–328. Original material: CC0 1.0. Editable sources and original figures.

Entire logarithms and the approximation of plurisubharmonic functions. Normalized logarithms of nonzero entire scalar functions approximate every proper plurisubharmonic function in local integral norm. Full dense-set convergence, bounded Cauchy–Riemann continuity and weighted interpolation proofs give the exact construction and closure. Classical source: Hörmander II, §15.1, Theorem 15.1.6 and Lemmas 15.1.7–15.1.8, printed pp. 277–278; 1983 edition, second revised printing 1990, reprint 2005. Original material: CC0 1.0. Editable sources and original figures.

Complex Fourier estimates and weak exponential representations. Three exact weighted complex Fourier estimates lead to a weak exponential representation against singular measures. Full support-cutoff, reflected-duality and compact-test extension proofs distinguish distributional integrals from pointwise values. Classical source: Hörmander II, §15.2, Lemma 15.2.2 and Theorem 15.2.4, printed pp. 280–281 and 286–287; 1983 edition, second revised printing 1990, reprint 2005. Original material: CC0 1.0. Editable sources and original figures.