Constant-coefficient equations and solvability
Polynomial symbols connect support geometry with regularity and solvability. The topics include fundamental solutions, operator strength, propagation, hyperbolic equations, Cauchy and boundary problems, polynomial division, global solvability and convolution systems.
A draft course with 214 lessons, worked examples, exercises and solutions. Individual lessons name their prerequisite assumptions, including supporting results whose proofs are still planned.
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Lessons
- Fundamental solutions and the directions an equation uses
- Measuring regularity with weighted Fourier spaces
- Local regularity, sharp embeddings, and compactness
- Regular kernels and changes in the equation
- Operator strength and local inverses
- Rescaled symbols and stable strength
- Freezing coefficients without losing strength
- The geometry of coefficients and strength
- Positive symbols and energy estimates
- Approximation and global solvability from support geometry
- Choosing polynomial and exponential approximants
- Singular supports and arbitrary distribution data
- Boundary distance and propagation
- Planar domains and directional solvability
- Wave kernels with complex coefficients
- Lorentz cones and domain solvability
- Cylindrical wedges and support
- Symbols at infinity
- Wavefronts of regular kernels
- Regularity and tempered growth
- Hypoellipticity and complex zeros
- Weighted interior estimates and operator strength
- Algebraic families of hypoelliptic operators
- Partial smoothing and polynomial coefficients
- From partial regularity to smooth solutions
- Subspace detection and singularity carriers
- Logarithmic inverses and smooth barriers
- Geometry of singular supports
- Uniform interior estimates with distance weights
- Directional growth and complex-zero geometry
- Anisotropic derivative classes and analyticity
- Petrowsky parabolicity and semielliptic regularity
- The wave Cauchy problem and Kirchhoff's formula
- Spacelike Cauchy surfaces and the Riesz formula
- Wave rays, caustics, and stationary amplitudes
- Simple wave caustics and the Airy transition
- Causal solvability forces hyperbolicity
- Hyperbolicity and lower order terms
- Multiple characteristics and allowed lower order terms
- Causal fundamental solutions and lower order expansions
- Cauchy data, regularity and spacelike initial surfaces
- Continuous functionals, test families and compact limits
- Lebesgue duality and the functionals on Fourier spaces
- Primitive factors and moving polynomial roots
- Cauchy bounds, root counts and analytic extensions
- Fixed-support derivatives and logarithmic Fourier graphs
- Polygonal regions and the directions on a circle
- Smooth convex sweeps and first contact
- Principal roots and uniform time kernels
- Small Gevrey classes and compact Fourier decay
- Small Gevrey solutions of the full Cauchy problem
- Small Gevrey fundamental solutions in the principal polar cone
- Fourier windows, sector growth and branched vanishing
- Uniqueness in a slab with bounded support
- Compact Cauchy data and local coherence
- Sharp support for irreducible Cauchy equations
- Regular balls, moving roots and complex gauges
- Weighted inversion on a half-line
- Analytic norms and propagation on complex balls
- Uniform bounds for algebraic analytic branches
- Root factors and the full symbol norm
- Weighted estimates from analytic root branches
- Disintegrating half-space restriction norms
- Weighted control on the negative half-space
- Local supported solutions with the exact symbol gain
- Supported smooth approximation
- Global supported solvability on countably many scales
- Supported fundamental solutions and test estimates
- Analytic root barriers and supported solvability
- Evolution operators in a component of equal strength
- Two-dimensional evolution roots and model components
- Incoming normal roots at a flat boundary
- Boundary data for a decaying half-line equation
- Boundary determinants as causal Fourier kernels
- Missing boundary rank produces a causal solution
- Boundary determinants annihilate causal solutions
- The principal boundary symbol at high frequency
- Degenerate boundary symbols and smooth nonuniqueness
- Uniqueness from the principal boundary symbol
- A necessary time strip for smooth mixed solvability
- Extending a boundary time strip to its propagation cone
- Polynomial reciprocal bounds inside a mixed boundary tube
- Boundary fundamental kernels and their propagation support
- Compatible smooth mixed data and the data that determine a solution
- Causal mixed solutions with distribution-valued data
- Complex oblique derivatives for the wave equation
- A reflected wave and its faster boundary pulse
- Constant-strength inverses with causal support
- Constant strength, regularity, and the directions of singularities
- Fourier limits and the obstructions that descend to an open set
- Global solvability and finite-dimensional adjoint obstructions
- Real Laurent paths and the growth of normal windows
- Flat half-space solutions from real frequency rays
- Two flat boundaries and a nonzero phase bridge
- Characteristic rays and directional strength
- Curved support boundaries and principal preservation
- Simple-root phases and flat switching errors
- Rational complex frequency paths without real poles
- Moving complex frequency windows to a linear limit
- Physical transport scales and curved phases
- Joining modes through a vanishing operator image
- Periodic flat solutions and uniform integer frequencies
- Compact elliptic kernels and adjoint obstructions
- Complex frequency windows and exact half-space support
- Every normal from a polynomial strength imbalance
- Splitting normal roots and exceptional-factor perturbations
- Elliptic nonuniqueness from complex frequency windows
- Normal root distances and compatible complex scales
- Small weaker perturbations and hyperbolic uniqueness
- Flat half-space solutions with perturbations of arbitrary order
- Smooth solutions with an exact characteristic halfspace as support
- Null-section curvature and the Riesz comparison
- Fundamental solutions in shrinking cones
- Root growth and Cauchy evolution in Schwartz spaces
- Support cones force reciprocal bounds
- When zero smooth periods mean that a cycle bounds
- Logarithmic Fourier graphs construct a cone-supported inverse
- Tangent cones that permit an imaginary push
- A wave sphere that bounds rationally but not integrally
- Averaging an entire function while avoiding polynomial zeros
- Counting cycles with rational periods
- Compact Fourier division and multiplicity-sensitive annihilators
- Polynomial tests and a convergent local Fourier quotient
- Projective exhaustion and the finite-chain tube receiver
- Ordinary finite-chain Morse handles and the exhaustion bound
- Compact supports, a closed submanifold and the normal circle
- Compact forms and the unshifted right cap
- Rational compact cohomology and the normal-circle connecting map
- The canonical rational compact connecting map and normal circle
- Smooth-wavefront convolution with a kernel singular only at zero
- Local Newtonian potentials and subharmonic regularity
- Poisson extension and subharmonic comparison
- Horizontal envelopes and their limiting slope
- Distributional limits of subharmonic functions
- Boundary measures and Green potentials in a half-space
- A linear profile under dilation
- Fourier endpoints and the asymptotic density of zeros
- Compact support and the Carleman condition
- Plurisubharmonic envelopes and support functions
- Constant upper envelopes and scaled plurisubharmonic averages
- Directional averages and additivity of growth indicators
- Fourier indicators and the convex hull of a measure's support
- Local compactness and Hartogs bounds
- Solving the Cauchy–Riemann equations with a weight
- Weighted holomorphic extensions and their growth
- Fourier transforms of analytic functionals on a real convex carrier
- Entire logarithms and the approximation of plurisubharmonic functions
- Complex Fourier estimates and weak exponential representations
- Four-condition complex weights and the topology of compact Fourier tests
- Weak solution representations near a characteristic variety
- Hermite interpolation and local division with a discriminant
- Exponential-polynomial representations on the characteristic hypersurfaces
- Compact smooth-test topology and its Fourier neighborhoods
- From complex solution representations to real regularity estimates
- Integral representations for arbitrary distributions
- Why complex Fourier weights need a construction
- Joint logarithmic-frequency limits
- Locating singularities through logarithmic Fourier strips
- Singularities and nonconvex logarithmic carriers
- Convolution and joint frequency carriers
- Changing centers in Fourier windows
- Frequency-selective singularities and smooth convolutions
- Slow decrease and entire Fourier division
- Recovering singularities from convolution profiles
- Isolated atoms and separated singularities
- Curved boundaries and directional Fourier carriers
- Exponential-polynomial solutions and convex approximation
- Analytic forcing and smooth convolution solutions
- Smooth forcing, invertible kernels and support confinement
- Support distances and admissible convolution domains
- Fréchet duality and smooth convolution solvability
- Convolution modulo smooth functions and compact singularity bounds
- Singular Fourier profiles and convex equation domains
- Solving convolution equations for arbitrary distribution data
- Finite-order convolution solutions and logarithmic windows
- Logarithmic zero retreat and reflected singularities
- Causal inverses and stability at a cone's vertex
- Zero-free cones turn slow decrease into reciprocal bounds
- Radial boundary kernels and regularity of convolution
- Compact multipliers and exponential solutions of convolution systems
- Analytic forcing without global analytic solutions
- Rational top forms detect cycles in a hypersurface complement
- Finite covers and normal circles in projective complements
- Equatorial cycles and the projective period test
- Angular distributions and their full Fourier transforms
- Analytic directions of holomorphic boundary values
- Dilation equations and tangential analytic singularities
- Pulling back and combining analytic singularities
- Gaussian packets and homogeneous Fourier directions
- Derivative sequences and microlocal regularity
- Analytic coefficients and one-sided uniqueness
- Real roots and their convex component
- One-sided root bounds and Taylor order
- Support normals and analytic singularities
- Holomorphic reciprocals and convolution regularity
- Harmonic potentials and real carriers
- Splitting real carriers and gluing hyperfunctions
- Polynomial reciprocals and tempered fundamental solutions
- Noncharacteristic directions and microlocal regularity
- Characteristic rays and directional kernels
- Resolving singular directions at a holomorphic boundary
- Why the singular carrier leaves an open set
- One contour for a compact family of equations
- Contour spaces and solvability for families of equations
- Power series and analytic lifting in Banach spaces
- Free resolutions for analytic operator kernels
- Banach-valued Dolbeault equations and polynomial approximation
- Finite cohomology bounds for analytic kernels
- Factorization and free presentations on convex products
- Reciprocal graphs and global analytic selection
- Holomorphic solutions on expanding contour scales
- Characteristic supporting surfaces and boundary curvature
- Homogeneous elliptic kernels and logarithmic scaling
- Quadratic boundary kernels and signature phases
Supporting readings and further routes
- Constant-coefficient equations and solvability
- Library
- Boundary flux and weak identities
- Versions, exact downloads and terms
- Complete distribution foundation readings