{ "readings": [ { "id": "AN03-U050", "title": "Reducing arbitrary boundary data to a boundary system", "source": "src/arbitrary-boundary-data-reduction.md", "reader": "arbitrary-boundary-data-reduction.html", "source_sha256": "34530ea0aed97b2fb2ca573eb9db6bf966e2c7f57b1c3d52c94356745d59298b", "original_current_source_sha256": "b7dc6a365c16fd662cbf44150e8852e57c79111ce1f858e49ddcf05af1b5d090", "license": "CC0-1.0", "notice": "CC0 1.0", "scope": "Complete current supporting reading, with its full theorem ranges and proofs.", "heading_aliases": [ { "line": 1, "heading": "Reducing arbitrary boundary data to a boundary system", "ids": [ "reducing-arbitrary-boundary-data-to-a-boundary-system" ], "level": 1 }, { "line": 27, "heading": "A compact cylinder with a missing measurement direction", "ids": [ "AN03-ABR-CYLINDER-001", "a-compact-cylinder-with-a-missing-measurement-direction" ], "level": 2 }, { "line": 399, "heading": "1. The reference problem and the comparison measurements", "ids": [ "AN03-ABR-001", "1-the-reference-problem-and-the-comparison-measurements", "the-reference-problem-and-the-comparison-measurements" ], "level": 2 }, { "line": 465, "heading": "2. The complete reference parametrix", "ids": [ "AN03-ABR-002", "2-the-complete-reference-parametrix", "the-complete-reference-parametrix" ], "level": 2 }, { "line": 590, "heading": "3. The boundary pseudodifferential system", "ids": [ "AN03-ABR-003", "3-the-boundary-pseudodifferential-system", "the-boundary-pseudodifferential-system" ], "level": 2 }, { "line": 643, "heading": "4. The coupled ansatz and its exact scope", "ids": [ "AN03-ABR-004", "4-the-coupled-ansatz-and-its-exact-scope", "the-coupled-ansatz-and-its-exact-scope" ], "level": 2 }, { "line": 700, "heading": "The full reference and comparison block", "ids": [ "AN03-ABR-BLOCK-001", "the-full-reference-and-comparison-block" ], "level": 3 }, { "line": 751, "heading": "5. Equivalence of regularity gains", "ids": [ "AN03-ABR-005", "5-equivalence-of-regularity-gains", "equivalence-of-regularity-gains" ], "level": 2 }, { "line": 824, "heading": "6. Equivalence of solvability modulo smooth errors", "ids": [ "AN03-ABR-006", "6-equivalence-of-solvability-modulo-smooth-errors", "equivalence-of-solvability-modulo-smooth-errors" ], "level": 2 }, { "line": 897, "heading": "7. Endpoints, order checks, and a nonelliptic model", "ids": [ "AN03-ABR-007", "7-endpoints-order-checks-and-a-nonelliptic-model", "endpoints-order-checks-and-a-nonelliptic-model" ], "level": 2 }, { "line": 934, "heading": "8. Exercises with complete solutions", "ids": [ "AN03-ABR-008", "8-exercises-with-complete-solutions", "exercises-with-complete-solutions" ], "level": 2 }, { "line": 992, "heading": "9. Reading notes and scope", "ids": [ "AN03-ABR-009", "9-reading-notes-and-scope", "reading-notes-and-scope" ], "level": 2 }, { "line": 1008, "heading": "The exact cotangent-space class", "ids": [ "AN03-ABR-COTANGENT-001", "the-exact-cotangent-space-class" ], "level": 3 } ], "author": "Codex; retained course contributions preserve their original AI author notices", "author_self_checked": true, "independent_review_claimed": false, "tex": "tex/arbitrary-boundary-data-reduction.tex", "pdf": "pdf/arbitrary-boundary-data-reduction.pdf", "historical_reader_source_sha256": "b7dc6a365c16fd662cbf44150e8852e57c79111ce1f858e49ddcf05af1b5d090", "historical_reader_source_hash_provenance": "Retained source binding recorded for the prior reader edition; no new source-reader equality claim.", "current_source_sha256": "34530ea0aed97b2fb2ca573eb9db6bf966e2c7f57b1c3d52c94356745d59298b", "source_edition_status": "Current complete HTML reading; current PDF available.", "reader_pending": false, "current_reader_source_policy_refresh_pending": false, "source_reader_equality_claimed": true, "accepted_writer_proof_record_sha256": "118cc246b4a9cc097f15eacdb613a27b69eca9f5ea394cd66a59af8e0fa634a6", "HTML_source_equality_claimed": true, "tex_sha256": "b410d22ca155c403440d1bb22d6c60dc6e51d28333e80de436d6d5db29b0d8dd", "pdf_sha256": "5316e9b6a457924f51a5a4a303e0edd7d0a189b61acdfdcc5ef839a4d020604a", "pdf_pages": 17, "reader_sha256": "dbd70cd10e3bc18e0a0d3c490a57aac9cfc32ecc9e9278263e36f995332a89fa", "public_PDF_prerequisite_routes_current": true, "current_HTML_source_equality_claimed": true, "current_PDF_source_equality_claimed": true }, { "id": "AN03-P004", "title": "Banach estimates, quotient spaces and compact parameter arguments", "source": "src/banach-foundation-bridges.md", "reader": "banach-foundation-bridges.html", "source_sha256": "69437bb0af3abe6ceaae6f7e22678eb06b2f3f00adcdd36d1fb655f6ce71edf4", "original_current_source_sha256": "69437bb0af3abe6ceaae6f7e22678eb06b2f3f00adcdd36d1fb655f6ce71edf4", "license": "CC0-1.0", "notice": "CC0 1.0", "scope": "Complete current supporting reading, with its full theorem ranges and proofs.", "heading_aliases": [ { "line": 1, "ids": [ "banach-estimates-quotient-spaces-and-compact-parameter-arguments" ], "heading": "Banach estimates, quotient spaces and compact parameter arguments" }, { "line": 7, "ids": [ "AN03-BFD-001", "an03-bfd-001-entry-assumptions-and-exact-interfaces", "1-the-assumptions-used-here", "the-assumptions-used-here" ], "heading": "1. The assumptions used here" }, { "line": 17, "ids": [ "AN03-BFD-002", "an03-bfd-002-finite-dimensional-norm-control-and-exact-sequences", "2-finite-dimensional-norms-and-exact-sequences", "finite-dimensional-norms-and-exact-sequences" ], "heading": "2. Finite-dimensional norms and exact sequences" }, { "line": 43, "ids": [ "AN03-BFD-003", "an03-bfd-003-completeness-of-the-spaces-actually-used", "3-completeness-of-operator-spaces", "completeness-of-operator-spaces" ], "heading": "3. Completeness of operator spaces" }, { "line": 65, "ids": [ "AN03-BFD-004", "an03-bfd-004-quotients-with-all-completeness-assumptions-visible", "4-quotient-norms-and-completeness", "quotient-norms-and-completeness" ], "heading": "4. Quotient norms and completeness" }, { "line": 82, "ids": [ "AN03-BFD-005", "an03-bfd-005-norm-preserving-complex-extension-and-scalar-detection", "5-hahn-banach-and-scalar-norm-tests", "hahnbanach-and-scalar-norm-tests" ], "heading": "5. Hahn–Banach and scalar norm tests" }, { "line": 130, "ids": [ "AN03-BFD-006", "an03-bfd-006-the-existing-complete-metric-baire-reading", "6-baire-s-theorem-for-complete-metric-spaces", "baires-theorem-for-complete-metric-spaces" ], "heading": "6. Baire's theorem for complete metric spaces" }, { "line": 150, "ids": [ "AN03-BFD-007", "an03-bfd-007-uniform-boundedness-with-an-arbitrary-index-set", "7-uniform-boundedness-for-any-operator-family", "uniform-boundedness-for-any-operator-family" ], "heading": "7. Uniform boundedness for any operator family" }, { "line": 168, "ids": [ "AN03-BFD-008", "an03-bfd-008-open-mapping-and-bounded-inverse-by-summable-errors", "8-open-mapping-through-summable-corrections", "open-mapping-through-summable-corrections" ], "heading": "8. Open mapping through summable corrections" }, { "line": 201, "ids": [ "AN03-BFD-009", "an03-bfd-009-nets-final-segments-and-neighborhood-tests", "9-nets-convergence-and-connectedness", "nets-convergence-and-connectedness" ], "heading": "9. Nets, convergence and connectedness" }, { "line": 217, "ids": [ "AN03-BFD-010", "an03-bfd-010-compactness-for-arbitrary-parameter-spaces", "10-compactness-without-a-countability-assumption", "compactness-without-a-countability-assumption" ], "heading": "10. Compactness without a countability assumption" }, { "line": 255, "ids": [ "AN03-BFD-011", "an03-bfd-011-square-summable-sequences-and-their-truncations", "11-square-summable-sequences", "square-summable-sequences" ], "heading": "11. Square-summable sequences" }, { "line": 272, "ids": [ "AN03-BFD-012", "an03-bfd-012-three-worked-uses", "12-worked-examples", "worked-examples" ], "heading": "12. Worked examples" }, { "line": 286, "ids": [ "AN03-BFD-013", "an03-bfd-013-exercises-with-complete-solutions", "13-exercises-and-solutions", "exercises-and-solutions" ], "heading": "13. Exercises and solutions" }, { "line": 321, "ids": [ "AN03-BFD-014", "an03-bfd-014-reading-comparisons-and-further-routes", "14-source-comparisons-and-limits", "source-comparisons-and-limits" ], "heading": "14. Source comparisons and limits" }, { "line": 366, "ids": [ "AN03-BFD-FRECHET-001", "the-original-seminorms-and-a-complete-metric" ], "heading": "14.1. The original seminorms and a complete metric" }, { "line": 386, "ids": [ "AN03-BFD-FRECHET-002", "completeness-of-the-smooth-spaces" ], "heading": "14.2. Completeness of the smooth spaces" }, { "line": 420, "ids": [ "AN03-BFD-FRECHET-003", "a-common-finite-order-for-an-arbitrary-family" ], "heading": "14.3. A common finite order for an arbitrary family" }, { "line": 461, "ids": [ "AN03-BFD-FRECHET-004", "open-mapping-with-every-seminorm-retained" ], "heading": "14.4. Open mapping with every seminorm retained" }, { "line": 490, "ids": [ "AN03-BFD-FRECHET-005", "the-closed-graph-and-its-exact-estimate" ], "heading": "14.5. The closed graph and its exact estimate" }, { "line": 509, "ids": [ "AN03-BFD-FRECHET-006", "the-distributional-graph-used-for-a-smooth-kernel" ], "heading": "14.6. The distributional graph used for a smooth kernel" }, { "line": 560, "ids": [ "AN03-BFD-SEMINORM-001", "extending-a-functional-with-its-original-seminorm-bound" ], "heading": "14.7. Extending a functional with its original seminorm bound", "level": 3 }, { "line": 585, "ids": [ "AN03-BFD-FRECHET-QUOTIENT-001", "complete-quotients-with-every-original-seminorm" ], "heading": "14.8. Complete quotients with every original seminorm", "level": 3 }, { "line": 643, "ids": [ "integration-approximation-and-weak-derivatives" ], "heading": "15. Integration, approximation and weak derivatives", "level": 2 }, { "line": 647, "ids": [ "AN03-BFD-MEASURE-001", "constructing-the-measure-without-importing-a-convergence-theorem" ], "heading": "15.0. Constructing the measure without importing a convergence theorem", "level": 3 }, { "line": 705, "ids": [ "AN03-BFD-LP-001", "convergence-product-integration-and-the-full-linear-jacobian" ], "heading": "15.1. Convergence, product integration and the full linear Jacobian", "level": 3 }, { "line": 749, "ids": [ "AN03-BFD-LP-002", "hölder-minkowski-and-all-young-endpoints" ], "heading": "15.2. Hölder, Minkowski and all Young endpoints", "level": 3 }, { "line": 782, "ids": [ "AN03-BFD-LP-003", "completeness-and-compact-smooth-density" ], "heading": "15.3. Completeness and compact smooth density", "level": 3 }, { "line": 817, "ids": [ "AN03-BFD-LP-004", "the-original-scaled-approximate-identity" ], "heading": "15.4. The original scaled approximate identity", "level": 3 }, { "line": 846, "ids": [ "AN03-BFD-LP-005", "complete-weak-derivative-limits-and-sobolev-approximation" ], "heading": "15.5. Complete weak-derivative limits and Sobolev approximation", "level": 3 }, { "line": 887, "ids": [ "AN03-BFD-POLAR-001", "the-full-polar-formula-on-the-original-plane" ], "heading": "15.6. The full polar formula on the original plane", "level": 3 }, { "line": 1008, "ids": [ "AN03-BFD-POLAR-002", "the-exact-gaussian-factor-used-by-fourier-inversion" ], "heading": "15.7. The exact Gaussian factor used by Fourier inversion", "level": 3 }, { "line": 1044, "ids": [ "AN03-BFD-GENERAL-000", "general-measurable-functions-and-sigma-finite-product-integration" ], "heading": "16. General measurable functions and sigma-finite product integration", "level": 2 }, { "line": 1046, "ids": [ "AN03-BFD-GENERAL-001", "measures-completions-measurable-limits-and-the-full-nonnegative-integral" ], "heading": "16.1. Measures, completions, measurable limits and the full nonnegative integral", "level": 3 }, { "line": 1102, "ids": [ "AN03-BFD-GENERAL-002", "the-exact-generating-class-argument-and-measurable-sections" ], "heading": "16.2. The exact generating-class argument and measurable sections", "level": 3 }, { "line": 1119, "ids": [ "AN03-BFD-GENERAL-003", "constructing-the-original-product-measure-and-proving-uniqueness" ], "heading": "16.3. Constructing the original product measure and proving uniqueness", "level": 3 }, { "line": 1145, "ids": [ "AN03-BFD-GENERAL-004", "full-tonelli-and-absolutely-integrable-complex-fubini" ], "heading": "16.4. Full Tonelli and absolutely integrable complex Fubini", "level": 3 }, { "line": 1167, "ids": [ "AN03-BFD-GENERAL-005", "completed-products-exceptional-sections-and-the-euclidean-receiving-map" ], "heading": "16.5. Completed products, exceptional sections and the Euclidean receiving map", "level": 3 }, { "line": 1191, "ids": [ "AN03-BFD-BOCHNER-000", "strong-measurability-and-complete-banach-valued-integration" ], "heading": "17. Strong measurability and complete Banach-valued integration", "level": 2 }, { "line": 1195, "ids": [ "AN03-BFD-BOCHNER-001", "strong-measurability-and-finite-valued-approximation" ], "heading": "17.1. Strong measurability and finite-valued approximation", "level": 3 }, { "line": 1241, "ids": [ "AN03-BFD-BOCHNER-002", "the-actual-vector-integral-norm-bound-and-continuous-maps" ], "heading": "17.2. The actual vector integral, norm bound and continuous maps", "level": 3 }, { "line": 1284, "ids": [ "AN03-BFD-BOCHNER-003", "complete-bochner-spaces-including-nonseparable-hilbert-spaces" ], "heading": "17.3. Complete Bochner spaces, including nonseparable Hilbert spaces", "level": 3 }, { "line": 1345, "ids": [ "AN03-BFD-BOCHNER-004", "both-original-fubini-orders-and-the-completed-product-exception" ], "heading": "17.4. Both original Fubini orders and the completed-product exception", "level": 3 }, { "line": 1389, "ids": [ "AN03-BFD-BOCHNER-005", "the-exact-fourier-and-operator-receivers" ], "heading": "17.5. The exact Fourier and operator receivers", "level": 3 }, { "line": 1539, "ids": [ "AN03-BFD-SCHWARTZ-001", "the-original-monomial-schwartz-entry" ], "heading": "18. The original monomial Schwartz entry", "level": 2 }, { "line": 1608, "ids": [ "AN03-BFD-CHOICE-001", "the-selected-zorn-entry-and-the-original-baire-recursion" ], "heading": "19. The selected Zorn entry and the original Baire recursion", "level": 2 }, { "line": 1630, "ids": [ "AN03-BFD-ENTRY-001", "exact-entry-and-receiving-scope" ], "heading": "20. Exact entry and receiving scope", "level": 2 }, { "line": 1638, "ids": [ "AN03-BFD-LATTICE-000", "exact-lattice-cutoffs-and-the-original-derivative-map" ], "heading": "21. Exact lattice cutoffs and the original derivative map", "level": 2 }, { "line": 1642, "ids": [ "AN03-BFD-LATTICE-001", "the-full-original-lattice-construction" ], "heading": "21.1. The full original lattice construction", "level": 3 }, { "line": 1729, "ids": [ "AN03-BFD-LATTICE-002", "exact-prerequisite-scope" ], "heading": "21.2. Exact prerequisite scope", "level": 3 } ], "author": "Codex; retained course contributions preserve their original AI author notices", "author_self_checked": true, "independent_review_claimed": false, "tex": "tex/banach-foundation-bridges.tex", "pdf": "pdf/banach-foundation-bridges.pdf", "historical_reader_source_sha256": "69437bb0af3abe6ceaae6f7e22678eb06b2f3f00adcdd36d1fb655f6ce71edf4", "historical_reader_source_hash_provenance": "Retained source binding recorded for the prior reader edition; no new source-reader equality claim.", "current_source_sha256": "69437bb0af3abe6ceaae6f7e22678eb06b2f3f00adcdd36d1fb655f6ce71edf4", "source_edition_status": "Current complete HTML reading; current PDF available.", "reader_pending": false, "current_reader_source_policy_refresh_pending": false, "source_reader_equality_claimed": true, "accepted_writer_proof_record_sha256": "dd7416b9c2ced9bbeedc5cc95f483435f6b08706db6cc357eda87d580d35ae86", "HTML_source_equality_claimed": true, "tex_sha256": "d9e1ec0fe66fdcd7bada49658fdb919d973fe688ca233943725a922e34df0572", "pdf_sha256": "aab5351e5aaa4c9343f38a38b7eb9f8939802f6b604598a79ba388ab4d28fb03", "pdf_pages": 50, "reader_sha256": "97971e547fdacb5b4bf72a7359ce73ad25ad2c03e9b9970ec4e97c6701c38c54", "public_PDF_prerequisite_routes_current": true, "current_HTML_source_equality_claimed": true, "current_PDF_source_equality_claimed": true }, { "id": "AN03-U024", "title": "Cauchy data from jumps and residues", "source": "src/calderon-cauchy-data.md", "reader": "calderon-cauchy-data.html", "source_sha256": "dcc1150c0fc436abad04eed38bfe68aeda2b86f4a6f62d0c93e2d7b0fc9f2fe6", "original_current_source_sha256": "dcc1150c0fc436abad04eed38bfe68aeda2b86f4a6f62d0c93e2d7b0fc9f2fe6", "license": "CC0-1.0", "notice": "CC0 1.0", "scope": "Complete current supporting reading, with its full theorem ranges and proofs.", "heading_aliases": [ { "line": 1, "ids": [ "cauchy-data-from-jumps-and-residues" ], "heading": "Cauchy data from jumps and residues" }, { "line": 9, "ids": [ "AN03-CCD-001", "an03-ccd-001-the-weighted-space-of-boundary-jets", "1-the-weighted-space-of-boundary-jets", "the-weighted-space-of-boundary-jets" ], "heading": "1. The weighted space of boundary jets" }, { "line": 56, "ids": [ "AN03-CCD-002", "an03-ccd-002-division-in-the-normal-derivative", "2-division-in-the-normal-derivative", "division-in-the-normal-derivative" ], "heading": "2. Division in the normal derivative" }, { "line": 95, "ids": [ "AN03-CCD-003", "an03-ccd-003-the-frozen-test-and-its-coordinate-invariance", "3-the-frozen-test-and-its-coordinate-invariance", "the-frozen-test-and-its-coordinate-invariance" ], "heading": "3. The frozen test and its coordinate invariance" }, { "line": 117, "ids": [ "AN03-CCD-004", "an03-ccd-004-a-neighborhood-where-the-calculus-is-available", "4-a-neighborhood-where-the-calculus-is-available", "a-neighborhood-where-the-calculus-is-available" ], "heading": "4. A neighborhood where the calculus is available" }, { "line": 128, "ids": [ "AN03-CCD-EXT-001", "prescribing-every-original-boundary-jet" ], "heading": "4.1. Prescribing every original boundary jet", "level": 3 }, { "line": 211, "ids": [ "AN03-CCD-EXT-002", "constructing-one-globally-injective-inward-collar" ], "heading": "4.2. Constructing one globally injective inward collar", "level": 3 }, { "line": 244, "ids": [ "AN03-CCD-EXT-003", "bundle-transport-and-the-actual-ambient-manifold" ], "heading": "4.3. Bundle transport and the actual ambient manifold", "level": 3 }, { "line": 285, "ids": [ "AN03-CCD-EXT-004", "extending-every-differential-coefficient-with-its-support" ], "heading": "4.4. Extending every differential coefficient with its support", "level": 3 }, { "line": 303, "ids": [ "AN03-CCD-EXT-005", "the-full-ellipticity-bound-and-the-receiving-boundary-distributions" ], "heading": "4.5. The full ellipticity bound and the receiving boundary distributions", "level": 3 }, { "line": 368, "ids": [ "AN03-CCD-005", "an03-ccd-005-transmission-of-the-entire-inverse-symbol", "5-transmission-of-the-entire-inverse-symbol", "transmission-of-the-entire-inverse-symbol" ], "heading": "5. Transmission of the entire inverse symbol" }, { "line": 383, "ids": [ "AN03-CCD-006", "an03-ccd-006-a-jump-as-a-triangular-distributional-operator", "6-a-jump-as-a-triangular-distributional-operator", "a-jump-as-a-triangular-distributional-operator" ], "heading": "6. A jump as a triangular distributional operator" }, { "line": 409, "ids": [ "AN03-CCD-007", "an03-ccd-007-a-green-potential-and-its-boundary-operator", "7-a-green-potential-and-its-boundary-operator", "a-green-potential-and-its-boundary-operator" ], "heading": "7. A Green potential and its boundary operator" }, { "line": 435, "ids": [ "AN03-CCD-008", "an03-ccd-008-the-complete-weighted-matrix-and-its-residues", "8-the-complete-weighted-matrix-and-its-residues", "the-complete-weighted-matrix-and-its-residues" ], "heading": "8. The complete weighted matrix and its residues" }, { "line": 468, "ids": [ "AN03-CCD-009", "an03-ccd-009-identifying-the-projection-by-a-jump-without-choosing-roots", "9-identifying-the-projection-by-a-jump-without-choosing-roots", "identifying-the-projection-by-a-jump-without-choosing-roots" ], "heading": "9. Identifying the projection by a jump, without choosing roots" }, { "line": 645, "ids": [ "AN03-CCD-010", "an03-ccd-010-why-the-full-defect-has-a-smooth-kernel", "10-why-the-full-defect-has-a-smooth-kernel", "why-the-full-defect-has-a-smooth-kernel" ], "heading": "10. Why the full defect has a smooth kernel" }, { "line": 683, "ids": [ "AN03-CCD-011", "an03-ccd-011-the-positive-trace-of-an-interior-particular-solution", "11-the-positive-trace-of-an-interior-particular-solution", "the-positive-trace-of-an-interior-particular-solution" ], "heading": "11. The positive trace of an interior particular solution" }, { "line": 701, "ids": [ "AN03-CCD-012", "an03-ccd-012-what-the-boundary-reduction-actually-says", "12-what-the-boundary-reduction-actually-says", "what-the-boundary-reduction-actually-says" ], "heading": "12. What the boundary reduction actually says" }, { "line": 722, "ids": [ "AN03-CCD-EX-001", "an03-ccd-ex-001-four-calculations-that-expose-different-constraints", "13-four-calculations-that-expose-different-constraints", "four-calculations-that-expose-different-constraints" ], "heading": "13. Four calculations that expose different constraints" }, { "line": 755, "ids": [ "AN03-CCD-PS-001", "an03-ccd-ps-001-exercises-with-complete-solutions", "14-exercises-with-complete-solutions", "exercises-with-complete-solutions" ], "heading": "14. Exercises with complete solutions" }, { "line": 800, "ids": [ "AN03-CCD-EXT-006", "the-exact-receiving-boundary-distribution" ], "heading": "15. 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Every ordered normal coefficient and both inverse recursions", "ids": [ "AN03-GCF-FOUNDATION-015", "boundary-foundation-15" ], "level": 2 }, { "line": 1390, "heading": "16. The actual inverse on both real normal tails", "ids": [ "AN03-GCF-FOUNDATION-016", "boundary-foundation-16" ], "level": 2 }, { "line": 1570, "heading": "17. The full paired normal class and its finite products", "ids": [ "AN03-GCF-FOUNDATION-017", "boundary-foundation-17" ], "level": 2 }, { "line": 1572, "heading": "A paired real-normal class with its original weights", "ids": [ "AN03-GCF-FOUNDATION-029", "paired-normal-definition" ], "level": 3 }, { "line": 1671, "heading": "18. Exact composition moments and differentiated remainders", "ids": [ "AN03-GCF-FOUNDATION-018", "boundary-foundation-18" ], "level": 2 }, { "line": 1799, "heading": "19. The full normal-jet source and both resolvent errors", "ids": [ "AN03-GCF-FOUNDATION-019", "boundary-foundation-19" ], "level": 2 }, { "line": 2138, "heading": "20. Half-space mapping and the entire supported-source norm", "ids": [ "AN03-GCF-FOUNDATION-020", "boundary-foundation-20" ], "level": 2 }, { "line": 2398, "heading": "21. All one-sided traces and original boundary orders", "ids": [ "AN03-GCF-FOUNDATION-021", "boundary-foundation-21" ], "level": 2 }, { "line": 2548, "heading": "22. Keeping source freezing and interior freezing separate", "ids": [ "AN03-GCF-FOUNDATION-022", "boundary-foundation-22" ], "level": 2 }, { "line": 2777, "heading": "23. The exact nonlocal correction of the elliptic estimate", "ids": [ "AN03-GCF-FOUNDATION-023", "boundary-foundation-23" ], "level": 2 }, { "line": 2870, "heading": "24. The actual Cauchy principal symbol and global dual spaces", "ids": [ "AN03-GCF-FOUNDATION-024", "boundary-foundation-24" ], "level": 2 }, { "line": 2872, "heading": "1. Comparing the actual inverse with the full principal polynomial", "ids": [ "AN03-GCF-FOUNDATION-026", "pc-inverse-comparison" ], "level": 3 }, { "line": 2908, "heading": "2. The exact top trace term and its residue projection", "ids": [ "AN03-GCF-FOUNDATION-027", "pc-cauchy-residue" ], "level": 3 }, { "line": 2944, "heading": "3. The global dual bootstrap with the correct supported spaces", "ids": [ "AN03-GCF-FOUNDATION-028", "pc-global-dual" ], "level": 3 }, { "line": 2975, "ids": [ "graded-boundary-defect-bridge", "25-a-full-matrix-defect-on-two-boundary-topologies" ], "heading": "25. A full matrix defect on two boundary topologies" }, { "line": 3119, "heading": "26. 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Local estimates and amplitude reduction" }, { "line": 48, "ids": [ "AN03-GEO-002", "an03-geo-002-kernels-exponential-tests-and-coordinate-transport", "2-kernels-exponential-tests-and-coordinate-transport", "kernels-exponential-tests-and-coordinate-transport" ], "heading": "2. Kernels, exponential tests, and coordinate transport" }, { "line": 120, "ids": [ "AN03-GEO-003", "an03-geo-003-the-second-symbol-term-and-the-density-it-measures", "3-the-second-symbol-term-and-the-density-it-measures", "the-second-symbol-term-and-the-density-it-measures" ], "heading": "3. The second symbol term and the density it measures" }, { "line": 162, "ids": [ "AN03-GEO-004", "an03-geo-004-assembling-an-operator-and-its-principal-symbol", "4-assembling-an-operator-and-its-principal-symbol", "assembling-an-operator-and-its-principal-symbol" ], "heading": "4. 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Sobolev and Besov spaces in a moving chart" }, { "line": 385, "ids": [ "AN03-GEO-012", "an03-geo-012-microlocal-scales-and-their-unattained-thresholds", "12-microlocal-scales-and-their-unattained-thresholds", "microlocal-scales-and-their-unattained-thresholds" ], "heading": "12. Microlocal scales and their unattained thresholds" }, { "line": 623, "ids": [ "AN03-GEO-013", "an03-geo-013-bundles-anti-duals-and-kernels-between-different-manifolds", "13-bundles-anti-duals-and-kernels-between-different-manifolds", "bundles-anti-duals-and-kernels-between-different-manifolds" ], "heading": "13. Bundles, anti-duals, and kernels between different manifolds" }, { "id": "AN03-GEO-KERNEL-001", "heading": "13.1. 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Smooth parameter pairings and distributional approximation", "level": 3 }, { "line": 1037, "ids": [ "AN03-GEO-CONV-004", "compact-fourier-pairings-and-the-receiving-inverse-identities" ], "heading": "13.10. Compact Fourier pairings and the receiving inverse identities", "level": 3 }, { "line": 1140, "ids": [ "AN03-GEO-EX-001", "an03-geo-ex-001-four-calculations-that-separate-the-assertions", "14-four-calculations-that-separate-the-assertions", "four-calculations-that-separate-the-assertions" ], "heading": "14. Four calculations that separate the assertions" }, { "line": 1152, "ids": [ "AN03-GEO-PS-001", "an03-geo-ps-001-exercises-with-complete-solutions", "15-exercises-with-complete-solutions", "exercises-with-complete-solutions" ], "heading": "15. Exercises with complete solutions" }, { "line": 1178, "ids": [ "AN03-GEO-FOUNDATION-000", "original-chart-bundle-and-inverse-maps" ], "heading": "16. Original chart, bundle and inverse maps", "level": 2 }, { "line": 1182, "ids": [ "AN03-GEO-FOUNDATION-001", "the-actual-smooth-partition-and-every-denominator-derivative" ], "heading": "16.1. The actual smooth partition and every denominator derivative", "level": 3 }, { "line": 1213, "ids": [ "AN03-GEO-FOUNDATION-002", "gluing-the-actual-finite-rank-bundle" ], "heading": "16.2. Gluing the actual finite-rank bundle", "level": 3 }, { "line": 1265, "ids": [ "AN03-GEO-FOUNDATION-003", "every-tensor-dual-metric-and-density-factor" ], "heading": "16.3. Every tensor, dual, metric and density factor", "level": 3 }, { "line": 1308, "ids": [ "AN03-GEO-FOUNDATION-004", "constructing-the-inverse-without-changing-the-map-or-norms" ], "heading": "16.4. Constructing the inverse without changing the map or norms", "level": 3 }, { "line": 1398, "ids": [ "AN03-GEO-FOUNDATION-005", "every-higher-inverse-derivative-and-the-implicit-map" ], "heading": "16.5. Every higher inverse derivative and the implicit map", "level": 3 }, { "line": 1452, "ids": [ "AN03-GEO-FOUNDATION-006", "exact-receiving-scope-and-remaining-work" ], "heading": "16.6. Exact receiving scope and remaining work", "level": 3 }, { "line": 1458, "ids": [ "AN03-GEO-FLOW-000", "original-finite-coordinate-flows-and-their-geodesic-receiving-map" ], "heading": "17. Original finite-coordinate flows and their geodesic receiving map", "level": 2 }, { "line": 1462, "ids": [ "AN03-GEO-FLOW-001", "the-original-integral-equation-and-its-complete-local-solution" ], "heading": "17.1. The original integral equation and its complete local solution", "level": 3 }, { "line": 1524, "ids": [ "AN03-GEO-FLOW-002", "ordered-linear-transport-on-the-original-two-sided-interval" ], "heading": "17.2. Ordered linear transport on the original two-sided interval", "level": 3 }, { "line": 1584, "ids": [ "AN03-GEO-FLOW-003", "all-parameter-derivatives-of-the-linear-inverse" ], "heading": "17.3. All parameter derivatives of the linear inverse", "level": 3 }, { "line": 1616, "ids": [ "AN03-GEO-FLOW-004", "the-first-actual-nonlinear-parameter-derivative" ], "heading": "17.4. The first actual nonlinear parameter derivative", "level": 3 }, { "line": 1657, "ids": [ "AN03-GEO-FLOW-005", "every-nonlinear-parameter-derivative-including-its-original-blocks" ], "heading": "17.5. Every nonlinear parameter derivative, including its original blocks", "level": 3 }, { "line": 1691, "ids": [ "AN03-GEO-FLOW-006", "variable-initial-times-without-replacing-the-original-time-coordinate" ], "heading": "17.6. Variable initial times without replacing the original time coordinate", "level": 3 }, { "line": 1727, "ids": [ "AN03-GEO-FLOW-007", "continuation-the-exact-composition-law-and-coordinate-transport" ], "heading": "17.7. Continuation, the exact composition law and coordinate transport", "level": 3 }, { "line": 1775, "ids": [ "AN03-GEO-FLOW-008", "the-actual-geodesic-vector-field-and-smooth-endpoint-map" ], "heading": "17.8. The actual geodesic vector field and smooth endpoint map", "level": 3 }, { "line": 1853, "ids": [ "AN03-GEO-FLOW-009", "the-unaltered-metric-identity-all-first-jets-and-minimizing-radius" ], "heading": "17.9. The unaltered metric identity, all first jets and minimizing radius", "level": 3 }, { "line": 1910, "ids": [ "AN03-GEO-FLOW-010", "every-transformed-divergence-and-density-contribution" ], "heading": "17.10. Every transformed divergence and density contribution", "level": 3 }, { "line": 1948, "ids": [ "AN03-GEO-FLOW-011", "exact-course-use-and-its-remaining-interfaces" ], "heading": "17.11. Exact course use and its remaining interfaces", "level": 3 }, { "line": 1965, "ids": [ "AN03-GEO-WAVEFRONT-000", "original-wavefront-and-pullback-foundations" ], "heading": "18. 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Facts used from earlier lessons" }, { "line": 19, "ids": [ "AN03-SPR-002", "an03-spr-002-the-scalar-polynomial-reading-contract", "2-scalar-polynomial-division-and-bezout", "scalar-polynomial-division-and-bézout" ], "heading": "2. Scalar polynomial division and Bézout" }, { "line": 41, "ids": [ "AN03-SPR-003", "an03-spr-003-from-coprime-factors-to-projections", "3-from-coprime-factors-to-projections", "from-coprime-factors-to-projections" ], "heading": "3. From coprime factors to projections" }, { "line": 74, "ids": [ "AN03-SPR-004", "an03-spr-004-cayleyhamilton-from-an-invariant-flag", "4-cayleyhamilton-from-an-invariant-flag", "cayleyhamilton-from-an-invariant-flag" ], "heading": "4. Cayley–Hamilton from an invariant flag" }, { "line": 106, "ids": [ "AN03-SPR-005", "an03-spr-005-the-exact-entire-series-contour-moments", "5-contour-moments-from-entire-series", "contour-moments-from-entire-series" ], "heading": "5. Contour moments from entire series" }, { "line": 154, "ids": [ "AN03-SPR-006", "an03-spr-006-indices-of-rectangles-and-circles", "6-indices-of-rectangles-and-circles", "indices-of-rectangles-and-circles" ], "heading": "6. Indices of rectangles and circles" }, { "line": 206, "ids": [ "AN03-SPR-007", "an03-spr-007-differentiation-along-a-fixed-contour", "7-differentiation-along-a-fixed-contour", "differentiation-along-a-fixed-contour" ], "heading": "7. Differentiation along a fixed contour" }, { "line": 231, "ids": [ "AN03-SPR-008", "an03-spr-008-a-repeated-pole-and-a-scope-diagnostic", "8-a-repeated-pole-and-why-the-hypothesis-matters", "a-repeated-pole-and-why-the-hypothesis-matters" ], "heading": "8. A repeated pole and why the hypothesis matters" }, { "line": 270, "ids": [ "complex-roots-with-all-original-coefficients-retained" ], "heading": "9. Complex roots with all original coefficients retained", "level": 2 }, { "line": 276, "ids": [ "AN03-SPR-ROOT-001", "the-actual-exponential-and-a-direction-with-any-prescribed-integer-power" ], "heading": "9.1. The actual exponential and a direction with any prescribed integer power", "level": 3 }, { "line": 303, "ids": [ "AN03-SPR-ROOT-002", "a-minimum-of-the-original-polynomial-modulus" ], "heading": "9.2. A minimum of the original polynomial modulus", "level": 3 }, { "line": 331, "ids": [ "AN03-SPR-ROOT-003", "the-exact-translated-coefficients-force-a-zero" ], "heading": "9.3. The exact translated coefficients force a zero", "level": 3 }, { "line": 357, "ids": [ "AN03-SPR-ROOT-004", "full-factorization-leading-coefficient-and-multiplicities" ], "heading": "9.4. Full factorization, leading coefficient and multiplicities", "level": 3 }, { "line": 373, "ids": [ "AN03-SPR-ROOT-005", "scalar-units-and-the-exact-bézout-and-annihilator-receiving-maps" ], "heading": "9.5. Scalar units and the exact Bézout and annihilator receiving maps", "level": 3 }, { "line": 409, "ids": [ "AN03-SPR-ROOT-006", "original-characteristic-polynomial-and-explicit-root-to-kernel-map" ], "heading": "9.6. Original characteristic polynomial and explicit root-to-kernel map", "level": 3 }, { "line": 458, "ids": [ "AN03-SPR-ROOT-007", "eigenline-invariant-range-and-both-receiving-quotient-maps" ], "heading": "9.7. Eigenline, invariant range and both receiving quotient maps", "level": 3 }, { "line": 494, "ids": [ "AN03-SPR-ROOT-008", "the-original-spectral-bound-and-contour-receiver" ], "heading": "9.8. The original spectral bound and contour receiver", "level": 3 }, { "line": 505, "ids": [ "full-finite-linear-algebra-foundations" ], "heading": "10. Full finite linear-algebra foundations", "level": 2 }, { "line": 511, "ids": [ "AN03-SPR-LINALG-001", "finite-spanning-sets-exchange-and-basis-extension" ], "heading": "10.1. Finite spanning sets, exchange and basis extension", "level": 3 }, { "line": 568, "ids": [ "AN03-SPR-LINALG-002", "coordinates-matrix-products-and-basis-comparisons" ], "heading": "10.2. Coordinates, matrix products and basis comparisons", "level": 3 }, { "line": 612, "ids": [ "AN03-SPR-LINALG-003", "rank-nullity-quotients-and-direct-sums" ], "heading": "10.3. Rank-nullity, quotients and direct sums", "level": 3 }, { "line": 636, "ids": [ "AN03-SPR-LINALG-004", "determinants-and-the-full-ordered-elimination-maps" ], "heading": "10.4. Determinants and the full ordered elimination maps", "level": 3 }, { "line": 705, "ids": [ "AN03-SPR-LINALG-005", "the-original-inner-product-and-the-full-gram-residuals" ], "heading": "10.5. The original inner product and the full Gram residuals", "level": 3 }, { "line": 768, "ids": [ "AN03-SPR-LINALG-006", "exact-finite-inner-product-bounds-and-the-unchanged-original-norm" ], "heading": "10.6. Exact finite inner-product bounds and the unchanged original norm", "level": 3 }, { "line": 817, "ids": [ "AN03-SPR-LINALG-007", "full-adjoints-in-the-original-gram-matrices" ], "heading": "10.7. Full adjoints in the original Gram matrices", "level": 3 }, { "line": 868, "ids": [ "AN03-SPR-LINALG-008", "trace-projection-rank-and-positivity-in-the-receiving-coordinates" ], "heading": "10.8. Trace, projection rank and positivity in the receiving coordinates", "level": 3 }, { "line": 885, "ids": [ "AN03-SPR-LINALG-009", "the-full-original-characteristic-criterion-and-invariant-blocks" ], "heading": "10.9. The full original characteristic criterion and invariant blocks", "level": 3 }, { "line": 937, "ids": [ "AN03-SPR-LINALG-010", "actual-receiving-maps-for-the-existing-spectral-and-metric-lessons" ], "heading": "10.10. Actual receiving maps for the existing spectral and metric lessons", "level": 3 }, { "line": 947, "ids": [ "AN03-SPR-DOMINATED-000", "matrix-entry-differentiation-under-the-integral" ], "heading": "11. Matrix-entry differentiation under the integral", "level": 2 }, { "line": 951, "ids": [ "AN03-SPR-DOMINATED-001", "the-exact-entry-map-and-its-original-operator-norm" ], "heading": "11.1. The exact entry map and its original operator norm", "level": 3 }, { "line": 982, "ids": [ "AN03-SPR-DOMINATED-002", "every-parameter-derivative-and-its-full-integral-map" ], "heading": "11.2. Every parameter derivative and its full integral map", "level": 3 }, { "line": 1043, "ids": [ "AN03-SPR-DOMINATED-003", "an-integrable-anchor-supplies-the-missing-zeroth-order-bound" ], "heading": "11.3. An integrable anchor supplies the missing zeroth-order bound", "level": 3 }, { "line": 1056, "ids": [ "AN03-SPR-DOMINATED-004", "the-riemann-integral-each-path-piece-and-every-cycle-coefficient" ], "heading": "11.4. The Riemann integral, each path piece and every cycle coefficient", "level": 3 }, { "line": 1107, "ids": [ "AN03-SPR-DOMINATED-005", "the-actual-ordered-resolvent-time-derivatives-and-upper-contour-bound" ], "heading": "11.5. The actual ordered resolvent, time derivatives and upper-contour bound", "level": 3 } ], "author": "Codex; retained course contributions preserve their original AI author notices", "author_self_checked": true, "independent_review_claimed": false, "tex": "tex/stable-prerequisite-bridges.tex", "pdf": "pdf/stable-prerequisite-bridges.pdf", "historical_reader_source_sha256": "94a62c4be0f39475db008e15c10467b9755161371bf418b7b09a55263c5a5dd6", "historical_reader_source_hash_provenance": "Retained source binding recorded for the prior reader edition; no new source-reader equality claim.", "current_source_sha256": "94a62c4be0f39475db008e15c10467b9755161371bf418b7b09a55263c5a5dd6", "source_edition_status": "Current complete HTML reading; current PDF available.", "reader_pending": false, "current_reader_source_policy_refresh_pending": false, "source_reader_equality_claimed": true, "accepted_writer_proof_record_sha256": "85bf4169f35ae6dcea2c46a9de053c785fed61d0ce6b1841f60e9780e9abd22a", "HTML_source_equality_claimed": true, "tex_sha256": "00a74a467723ed70b74810bdf31aa42dd14b62e1e0c1a183d72a95c43896a1ea", "pdf_sha256": "9423a964d3e29ebc090effb9d75707ccb16618dbefce057386cd1df1eee5c3cb", "pdf_pages": 32, "reader_sha256": "a068219a7cd383f3136c0a4a5629d74967dd107ec8142493d3b632c8426f21c6", "public_PDF_prerequisite_routes_current": true, "current_HTML_source_equality_claimed": true, "current_PDF_source_equality_claimed": true }, { "id": "AN03-U013", "title": "Curved weights and the directions in which support can end", "source": "src/support-continuation.md", "reader": "support-continuation.html", "source_sha256": "6e40fd6cdc17cea25d50c78a35d75e4fcf2e836568f6969fc1198253e8fe967b", "original_current_source_sha256": "dbad82e2f9ed8cb85480c4d8003a387f58902104c05b64fd39cec2c72a57fff0", "license": "CC0-1.0", "notice": "CC0 1.0", "scope": "Complete current supporting reading, with its full theorem ranges and proofs.", "heading_aliases": [ { "line": 1, "ids": [ "curved-weights-and-the-directions-in-which-support-can-end" ], "heading": "Curved weights and the directions in which support can end" }, { "line": 7, "ids": [ "AN03-SUC-001", "an03-suc-001-support-geometry-coefficients-and-entry-assumptions", "1-support-geometry-coefficients-and-entry-assumptions", "support-geometry-coefficients-and-entry-assumptions" ], "heading": "1. Support geometry, coefficients, and entry assumptions" }, { "line": 76, "ids": [ "AN03-SUC-002", "an03-suc-002-a-frequency-inequality-with-one-normal-derivative", "2-a-frequency-inequality-with-one-normal-derivative", "a-frequency-inequality-with-one-normal-derivative" ], "heading": "2. A frequency inequality with one normal derivative" }, { "line": 106, "ids": [ "AN03-SUC-003", "an03-suc-003-a-finite-norm-identity-for-a-signed-quadratic-weight", "3-a-finite-norm-identity-for-a-signed-quadratic-weight", "a-finite-norm-identity-for-a-signed-quadratic-weight" ], "heading": "3. A finite norm identity for a signed quadratic weight" }, { "line": 161, "ids": [ "AN03-SUC-004", "an03-suc-004-exchanging-derivatives-across-a-lipschitz-multiplier", "4-exchanging-derivatives-across-a-lipschitz-multiplier", "exchanging-derivatives-across-a-lipschitz-multiplier" ], "heading": "4. Exchanging derivatives across a Lipschitz multiplier" }, { "line": 221, "ids": [ "AN03-SUC-005", "an03-suc-005-from-a-linear-weight-to-a-curved-one", "5-from-a-linear-weight-to-a-curved-one", "from-a-linear-weight-to-a-curved-one" ], "heading": "5. From a linear weight to a curved one" }, { "line": 274, "ids": [ "AN03-SUC-006", "an03-suc-006-the-uniform-estimate-with-lipschitz-coefficients", "6-the-uniform-estimate-with-lipschitz-coefficients", "the-uniform-estimate-with-lipschitz-coefficients" ], "heading": "6. The uniform estimate with Lipschitz coefficients" }, { "line": 365, "ids": [ "AN03-SUC-007", "an03-suc-007-the-support-theorem-at-the-initial-sobolev-regularity", "7-the-support-theorem-at-the-initial-sobolev-regularity", "the-support-theorem-at-the-initial-sobolev-regularity" ], "heading": "7. The support theorem at the initial Sobolev regularity" }, { "line": 452, "ids": [ "AN03-SUC-008", "an03-suc-008-complex-quadratic-forms-and-the-dimension-distinction", "8-complex-quadratic-forms-and-the-dimension-distinction", "complex-quadratic-forms-and-the-dimension-distinction" ], "heading": "8. Complex quadratic forms and the dimension distinction" }, { "line": 542, "ids": [ "AN03-SUC-009", "an03-suc-009-worked-models-and-limits-of-the-criterion", "9-worked-models-and-limits-of-the-criterion", "worked-models-and-limits-of-the-criterion" ], "heading": "9. Worked models and limits of the criterion" }, { "line": 572, "ids": [ "AN03-SUC-010", "an03-suc-010-problems-with-complete-solutions", "10-problems-with-complete-solutions", "problems-with-complete-solutions" ], "heading": "10. 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Conventions and background", "level": 2 }, { "line": 19, "heading": "Notation", "ids": [ "notation" ], "level": 3 }, { "line": 40, "heading": "Proofs used from earlier lessons", "ids": [ "proofs-used-from-earlier-lessons" ], "level": 3 }, { "line": 88, "ids": [ "AN03-TC-SCHUR-001", "schurs-bound-on-the-original-measure-spaces" ], "heading": "1.1. Schur's bound on the original measure spaces", "level": 3 }, { "line": 101, "heading": "A measurable counterexample with both bounds at every point", "ids": [ "a-measurable-counterexample-with-both-bounds-at-every-point" ], "level": 4 }, { "line": 141, "heading": "The exact \\(L^2\\) operator without measure restrictions", "ids": [ "the-exact-l2-operator-without-measure-restrictions" ], "level": 4 }, { "line": 200, "heading": "The raw integral on every finite-measure piece", "ids": [ "the-raw-integral-on-every-finite-measure-piece" ], "level": 4 }, { "line": 243, "heading": "The space defined by this defect and its complete quotient map", "ids": [ "the-space-defined-by-this-defect-and-its-complete-quotient-map" ], "level": 4 }, { "line": 284, "heading": "When the raw-integral assertion is recovered", "ids": [ "when-the-raw-integral-assertion-is-recovered" ], "level": 4 }, { "line": 314, "ids": [ "AN03-TC-002", "2-totally-characteristic-differential-operators", "totally-characteristic-differential-operators" ], "heading": "2. Totally characteristic differential operators", "level": 2 }, { "line": 348, "ids": [ "AN03-TC-003", "3-function-spaces-on-the-half-space", "function-spaces-on-the-half-space" ], "heading": "3. Function spaces on the half space", "level": 2 }, { "line": 350, "heading": "Restrictions and supports", "ids": [ "restrictions-and-supports" ], "level": 3 }, { "line": 363, "heading": "Restricted Schwartz functions", "ids": [ "restricted-schwartz-functions" ], "level": 3 }, { "line": 401, "ids": [ "AN03-TC-004", "4-symbols-compressed-quantization-and-lacunarity", "symbols-compressed-quantization-and-lacunarity" ], "heading": "4. Symbols, compressed quantization and lacunarity", "level": 2 }, { "line": 403, "heading": "The symbol class and its quantization", "ids": [ "the-symbol-class-and-its-quantization" ], "level": 3 }, { "line": 435, "heading": "Lacunary symbols", "ids": [ "lacunary-symbols" ], "level": 3 }, { "line": 470, "heading": "Every symbol is lacunary up to a residual symbol", "ids": [ "every-symbol-is-lacunary-up-to-a-residual-symbol" ], "level": 3 }, { "line": 510, "ids": [ "AN03-TC-005", "5-action-on-restricted-schwartz-functions", "action-on-restricted-schwartz-functions" ], "heading": "5. Action on restricted Schwartz functions", "level": 2 }, { "line": 512, "heading": "Continuity, commutators and boundary jets", "ids": [ "continuity-commutators-and-boundary-jets" ], "level": 3 }, { "line": 566, "heading": "Composition with totally characteristic derivatives", "ids": [ "composition-with-totally-characteristic-derivatives" ], "level": 3 }, { "line": 582, "ids": [ "AN03-TC-006", "6-kernels-near-the-corner", "kernels-near-the-corner" ], "heading": "6. Kernels near the corner", "level": 2 }, { "line": 584, "heading": "Coordinates at the corner", "ids": [ "coordinates-at-the-corner" ], "level": 3 }, { "line": 609, "heading": "Residual kernels", "ids": [ "residual-kernels" ], "level": 3 }, { "line": 679, "heading": "A kernel bound at finite negative order", "ids": [ "a-kernel-bound-at-finite-negative-order" ], "level": 3 }, { "line": 694, "heading": "Examples of residual kernels", "ids": [ "examples-of-residual-kernels" ], "level": 3 }, { "line": 706, "heading": "Residual kernels are polyhomogeneous conormal distributions", "ids": [ "residual-kernels-are-polyhomogeneous-conormal-distributions" ], "level": 3 }, { "line": 767, "ids": [ "AN03-TC-007", "7-adjoints", "adjoints" ], "heading": "7. Adjoints", "level": 2 }, { "line": 771, "heading": "The adjoint of a strongly lacunary residual operator", "ids": [ "the-adjoint-of-a-strongly-lacunary-residual-operator" ], "level": 3 }, { "line": 796, "heading": "The adjoint transform", "ids": [ "the-adjoint-transform" ], "level": 3 }, { "line": 831, "heading": "Adjoints of lacunary operators", "ids": [ "adjoints-of-lacunary-operators" ], "level": 3 }, { "line": 852, "ids": [ "AN03-TC-008", "8-composition", "composition" ], "heading": "8. Composition", "level": 2 }, { "line": 924, "heading": "Complex cutoffs in the differential example", "ids": [ "AN03-TC-ADJOINT-CUTOFF-001", "complex-cutoffs-in-the-differential-example" ], "level": 3 }, { "line": 1000, "ids": [ "AN03-TC-009", "9-extension-to-distributions", "extension-to-distributions" ], "heading": "9. Extension to distributions", "level": 2 }, { "line": 1004, "heading": "Supported and restricted distributions", "ids": [ "supported-and-restricted-distributions" ], "level": 3 }, { "line": 1043, "heading": "Residual operators produce conormal distributions", "ids": [ "residual-operators-produce-conormal-distributions" ], "level": 3 }, { "line": 1060, "ids": [ "AN03-TC-010", "10-boundedness-on-l-2-sobolev-and-besov-spaces", "boundedness-on-l2-sobolev-and-besov-spaces" ], "heading": "10. 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