Alphabeta MathPersonal research
Exterior geometry and function spaces
Specify the obstacle and generalized-ray conventions, Sobolev and time-space notation, Dirichlet realization and spectral scales; connect bounded fractional inputs to exterior localization, traces, compactness and continuation.
1 · Prerequisites
- The Axiom of Choice
- Real-order Hˢ as weighted Fourier distributions
- Integer-order W^k,2 and Hᵏ agree with equivalent norms
- Multiplication operators: domain, spectral measure and spectrum
- Plancherel theorem
- Positive spectral Hilbert scale: duality and interpolation
- A quotient of a Banach space by a closed subspace is Banach
- Integer-order Sobolev spaces and their norms
- Bounded restriction and cutoff localisation in Sobolev spaces
- Transport of spectral projection-valued measures along isometric bijections
- Unbounded Borel functional calculus: domains, products, spectral mapping
- Cᵏ boundary flattening preserves local W^k,p
- Weak Leibniz rule with a smooth factor
- Compactly supported Sobolev functions extend by zero in every integer order
- Cᵏ Euclidean maps and diffeomorphisms
- Complex Lp classes and Euclidean test-function conventions
- A C¹ diffeomorphism satisfies the change-of-variables formula for L¹ functions
- A C¹ diffeomorphism maps Lebesgue null sets to Lebesgue null sets
- Fourier transform agrees with l one and plancherel transforms
- Test function space d of an open set
- Well-defined strip norm and the action of a common endpoint operator
- Integer-order Sobolev spaces are Banach
- Codimension and hypersurfaces
- Smooth atlases
- Slice-chart restrictions form a smooth atlas
- The inclusion of an embedded submanifold is a smooth embedding
- Chart and partition independence of surface measure
- Finite ambient partitions near compact sets
- Surface integration on compact C1 hypersurfaces
- Distributional derivatives are polynomial Fourier multipliers
- Complex Lp completeness and almost-everywhere subsequences
- Smooth partitions of unity subordinate to an open cover
- Weak derivative of a locally integrable function
- Fourier inversion on Schwartz space
- Tonelli and Fubini for the completed product, with only almost-everywhere section measurability
- Holder's inequality for integrals, including the endpoint cases
- The fundamental theorem of calculus and integration by parts
- Test function cutoffs and euclidean localization
- Schwartz space and its seminorms
- Fourier transform acts continuously on Schwartz space
- Schwartz derivatives are integrable
- Dominated convergence
2 · Logical flowchart
Arrows show item dependencies. A grouped prerequisite node points to items that use one or more of its members; their exact links are listed above.
3 · Items
Lemma
Whole-space interpolation of one extension–restriction operator
Lemma
Localized smooth coordinate changes and cutoffs on Fourier Hʳ
Lemma
Restriction-quotient interpolation from a common endpoint retraction
Lemma
Atlas-independent hypersurface Sobolev norms and H^3/2 tangential derivatives
Lemma