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.