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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
- A compactly supported common L²/H² extension from a bounded smooth domain
- Whole-space interpolation of one extension–restriction operator
- Restriction-quotient interpolation from a common endpoint retraction
- Test function cutoffs and euclidean localization
- Weak Leibniz rule with a smooth factor
- Compactly supported Sobolev functions extend by zero in every integer order
- Well-defined strip norm and the action of a common endpoint operator
- Zero-boundary Sobolev space as a norm closure
- Compactly supported smooth functions are dense in W^k,p(Rⁿ)
- Integer-order W^k,2 and Hᵏ agree with equivalent norms
- Bounded restriction and cutoff localisation in Sobolev spaces
- Dirichlet spectral Sobolev scale and duality
- Exterior Dirichlet realization, H² domain and form domain
- Positive spectral Hilbert scale: duality and interpolation
- Bounded smooth fractional Dirichlet domain contract
- The kernel of the trace is the closure of the test functions
- Real-order Hˢ as weighted Fourier distributions
- Plancherel theorem
- Translation, modulation, linear dilation and reflection laws
- The Fréchet–Kolmogorov compactness criterion in Lᵖ(ℝⁿ)
- Uniformly supported families have vanishing tails
- Integer-order Sobolev spaces and their norms
- Complex completeness, density, and inner product: the consumer interface
- The fundamental lemma of the calculus of variations
- The zero-order Bessel completion is exactly L²
- Fourier transform agrees with l one and plancherel transforms
- Real powers of the Japanese bracket act on Schwartz space
- Tempered distributions embed continuously in distributions
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
A common exterior extension and explicit collar–tail retraction
Lemma
Simultaneous Sobolev extension and localization
Lemma
Exterior spectral domains and the critical trace index
Lemma