Fourier-integral calculus and symbol foundations

Six components prove canonical geometry, Maslov composition, ordinary analytic composition, Sobolev mapping, graph compactness and essential norms, and general homogeneous symbol transport. Twenty-four solved exercises check hypotheses, signs, constants, parameter ranges and convergence.

1. Geometry and half-densities

Clean intersections and proper fibre integration. Read the complete proofs · Editable source · Free sources and terms

2. Maslov lines and Gaussian factors

Quadratic phase changes and homogeneous degree. Read the complete proofs · Editable source · Free sources and terms

3. The analytic clean-composition theorem

Every ordinary-symbol remainder and the principal-symbol fibre formula. Read the complete proofs · Editable source · Free sources and terms

4. From canonical ranks to Sobolev bounds

Graph L2 estimates, graph slices, parameter integration and every real Sobolev order. Read the complete proofs · Editable source · Free sources and terms

5. Compactness and essential norms

Finite-rank approximation, packet lower bounds, graph density normalization and relative-frequency branch separation. Read the complete proofs · Editable source · Free sources and terms

6. Homogeneous maps and complete symbol spaces

All parameter ranges, proper normalized supports, local-section inverses, full asymptotics and cutoff topology. Read the complete proofs · Editable source · Free sources and terms

Exact proof dependencies · Earlier intrinsic proofs · Tangent and Gaussian symbols

Analytic FIO estimates in symbol classes with derivative losses, broader intrinsic characterization and distribution-space topology, involutivity, propagation and the remaining course are unfinished.