Course contents

One hundred and forty-four lessons treat distributions and their order, kernels, boundary values, singular powers, convolution, wave kernels, Fourier growth, derivative-growth classes, wavefront localization, holomorphic decompositions, complex kernel continuation, prescribed wavefront sets, holomorphic wedges, homogeneous Fourier correspondence, analytic pullbacks, distribution products, proper integration, conic convergence, general kernel composition, analytic support boundaries, convex contact, characteristic transport, closed-set invariance, analytic differential regularity, Holmgren uniqueness, Poisson-bracket tests for support, convergent ramified roots, exact characteristic-half-space solutions, their analytic wavefronts, convex-domain uniqueness, support criteria in half-spaces and proper cones, analytic parametrices with entire errors, propagation along straight characteristics at every derivative-growth scale, exact prescribed global singular rays with smooth forcing, analytic regularity for convolution equations on proper supports, microhyperbolic cones, boundary inverses, exact convex wavefront bounds, homogeneous waves with prescribed one-sign light rays, analytic functionals with real-carrier approximation, entire harmonic lifts, canonical Poisson fields, their full converse modulo entire harmonic functions, decay normalization, smallest real carriers, finite carrier splitting with exact overlap ambiguity, completeness modulo smaller carriers, arbitrary-cover hyperfunction gluing, analytic coordinate changes, atlas independence and extension with support in the relative closure, spherical hyperfunction recovery, analytic wavefronts, growth-free holomorphic boundaries, their exact converse, the opposite-cone extension theorem, angular localization, closed-cone splitting with exact intersection ambiguity, the holomorphic-germ quotient, analytic substitution for hyperfunctions, the inverse-Jacobian coordinate comparison, distributional restriction agreement, and exact coordinate and submersion wavefronts, full hyperfunction tensor wavefront bounds, admissible products, chain and Leibniz rules, compatible analytic pullbacks and finite products under joint noncancellation, proper hyperfunction integration, closed-ball angular recovery, zero-fiber-covector bounds, differentiation and iterated integration, analytic fiber-coordinate changes and analytic-input kernels, singular-input kernel action with output-axis terms, contraction through zero intermediate covectors, proper-support convolution, simultaneous compact replacement and convolution associativity, uniform Baire bounds for analytic fiber weights, Cauchy carrier separation, both hyperfunction support-normal signs, rough and convex boundary contact, and growth-free open-set uniqueness, fixed-fiber contour representatives, integration of canceling holomorphic sides, noncharacteristic analytic substitution through proper integrals, singular projection formulas, composed kernel action and finite kernel associativity with every zero intermediate covector, rectangular analytic primitives, weighted derivative and primitive bounds, convergent analytic perturbation with exact remainder and germ uniqueness, and thin-polydisc analytic Cauchy existence with arbitrary normal jets, analytic Cauchy germs along noncharacteristic hypersurfaces, continuation through real C1 boundaries, expanding convex domains, global analytic Cauchy solutions and complex-cone existence with continuation across the vertex, full noncharacteristic hyperfunction regularity, microlocal existence, directional quotient inverses, two-sided noncharacteristic traces and C1 supporting-surface uniqueness, supported hyperfunction division with exact support, canonical one-sided extensions and normal data, exact boundary-source matrices, distributional endpoint trace comparison and analytic-coordinate compatibility, Gaussian carrier estimates, exact Gaussian inversion and recovery of arbitrary compact hyperfunctions, and exponential characterization of analytic regularity. The reader includes editable sources, reproducible illustrations and the prerequisite readings used by its proofs. Gaussian rescaling, both tangent directions of fiber normals, affine sections with closure after projection, convex propagation within a fiber, analytic boundary regularity at all normal frequencies, and smooth-data counterexamples are proved with worked geometric examples. The Fourier–Laplace range for every compact convex real carrier is constructed by Laplace contours, moment recovery and rotated boxes; examples explain infinite point derivatives and nonconvex-carrier limitations. Gaussian reconstruction supplies global derivative-growth representatives modulo analytic functions and the global regular-difference characterization of wavefront sets. Smooth extension at every finite order, square-variable descent with parameters, coordinate parity, exchange invariants and smooth flat majorants are constructed explicitly. Exact distributional extension orders are determined for powers with inverse-power oscillation, and a real phase giving an order-one extension is constructed for every continuously differentiable amplitude. Concentration and oscillatory limits retain exact point masses, periodic coefficients, radial center contributions and the dependence on two boundary parameters; a uniform gradient bound turns distributional convergence into local uniform convergence. Products and singular limits include coordinate principal values, quartic finite parts throughout a complex half-plane, logarithmic concentration on crossing axes, and exact overlap laws for moving Cauchy kernels. Primitives on intervals give exact derivative-order bounds, finite-measure endpoint extensions and weighted Taylor subtractions; exponential phases have the sharp extension threshold including equality. Reciprocal compression distinguishes principal values from concentrated mass; finite-difference moments represent every pole derivative, and smooth boundary coefficients determine all surviving point jets. Reflected endpoints produce explicit point derivatives; finite-order zeros permit smooth division on intervals, with a complete description of the kernel and sharp order bounds. Degenerate differential equations are classified by their admissible exponential branches, finite point-jet inverses and freely prescribed constants between isolated zeros. Coordinate-power kernels have arbitrary tangential distribution coefficients; simultaneous polynomial constraints reduce to finite jet equations. Directional uniqueness and parabolic boundary poles have sharp criteria and exact order bounds. Sublevel areas characterize planar concentration at nondegenerate zeros; logarithmic traces define cut boundary values, and polygonal logarithmic potentials converge to a circular curvature measure. Boundary variation determines derivative order; a complete quadratic division criterion handles accumulating zeros, and exact interface and branch formulas determine distributional jumps. Finite-order complex-phase estimates combine cancellation and damping with exact weights, including integrable degeneracy at critical points. Finite stationary expansions retain precise phase and amplitude regularity, explicit differential coefficients and complex Gaussian branches; nilpotent quadratic exchange has an exact finite remainder and a sharp loss. Paired oscillatory variables have an explicit critical point and a mixed-derivative expansion with a precise amplitude budget, factorial coefficients and sharp rounded decay orders. Boundary interval bounds propagate by a positive harmonic envelope on connected upper domains, including domains with holes; the full asymptotic order follows from deeper truncation. Smooth complex critical graphs give explicit finite-derivative stationary coefficients, a fixed Gaussian branch and absolute remainders, including null imaginary-Hessian directions and parameters without real critical points. Curved surfaces have uniform Fourier asymptotics, convex bodies have two signed normal contributions, and a smooth summation argument gives a quantitative lattice-count remainder. Stationary phase on a half-space has exact endpoint signs, half-integer critical expansions and a Fresnel transition as a real critical point crosses the boundary. Finite-order real phases have complete asymptotic expansions with exact gamma coefficients, parity cancellations and nonlinear Jacobians; two canonical Airy profiles remain uniform across a cubic collision. Smooth polynomial normal coordinates give uniform cubic Airy expansions for arbitrary compact amplitudes, including one degenerate direction with nondegenerate transverse variables. Complex homogeneous phases define oscillatory distributions for every positive frequency gain and base loss less than one, with exact order bounds, projected stationary sets and normalized causal wave kernels. Fractional difference energies have exact Fourier constants and limiting values; weighted Fourier embeddings have sharp boundaries and explicit counterexamples. Averaged annular derivatives give weak first-endpoint, finite-exponent and Hölder multiplier estimates, interior elliptic regularity and an explicit continuous-forcing obstruction. Complex homogeneous phases with nonnegative imaginary part obey the full stationary covector bound, illustrated by quadratic damping with one singular direction and both directions for its real part. Compact distributions have an exact frequency projection of their smooth wavefront, and arbitrary smooth densities carry every conormal direction over their full support, including flat zeros of the weight. Transverse smooth densities restrict and multiply with an exact normal-angle determinant; curved intersections and vanishing restricted weights determine their full supports and conormal directions. Quadratic boundary powers and logarithms have exact point-source constants and complete wavefront sets, with signature-dependent vertex directions and all forward wave powers. Holomorphic boundary values converge along arbitrary smooth approaches to real submersions, with uniform finite-test bounds and exact examples at regular and critical levels. Any closed conic smooth wavefront set can be realized by a bounded integrable function with prescribed finite differentiability; symmetric direction sets admit real-valued realizations. Local holomorphic wedges with polynomial bounds on each smaller approach cone have compatible boundary distributions and the exact dual-cone restriction; smooth and analytic remainders have distinct effects. Real principal type polynomials admit a uniform estimate for arbitrary additional imaginary displacements, with exact linear and nonlinear examples. Summable convolution gives cutoffs with quantitative directional bounds in every finite-dimensional norm. Nonquasianalytic classes admit compact cutoffs and fixed Fourier witnesses, with explicit Gevrey estimates. The Denjoy–Carleman criterion characterizes unique infinite jets for arbitrary positive derivative budgets. Convex minorants, compact functions and the exact numerical condition for fixed Fourier tests have complete proofs and examples. Quadratic boundary distributions and causal wave powers have exact smooth, analytic and derivative-growth wavefront classifications, including all signatures and vertex directions. Tangential coefficients and normal deltas have exact wavefront formulas. Analytic submanifold densities and transverse intersections include both normal signs and zeros of the restricted weights. Regular simple arcs give minimal complex carriers. Polynomial approximation, exact endpoint tests and holomorphic weights distinguish minimal carriers from regular path images containing loops. Banach-space calculus gives local inverses, sections and retractions, differentiation from dense directions, symmetric higher derivatives and complete Taylor remainders. Symmetric convex bodies have a unique largest ellipsoid, a finite contact-point certificate and a sharp square-root-of-dimension comparison with a Euclidean norm. Slowly varying norms admit locally finite partitions with fixed or variable summable derivative budgets and smooth closed-set separators with explicit distance estimates. Smoothing across zero scale gives finite normal-jet extensions with local bounds; arbitrary smooth normal jets can be realized in a prescribed common compact support or over an open base.

Learning guide · Prerequisite readings · Offline reader and sources