Connecting maps and wavefront convolution
Read the complete canonical compact-cochain connecting-map proof and the smooth-wavefront convolution proof, with their examples, solutions, figures and reproducible sources.
Local Newtonian potentials
Read the complete lesson for the explicit Laplacian kernel, local positive-mass decomposition and exact-domain continuity proof. Original proof, learner text, native figures and reproducible sources.
Poisson extension and subharmonic comparison
Read the complete lesson for the explicit kernel, global harmonic polynomial approximation, compact comparison and exact supremum premise. Original proof, examples, solutions and reproducible native figures.
Horizontal envelopes and limiting slopes
Read the complete lesson for the compact maximum argument, exact harmonic slab barrier and secant proof of the slope and increment bounds. Original proof, examples, complete solutions, diagrams and editable sources.
Distributional limits of subharmonic functions
Read the complete lesson for the continuous-kernel cap, the exact local integrability range, and the measure test that identifies the upper limit. Original proof, learner text, native figures and reproducible sources.
Boundary measures and Green potentials
Read the complete lesson for the reflected Green kernel, compact interior cutoffs, positive harmonic sphere measures, the inversion which exposes linear height growth, and the exact slice integral giving the weak boundary trace. Original proof, learner text, native figures and reproducible sources.
A linear profile under dilation
Read the complete lesson for the exact horizontal Green integral, weighted signed-boundary estimates, and volume convergence under dilation. Original proof, solutions, native figures and reproducible sources.
Fourier endpoints and the asymptotic density of zeros
Read the complete lesson for exact endpoint slopes, the scaled positive zero measures and passage to disk counts with multiplicity. The explicit dilation input uses the half-space representation. Original exposition, solutions and reproducible native figures.
Compact support and the Carleman condition
Read the complete lesson for Gaussian Fourier injectivity, the logarithmic trace through zeros, the ordered derivative budget and the exact reciprocal integral. Original proof, learner text, solutions, native figures and reproducible sources.
Plurisubharmonic envelopes and support functions
Read the complete lesson for shifted complex-line slice finiteness, the recession increment bound, explicit dominated linear extension and log-annulus Liouville proof. Original arguments, examples, solutions and reproducible native figures.
Constant upper envelopes and scaled PSH averages
Read the complete lesson for positive translation averages, regularization of countable and arbitrary PSH suprema, the decreasing-tail proof, and the real-parameter scaled-average application. Original proof, learner text, native figures and reproducible sources.
Directional averages and additivity of growth indicators
Read the complete lesson for the all-line slope identification, fixed-ball submean, exact growing-radius qualifications and additive support sets. Original proof, examples, solutions and reproducible native figures.
Fourier indicators and the convex hull of a measure's support
Read the complete lesson. The Fourier transform of a compact complex measure has a PSH logarithm whose horizontal indicator equals its convex support function, including cancellation and nonatomic endpoints. Full proof · Reproduction sources.
Local compactness and Hartogs bounds
Read the complete lesson. A locally upper-bounded PSH sequence on a connected domain either collapses uniformly on compacts or has a proper local L1 limit along a subsequence. Explicit selection and moving maxima prove the compact Hartogs bound. Full proof · Reproduction sources.
Solving the Cauchy–Riemann equations with a weight
Read the complete lesson. Positive Levi curvature yields a weighted solution of closed Cauchy–Riemann data. Full graph-domain and weak-limit arguments prove the strict estimate and the general PSH estimate with exact factor 2 and square weight. Full proof · Reproduction sources.
Weighted holomorphic extensions and their growth
Read the complete lesson. An entire function on a complex linear subspace extends with the exact codimension-dependent weighted norm bound. Full restriction and iteration arguments give the precise polynomial growth transfer. Full proof · Reproduction sources.
Fourier transforms of analytic functionals on a real convex carrier
Read the complete lesson. The Fourier transform of an analytic functional has arbitrarily small exponential losses. A diagonal holomorphic extension constructs the functional and proves compatibility, uniqueness and local holomorphic test bounds. Full proof · Reproduction sources.
Entire logarithms and the approximation of plurisubharmonic functions
Read the complete lesson. Normalized logarithms of nonzero entire scalar functions approximate every proper plurisubharmonic function in local integral norm. Full dense-set convergence, bounded Cauchy–Riemann continuity and weighted interpolation proofs give the exact construction and closure. Full proof · Reproduction sources.
Complex Fourier estimates and weak exponential representations
Read the complete lesson. Three exact weighted complex Fourier estimates lead to a weak exponential representation against singular measures. Full support-cutoff, reflected-duality and compact-test extension proofs distinguish distributional integrals from pointwise values. Full proof · Reproduction sources.
Four-condition complex weights and the topology of compact Fourier tests
Read the complete lesson. A variable-scale smoothing and an exact curvature correction build a plurisubharmonic Fourier weight which dominates every continuous compact-test seminorm. Full compact-exhaustion, stabilized-maximum and separated-support arguments give all four weight conditions and the reflected weak representation. Full proof · Reproduction sources.
Weak solution representations near a characteristic variety
Read the complete lesson. A variable-scale polynomial correction and a weighted Cauchy–Riemann repair represent homogeneous solutions by weak integrals near their characteristic variety. Full support, compact-test and Hilbert-limit arguments prove the representation and transfer complex characteristic weight bounds to local regularity. Full proof · Reproduction sources.
Hermite interpolation and local division with a discriminant
Read the complete lesson. Full root-jet interpolation and a uniform analytic remainder on a complex ball give a discriminant-strength surface bound. Holomorphic root clusters, exact root-distance conventions and a complete weighted-mean argument retain control through central root collisions. Full proof · Reproduction sources.
Exponential-polynomial representations on the characteristic hypersurfaces
Read the complete lesson. An exact algebraic area bound and complete anisotropic division and weighted repair argument represent solutions by surface densities on their characteristic hypersurfaces. All repeated-factor jets, Fourier reflection and a proved polynomial loss are retained. Full proof · Reproduction sources.
Compact smooth-test topology and its Fourier neighborhoods
Read the complete lesson. Four constructive compact-test neighborhood criteria are equivalent to a convergent Fourier envelope and one strictly plurisubharmonic weighted integral. The complete logarithmic contour, successive exhaustion choices and whole-strip weight stabilization retain the exact source curvature. Full proof · Reproduction sources.
From complex solution representations to real regularity estimates
Read the complete lesson. Two complete exercise bridges identify the real polynomial-strength and complex characteristic weight criteria, and derive all directional derivative bounds from an actual convergent solution integral. Explicit heat solutions attain the spatial and time growth orders. Full proof · Reproduction sources.
Integral representations for arbitrary distributions
Read the complete lesson. Every distribution has a weak whole-complex frequency integral, and every homogeneous distributional solution has an actual characteristic-surface formula with all multiplicity jets. A complete finite-envelope smoothing proof accommodates unbounded local test orders. Full proof · Reproduction sources.
Why complex Fourier weights need a construction
Read the complete lesson. Exact negative Levi curvature explains the naive support weight obstruction. Strict seeds, controlled carrier enlargement and a cofinal family of Fourier norms recover the required topology. Holomorphic coordinate covariance and the scalar theorem retain their precise scope. Full proof · Reproduction sources.
Joint logarithmic-frequency limits
Read the complete lesson. Common logarithmic Fourier windows preserve singular profiles modulo smooth functions. Full PSH compactness and carrier bounds yield joint support-function families, surjective coordinate lifting and an arbitrary-family ultrafilter extension. Full proof · Reproduction sources.
Locating singularities through logarithmic Fourier strips
Read the complete lesson. A full fixed-order logarithmic-strip criterion locates the compact singular carrier. Gaussian contour inversion proves smoothness outside it,all logarithmic profiles recover its convex hull,and bounded polynomial division preserves that hull for compact distributions. Full proof · Reproduction sources.
Singularities and nonconvex logarithmic carriers
Read the complete lesson. Every compact distribution has singular support inside the closed union of its logarithmic profile carriers. Explicit finite-convolution frequency cells and a directional Taylor correction prove local smoothness in the gaps,with all derivative orders,empty cases and a disconnected carrier example. Full proof · Reproduction sources.
Convolution and joint frequency carriers
Read the complete lesson. Compact convolution adds logarithmic profiles and their convex carriers on one shared frequency sequence. The full joint-family identity includes proper and collapsed coordinates,and explains atomic cancellation and crossed singular factors whose convolution is smooth. Full proof · Reproduction sources.
Changing centers in Fourier windows
Read the complete lesson. An exact affine center change transports proper and collapsed logarithmic Fourier profiles. A shrinking-error diagonal gives uniform bounds throughout expanding frequency neighborhoods,including moving parameter points and common radii for finite or countable coordinate lists. Full proof · Reproduction sources.
Frequency-selective singularities and smooth convolutions
Read the complete lesson. Compact continuous functions with one isolated singularity can concentrate logarithmic Fourier activity in prescribed neighborhoods. The complete kernel and Baire construction characterizes distributions that smooth a nonsmooth convolution factor,with arbitrarily small support at any prescribed point. Full proof · Reproduction sources.
Slow decrease and entire Fourier division
Read the complete lesson. Five equivalent criteria connect logarithmic Fourier lower bounds to entire division by a compact distribution's transform. Complete half-plane and quotient-growth proofs include all transform zeros,a uniform Baire exponent and stability under finite-regularity compact perturbations. Full proof · Reproduction sources.
Recovering singularities from convolution profiles
Read the complete lesson. A chosen convolution profile is realized by a factor singular at one point. Complete proofs give the inverse singular-support criterion,universal singular-hull addition,the invertible difference-set bound,and polynomial profiles and point-supported operators. Full proof · Reproduction sources.
Isolated atoms and separated singularities
Read the complete lesson. Isolated support points belong to every logarithmic Fourier carrier. Full proofs construct positive atomic measures with any compact convex carrier and give the common-frequency maximum law for sums with separate singularities,including compact convolution singular-support calculus. Full proof · Reproduction sources.
Curved boundaries and directional Fourier carriers
Read the complete lesson. Positive curvature and uniform stationary phase give the exact segment carrier between opposite normal points. Full proofs handle logarithmic complex shifts,moving frequency directions,rapid remainders and a sharp directional wavefront bound. Full proof · Reproduction sources.
Exponential-polynomial solutions and convex approximation
Read the complete lesson. Full smooth approximation on convex domains for arbitrary nonzero compact convolution kernels. Entire annihilator division,convex analytic quotient carriers and Gaussian localization include noninvertible kernels. Full proof · Reproduction sources.
Analytic forcing and smooth convolution solutions
Read the complete lesson. Smooth solvability for analytic forcing and every nonzero compact convolution kernel on a convex domain. A uniform analytic-functional estimate and summable homogeneous corrections include the noninvertible case. Full proof · Reproduction sources.
Smooth forcing, invertible kernels and support confinement
Read the complete lesson. Necessary conditions for distributional convolution solvability on arbitrary compatible open domains. Compact smooth forcing requires invertibility;every smooth forcing also confines transpose-test supports. Full proof · Reproduction sources.
Support distances and admissible convolution domains
Read the complete lesson. Exact equivalence of support confinement and complement-distance preservation for compact convolution tests. Convex maximal equation domains and whole-component necessity include arbitrary nonconvex domain pairs. Full proof · Reproduction sources.
Fréchet duality and smooth convolution solvability
Read the complete lesson. General polar-section closure and adjoint surjectivity theorems yield smooth convolution solvability on compatible open domains. Uniform support and Fourier bounds establish compact inverse polars. Full proof · Reproduction sources.
Convolution modulo smooth functions and compact singularity bounds
Read the complete lesson. Canonical convolution on distribution classes modulo smooth functions. Isolated-singularity and increasing-order obstructions yield invertibility and compact singularity confinement,with an exact boundary-distance characterization. Full proof · Reproduction sources.
Singular Fourier profiles and convex equation domains
Read the complete lesson. Exact profile-carrier criteria for convex convolution domains. Full regular-boundary geometry establishes the reflected-normal condition and its converse,with examples separating point carriers from the combined singular hull. Full proof · Reproduction sources.
Solving convolution equations for arbitrary distribution data
Read the complete lesson. Full solvability for arbitrary distribution classes through proper local adjoints and two-domain exhaustion estimates. Smooth corrections give the ordinary criterion;finite-order extension and global representatives give bounded-region solution orders. Full proof · Reproduction sources.
Finite-order convolution solutions and logarithmic windows
Read the complete lesson. A fixed Fourier exponent in every logarithmic window gives one derivative loss and global finite-order convolution solutions. Full profile and complex-mean criteria,weighted exhaustion,domain correction and fundamental-solution necessity are proved. Full proof · Reproduction sources.
Logarithmic zero retreat and reflected singularities
Read the complete lesson. Complete compact-kernel hypoellipticity criteria, a common reciprocal exponent, a fully justified logarithmic-contour parametrix, exact reflected singularities and local quotient regularity. Full proof · Reproduction sources.
Causal inverses and stability at a cone's vertex
Read the complete lesson. Proper-cone causal inverse series,stability under changes away from a vertex,a finite differentiability threshold,and equivalent directional support geometry. Includes the exact preceding forward Fourier criterion and the explicitly qualified converse. Full proof · Reproduction sources.
Zero-free cones turn slow decrease into reciprocal bounds
Read the complete lesson. Slow decrease and angular zero-free logarithmic regions are equivalent to reciprocal polynomial/exponential bounds. A normalized harmonic-neighborhood argument gives a stronger exponential estimate;an explicit smooth compact kernel shows why zero-freeness alone is insufficient. Full proof · Reproduction sources.
Radial boundary kernels and regularity of convolution
Read the complete lesson. Spherical boundary-value kernels are hypoelliptic in every dimension through compact representatives modulo smooth functions. A derived radial Fourier phase gives zero-free logarithmic strips,full global-distribution regularity and compact parametrices with the reflected singular sphere. Full proof · Reproduction sources.
Compact multipliers and exponential solutions of convolution systems
Read the complete lesson. Six compact convolution kernels in every dimension at least two have a normalized entire solution but no exponential polynomial solution. Compact multiplier bounds,an entire graph and a proper ideal prove the construction. Full proof · Reproduction sources.
Analytic forcing without global analytic solutions
Read the complete lesson. One globally analytic forcing has smooth global solutions but no global analytic solution for either the heat operator with two spatial variables or the partial Laplacian with an analytic parameter. Full proof · Reproduction sources.
Rational top forms detect cycles in a hypersurface complement
Read the complete lesson. Rational top forms span smooth top-degree cohomology and detect finite rational or complex cycles in a polynomial complement, including singular and repeated divisors. Full proof · Reproduction sources.
Finite covers and normal circles in projective complements
Read the complete lesson. A finite cyclic cover gives the rational homology bound for projective hypersurface complements and the positive-normal-circle tube injection, including singular and repeated removed divisors. Full proof · Reproduction sources.
Equatorial cycles and the projective period test
Read the complete lesson. The exact affine/projective comparison fixes center multiplicity two and normal coefficient one, then proves rational component constancy at the stated analytic prerequisites, with the full angular-contour and polynomial-continuation reading. Full proof · Reproduction sources.
Angular distributions and their full Fourier transforms
Read the complete lesson. The full parity-sensitive angular Fourier identity for every integer exponent and arbitrary angular distributions, with exact finite parts, Radon test pairings and frequency-origin terms. Full proof · Reproduction sources.
Analytic directions of holomorphic boundary values
Read the complete lesson. Polynomially bounded holomorphic tube boundaries have analytic wavefront in the positive polar; controlled cutoff sequences prove analytic differential decrease and ordinary-wavefront inclusion. Full proof · Reproduction sources.
Dilation equations and tangential analytic singularities
Read the complete lesson. Complex-degree Euler equations on arbitrary open sets admit only tangential analytic wavefront covectors; analytic forcing and analytic zeroth-order potentials obey the same local bound. Full proof · Reproduction sources.
Pulling back and combining analytic singularities
Read the complete lesson. Transverse analytic pullback, tensor, nonopposing product, restriction and proper projection bounds retain exact covectors; normal continuity and hypocontinuity preserve fixed-wavefront limits. Full proof · Reproduction sources.
Gaussian packets and homogeneous Fourier directions
Read the complete lesson. Gaussian packet regularity and finite inverse heat expansions prove exact ordinary and analytic Fourier-wavefront interchange for arbitrary complex degrees and finite Euler chains, retaining origin terms. Full proof · Reproduction sources.
Derivative sequences and microlocal regularity
Read the complete lesson. Admissible derivative sequences, fixed-index localization and Gaussian criteria give exact class pullback, tensor, product, restriction, projection and homogeneous Fourier bounds, with precise uniform estimates for limits. Full proof · Reproduction sources.
Analytic coefficients and one-sided uniqueness
Read the complete lesson. A full derivative jet reduction and an analytic supporting paraboloid prove distributional Holmgren uniqueness for every scalar order and normal-invertible square systems across noncharacteristic C1 surfaces. Full proof · Reproduction sources.
Real roots and their convex component
Read the complete lesson. Real-root lines determine an open convex component of a homogeneous polynomial. The full proof treats complex coefficients, repeated roots, every component direction and the zero-free imaginary tube. Full proof · Reproduction sources.
One-sided root bounds and Taylor order
Read the complete lesson. A one-sided imaginary bound on local zeros for both real parameter signs forces the full Taylor order. The proof treats complex coefficients, repeated roots and convergent germs. Full proof · Reproduction sources.
Support normals and analytic singularities
Read the complete lesson. Both signed supporting normals of a distribution belong to its analytic wavefront set. A Cauchy-transform argument, proper tangential projection and nearest-point limits include local C1 supporting functions. Full proof · Reproduction sources.
Holomorphic reciprocals and convolution regularity
Read the complete lesson. A polynomially bounded holomorphic reciprocal on a complex frequency cone yields analytic convolution ellipticity. The theorem covers tempered kernels, compactly supported distributions and proper convolution with inputs of arbitrary growth. Full proof · Reproduction sources.
Harmonic potentials and real carriers
Read the complete lesson. Analytic functionals on arbitrary compact real carriers correspond to odd harmonic potentials modulo entire harmonic functions. Full reconstruction identifies the smallest real carrier and preserves derivatives and compact-distribution supports. Full proof · Reproduction sources.
Splitting real carriers and gluing hyperfunctions
Read the complete lesson. Harmonic approximation and Laplace correction split arbitrary compact real carriers. A convergent correction series proves quotient restriction,arbitrary hyperfunction gluing and support-preserving differential operations. Full proof · Reproduction sources.
Polynomial reciprocals and tempered fundamental solutions
Read the complete lesson. A polynomial differential functional equation constructs a tempered reciprocal for every nonzero complex polynomial. Full algebraic growth,meromorphic continuation and finite-part proofs give tempered fundamental solutions and compact-data solvability. Full proof · Reproduction sources.