Subellipticity and unique continuation · Self-checked by the writing AI

Subellipticity and unique continuation

These lessons develop subelliptic models, weighted uniqueness, bracket estimates, adapted inverses, quadratic energy, complex spectra, uniform quadratic families and microlocal spectral criteria. Fractional conic regularity, canonical transport and weighted Taylor geometry give full smooth-symbol rotated-bracket tests, exact complex conic preparation, the adverse odd-crossing obstruction, necessary finite-type conditions and the complete mixed-polynomial sign classification. Sharp scalar derivative scales lead to the full finite-time-type operator bound for nonnegative principal symbols, including spatial dependence, phase-space localization and complex lower terms. Condition(P) then gives a positive local germ, a type-preserving global extension and the sharp conic estimate at every real order. An anisotropic bracket scale and a small-transverse-gradient cell give the first stable neighborhoods for the general argument. Oriented sign geometry aligns the whole transverse-gradient family and controls replacement directions. An explicit symplectic orbit map then yields the full canonical-cell residual estimate, including a longer spatial radius and every mixed derivative. The linear frequency coefficient has controlled polynomial jets and supplies the needed bracket-scale bounds. Full weighted polynomial reduction compares both the bracket family and the residual jet with Wronskian data, uniformly over normalized coefficients. The full polynomial-Wronskian dichotomy either keeps the bracket scale uniformly large or produces an exact cancellation plane, on both spatial-radius branches. Admissible neighborhoods are then defined by the uniform lower-bound alternative. Every point in the large-gradient case has an admissible center at the same time within a uniform multiple of the smaller spatial radius. A finite family of curved Hamiltonian neighborhoods covers the large-gradient region, with bounded overlap and a partition whose transition also has a small-gradient cover. Both cutoff families satisfy full transverse derivative and normalized transport bounds. The scalar quotient theorem controls approximate roots with all time and denominator errors. Its full affine normalization retains exact canonical scales, both rectangle widths and every real frequency; an admissible example corrects the printed pointwise upper jet bound. A nonnegative transport coefficient yields the same fractional gain on both frequency half-lines, with its full potential and time-derivative norms; the global finite-jet extension and each cutoff commutator are proved. Polynomial commutators give a uniform parameter gain through exact smooth jet identities and a Fourier multiplier energy proof. A normalized compact packet identifies the correct scope of real weights, and a valid replacement handles the entire real-weight class. One-sided quotient bounds control the complete weighted polynomial equation: a Lipschitz intercept extension and exact frequency split remove the mixed derivative error and retain both original derivative norms. A four-leaf packet test establishes the precise corrected scope. The full smooth model follows from a positive polynomial approximation that preserves the exact residual phase. Quantitative quotient comparisons,global coefficient saturation and direct variable-coefficient cutoff/inversion estimates give both the weighted gain and the bracket minimum throughout the rectangle. The full two-variable estimate then follows from attained rooted scales,positive time anchors and exact integration over comparable cells. Its literal printed complex bracket leaf is tested by a complete constant-coefficient counterexample;the real-leaf repair is proved on the entire stated class. A finite operator reconstruction recovers both partition norms and controls the mixed weighted matrix and both equation commutator families,with every composition remainder retained. Both localized scalar and affine gain estimates then follow,including exact coordinate and gauge transfer,Fourier coefficient ordering and the repaired output support condition. The full general sufficient estimate follows by absorbing both partition costs at a fixed large auxiliary parameter and recovering the original localized norm from every center-scale lower bound. The initial compact pseudodifferential cutoff commutator has its full uniform remainder bound. Exact scalar scaling proves sharpness,and a normalized Gaussian shows the necessity of the sign orientation.

These self-checked lessons cover the complete assigned scope. The full finite-type subelliptic criterion is proved,including infinitely flat time zeros,nearby characteristic root geometry and the complete conic estimate. The complex first-factor and bounded-imaginary-part estimates and smooth ordered pair are proved. Both weighted second-order auxiliary elliptic estimates are proved,including the mixed inverse and time-support extension. The intact second-order positive estimate is proved without choosing smooth roots,including its frozen-time variant and a smooth exchanging-root example. Both first-order bracket-control branches are proved with the source constant three and their complete symbol errors. The full smooth-coefficient Calderon theorem now includes angular factorization,weak graph approximation and the C1 surface conclusion. Admissible conormals now include full real-root alternatives,complex double-root derivative control and openness. The mixed-factor lesson proves the finite local real-factor estimates, intact quadratic cofactor recovery, the exact fixed-scale weighted estimate and oriented weak unique continuation for admissible normals. The general Carleman lesson retains the full polynomial symbol and exact leading constant, repairs the Weyl cutoff compression, and proves the real-tangent necessary condition with its noncharacteristic distinction. Angular calculus and a mixed companion estimate prove simple-root continuation with Lipschitz principal coefficients in every finite dimension, at the stated smooth real characteristic geometry and weak graph hypotheses. Exact used prerequisites are proved at their declared selected entry.

Named graph-calculus and ordinary or homogeneous symplectic prerequisites are available as exact CC0 reference sources, with original authorship and draft status retained. Only their exact reference sources are bundled. The exact prerequisites used here are proved at the declared selected entry; the full prerequisite courses keep their own scopes and statuses.

The independently written AN-03 prerequisite chapters are dedicated to CC0 1.0; license and attribution notices.

Reference licenses.

  1. Degenerate energy and sharp gains of regularity
  2. Finite type and the sign of the symbol
  3. Transport, exponential weights, and local uniqueness
  4. Inverting an operator with unequal frequency scales
  5. Brackets, drift, and general hypoellipticity
  6. Nonnegative symbols and weighted brackets
  7. Quadratic energy and the positive trace
  8. Melin's lower bounds
  9. A complex subprincipal symbol and one derivative lost
  10. Concentrated packets and necessary quadratic models
  11. The spectrum of a complex quadratic polynomial
  12. Quadratic coercivity when some directions have zero energy
  13. Parameter inverses and slowly quantized quadratics
  14. Shrinking oscillators and uniform quadratic bounds
  15. From frozen quadratics to a local estimate
  16. Spectral gaps and one derivative lost
  17. Detecting a fractional gain in a cone
  18. Weighted packets and the limit of the gain
  19. Canonical transport of fractional regularity
  20. Sign-constrained weighted Taylor geometry
  21. Rotated brackets and finite-jet flow coordinates
  22. Smooth complex preparation and conic flow coordinates
  23. Crossing direction and the necessary finite-type bound
  24. Full sign conditions for mixed weighted polynomials
  25. A derivative scale for signed one-dimensional equations
  26. From scalar time estimates to an operator bound
  27. From a constant sign to a conic gain
  28. An adaptive scale for repeated brackets
  29. How sign orientation aligns transverse gradients
  30. An explicit canonical cell for a large transverse gradient
  31. The linear frequency coefficient and the bracket scale
  32. Polynomial brackets and the residual jet
  33. A cancellation plane or a uniform bracket lower bound
  34. Finding an admissible center nearby
  35. Covering by curved Hamiltonian neighborhoods
  36. Cutoffs transported through adaptive neighborhoods
  37. How sign orientation controls approximate roots
  38. Normalizing the two-variable affine model
  39. Two frequency signs for a nonnegative transport coefficient
  40. A parameter gain from polynomial commutators
  41. A weighted equation controls the polynomial model
  42. A smooth model from a positive polynomial approximation
  43. From normalized cells to a two-variable subelliptic estimate
  44. Recovering localized norms and controlling partition commutators
  45. Local gain in the scalar and affine cells
  46. Assembling the general subelliptic estimate
  47. Recovering a conic norm from scaled frequency bands
  48. From rotated endpoint signs to subellipticity
  49. Exponential weights for first-order Cauchy factors
  50. Weighted elliptic control for a second-order evolution factor
  51. A double-root estimate without choosing smooth roots
  52. When a bracket controls the lost curvature
  53. Simple characteristic roots and local Cauchy uniqueness
  54. Which conormals admit simple and double Cauchy factors?
  55. Mixed Cauchy factors and admissible unique continuation
  56. General Carleman estimates and real tangent necessity
  57. Oriented strong pseudoconvexity and weak unique continuation
  58. Weak convexity, one-sided approximation, and compact contact
  59. Why weak convexity needs the fiber hypothesis
  60. C² surfaces and compact-contact uniqueness
  61. Angular calculus with Lipschitz coefficients
  62. Simple-root uniqueness with Lipschitz principal coefficients

The added lesson proves the weak quadratic estimate, negative-side adapted coordinates and compact-contact neighborhood uniqueness with the surrounding strong fiber hypothesis retained. The omitted-fiber interpretation is refuted in the following counterexample lesson. The C2 compact-contact extension is complete at its declared strict smooth interface and standing invertible real fiber. The simple-root Lipschitz coefficient theorem is proved in the two final lessons in every finite dimension, retaining its original characteristic and weak graph hypotheses. The complete assigned scope and exact used prerequisites are carried by these lessons and their linked proofs.

The added lesson proves oriented strong pseudoconvexity, uniform C² stability, exponential convexification and weak one-sided unique continuation. The smooth weak-convexity and compact-contact conclusions, the omitted-fiber counterexample and the C² extension are developed in the subsequent lessons. The full simple-root Lipschitz coefficient result is developed in the final two lessons.