Constant-coefficient equations and solvability

Polynomial symbols connect the geometry of support with regularity and solvability. These lessons develop the analytic tools needed to follow that connection.

Draft course. The course is still being written.

148 lessons with proofs, worked examples, exercises and solutions. Some arguments state exact planned prerequisites; those remain visible as assumptions while their foundational proofs are developed.

Download the reader and editable sources · Read the supporting proof chapters

This public draft contains 148 lessons. The exact characteristic-halfspace prerequisite now has a complete written construction. The null-section curvature lesson compares the intrinsic and ambient wave representations, including every focal degeneracy. The shrinking-cone lesson constructs a fundamental solution in each closed cone of a decreasing sequence and proves that no general distribution fundamental solution can lie in their intersection. The Schwartz evolution lesson proves the exact logarithmic root criterion for autonomous finite-dimensional multipliers, gives all higher-time-order traces and forcing factors, and constructs an explicit causal delay inverse. The support-cone lesson proves the general forward support/reciprocal implication with its complete proof, seven solved problems and two exact reproducible figures. It preserves all conditional projective and coefficient premises. Four complete cached CC0 distribution foundations accompany the reader. It constructs one common distributional inverse under uniform closed-angular reciprocal bounds for a nonempty ordinarily convex open positive-dilation cone excluding zero. The local tangent-cone lesson includes the complete specialized LC035-1–6 proof, three examples, six solutions and two exact figures, relative to C1/D1 and D4. The integral wave-sphere lesson includes the complete ordinary integral chain and mapping-torus proof, three comparisons, six solutions and its exact diagram: the actual four-dimensional wave sphere has nonzero integral order two, vanishes over characteristic-zero fields and remains nonzero over the two-element field; the two-dimensional reduced-degree-zero case bounds integrally. It supplies L004’s finite-degree smooth averaging and denominator-differential receiver at its declared scalar scope; all other L004 prerequisites remain explicit. The rational-period lesson retains the full PR1–PR31 learner and formal proof, three examples, six solutions and three original diagrams. General projective and component receivers remain explicit. The compact Fourier division lesson includes the full CF1–CF31 proof, three worked examples, six complete solutions and three original diagrams. Local polynomial germ convergence and compact singular-support hulls remain separate. The course still has explicitly planned foundations.

  1. Fundamental solutions and the directions an equation uses
  2. Measuring regularity with weighted Fourier spaces
  3. Local regularity, sharp embeddings, and compactness
  4. Regular kernels and changes in the equation
  5. Operator strength and local inverses
  6. Rescaled symbols and stable strength
  7. Freezing coefficients without losing strength
  8. The geometry of coefficients and strength
  9. Positive symbols and energy estimates
  10. Approximation and global solvability from support geometry
  11. Choosing polynomial and exponential approximants
  12. Singular supports and arbitrary distribution data
  13. Boundary distance and propagation
  14. Planar domains and directional solvability
  15. Wave kernels with complex coefficients
  16. Lorentz cones and domain solvability
  17. Cylindrical wedges and support
  18. Symbols at infinity
  19. Wavefronts of regular kernels
  20. Regularity and tempered growth
  21. Hypoellipticity and complex zeros
  22. Weighted interior estimates and operator strength
  23. Algebraic families of hypoelliptic operators
  24. Partial smoothing and polynomial coefficients
  25. From partial regularity to smooth solutions
  26. Subspace detection and singularity carriers
  27. Logarithmic inverses and smooth barriers
  28. Geometry of singular supports
  29. Uniform interior estimates with distance weights
  30. Directional growth and complex-zero geometry
  31. Anisotropic derivative classes and analyticity
  32. Petrowsky parabolicity and semielliptic regularity
  33. The wave Cauchy problem and Kirchhoff's formula
  34. Spacelike Cauchy surfaces and the Riesz formula
  35. Wave rays, caustics, and stationary amplitudes
  36. Simple wave caustics and the Airy transition
  37. Causal solvability forces hyperbolicity
  38. Hyperbolicity and lower order terms
  39. Multiple characteristics and allowed lower order terms
  40. Causal fundamental solutions and lower order expansions
  41. Cauchy data, regularity and spacelike initial surfaces
  42. Continuous functionals, test families and compact limits
  43. Lebesgue duality and the functionals on Fourier spaces
  44. Primitive factors and moving polynomial roots
  45. Cauchy bounds, root counts and analytic extensions
  46. Fixed-support derivatives and logarithmic Fourier graphs
  47. Polygonal regions and the directions on a circle
  48. Smooth convex sweeps and first contact
  49. Principal roots and uniform time kernels
  50. Small Gevrey classes and compact Fourier decay
  51. Small Gevrey solutions of the full Cauchy problem
  52. Small Gevrey fundamental solutions in the principal polar cone
  53. Fourier windows, sector growth and branched vanishing
  54. Uniqueness in a slab with bounded support
  55. Compact Cauchy data and local coherence
  56. Sharp support for irreducible Cauchy equations
  57. Regular balls, moving roots and complex gauges
  58. Weighted inversion on a half-line
  59. Analytic norms and propagation on complex balls
  60. Uniform bounds for algebraic analytic branches
  61. Root factors and the full symbol norm
  62. Weighted estimates from analytic root branches
  63. Disintegrating half-space restriction norms
  64. Weighted control on the negative half-space
  65. Local supported solutions with the exact symbol gain
  66. Supported smooth approximation
  67. Global supported solvability on countably many scales
  68. Supported fundamental solutions and test estimates
  69. Analytic root barriers and supported solvability
  70. Evolution operators in a component of equal strength
  71. Two-dimensional evolution roots and model components
  72. Incoming normal roots at a flat boundary
  73. Boundary data for a decaying half-line equation
  74. Boundary determinants as causal Fourier kernels
  75. Missing boundary rank produces a causal solution
  76. Boundary determinants annihilate causal solutions
  77. The principal boundary symbol at high frequency
  78. Degenerate boundary symbols and smooth nonuniqueness
  79. Uniqueness from the principal boundary symbol
  80. A necessary time strip for smooth mixed solvability
  81. Extending a boundary time strip to its propagation cone
  82. Polynomial reciprocal bounds inside a mixed boundary tube
  83. Boundary fundamental kernels and their propagation support
  84. Compatible smooth mixed data and the data that determine a solution
  85. Causal mixed solutions with distribution-valued data
  86. Complex oblique derivatives for the wave equation
  87. A reflected wave and its faster boundary pulse
  88. Constant-strength inverses with causal support
  89. Constant strength, regularity, and the directions of singularities
  90. Fourier limits and the obstructions that descend to an open set
  91. Global solvability and finite-dimensional adjoint obstructions
  92. Real Laurent paths and the growth of normal windows
  93. Flat half-space solutions from real frequency rays
  94. Two flat boundaries and a nonzero phase bridge
  95. Characteristic rays and directional strength
  96. Curved support boundaries and principal preservation
  97. Simple-root phases and flat switching errors
  98. Rational complex frequency paths without real poles
  99. Moving complex frequency windows to a linear limit
  100. Physical transport scales and curved phases
  101. Joining modes through a vanishing operator image
  102. Periodic flat solutions and uniform integer frequencies
  103. Compact elliptic kernels and adjoint obstructions
  104. Complex frequency windows and exact half-space support
  105. Every normal from a polynomial strength imbalance
  106. Splitting normal roots and exceptional-factor perturbations
  107. Elliptic nonuniqueness from complex frequency windows
  108. Normal root distances and compatible complex scales
  109. Small weaker perturbations and hyperbolic uniqueness
  110. Flat half-space solutions with perturbations of arbitrary order
  111. Smooth solutions with an exact characteristic halfspace as support
  112. Null-section curvature and the Riesz comparison
  113. Fundamental solutions in shrinking cones
  114. Root growth and Cauchy evolution in Schwartz spaces
  115. Support cones force reciprocal bounds
  116. When zero smooth periods mean that a cycle bounds
  117. Logarithmic Fourier graphs construct a cone-supported inverse
  118. Tangent cones that permit an imaginary push
  119. A wave sphere that bounds rationally but not integrally
  120. Averaging an entire function while avoiding polynomial zeros
  121. Counting cycles with rational periods
  122. Compact Fourier division and multiplicity-sensitive annihilators
  123. Polynomial tests and a convergent local Fourier quotient
  124. Projective exhaustion and the finite-chain tube receiver
  125. Ordinary finite-chain Morse handles and the exhaustion bound
  126. Compact supports, a closed submanifold and the normal circle
  127. Compact forms and the unshifted right cap
  128. Rational compact cohomology and the normal-circle connecting map
  129. The canonical rational compact connecting map and normal circle
  130. Smooth-wavefront convolution with a kernel singular only at zero
  131. Local Newtonian potentials and subharmonic regularity
  132. Poisson extension and subharmonic comparison
  133. Horizontal envelopes and their limiting slope
  134. Distributional limits of subharmonic functions
  135. Boundary measures and Green potentials in a half-space
  136. A linear profile under dilation
  137. Fourier endpoints and the asymptotic density of zeros
  138. Compact support and the Carleman condition
  139. Plurisubharmonic envelopes and support functions
  140. Constant upper envelopes and scaled plurisubharmonic averages
  141. Directional averages and additivity of growth indicators
  142. Fourier indicators and the convex hull of a measure's support
  143. Local compactness and Hartogs bounds
  144. Solving the Cauchy–Riemann equations with a weight
  145. Weighted holomorphic extensions and their growth
  146. Fourier transforms of analytic functionals on a real convex carrier
  147. Entire logarithms and the approximation of plurisubharmonic functions
  148. Complex Fourier estimates and weak exponential representations

Projective exhaustion and the qualified normal-first tube receiver now links the complete ordinary finite-chain Morse handle proof. Both retain every example, six complete solutions each, original native figures and exact programme-source context. Geometric H and the selected Thom/tubular entries keep their hypotheses; real coefficient comparisons, rational spanning, affine C8 and lower recursive closure remain separately qualified.

The complete local polynomial-annihilator proof now supplies L011’s quotient germ and its convergence, with all multiplicities, three examples, six solutions and three original diagrams. Its local-versus-entire distinction and lower compact-transform, polynomial and Cauchy entries remain explicit.

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/GFDL 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.

Complete distribution foundation readings.

Compact supports and the normal circle, Compact forms and the unshifted right cap, and Rational compact cohomology give the real closed–open sequence, its normalized form/right-cap comparison and the rational coefficient map. The canonical compact connecting map proves the separate rational cochain comparison. Smooth-wavefront convolution supplies the compact-input estimate used in L021. Each lesson includes complete proofs, examples, solutions, figures and reproducible sources under its stated hypotheses. Integral descent, rational-form spanning, affine C8, component constancy and the remaining foundation hypotheses are separate.