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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
- Spectral cutoff cores and unitary identifications of completed scales
- Admissible spectral strip profiles and the exact interpolation upper bound
- Spectral cutoff dual tests and the exact interpolation lower bound
- Unbounded Borel functional calculus: domains, products, spectral mapping
- The Axiom of Choice
- Hilbert space
- Completion of a normed space
- Symmetric, self-adjoint and essentially self-adjoint operators
- Integer-order Sobolev spaces are Banach
- A single collar-reflection and free-tail exterior extension
- Well-defined strip norm and the action of a common endpoint operator
- A quotient of a Banach space by a closed subspace is Banach
- Strongly measurable Banach-valued function
- Bochner Lp is complete for a Banach value space
- Realification preserves a complex norm and its metric
- J. C. Bellido and G. García-Sáez, Bessel Potential Spaces and Complex Interpolation: Continuous embeddings, arXiv:2503.04310v6 · Compatible couples and sums: PDF pp.4–6; §2.2, PDF pp.9–10 (strip definition, Banach property and exact operator bound); Example 2.7(i), PDF p.11 (Bochner interpolation); Definition 3.1 and Remark 3.2, PDF pp.15–16 (tempered-distribution multiplier and well-definedness); Definitions 3.3, 3.6 and Theorem 3.7, PDF pp.16–17 (Bessel space and identity).
- Exterior and truncated obstacle domains
- Complex integer-order Sobolev notation
- Integer-order Sobolev spaces and their norms
- Zero-boundary Sobolev space as a norm closure
- Exterior Dirichlet realization, H² domain and form domain
- Spectral theorem for unbounded self-adjoint operators (PVM form)
- Strongly continuous one-parameter unitary group
- A self-adjoint operator generates a strongly continuous unitary group
- Spectral form domain and core of a semibounded operator
- Banach space
- A bounded linear operator between normed spaces
- J. C. Bellido and G. García-Sáez, Bessel Potential Spaces and Complex Interpolation: Continuous embeddings, arXiv:2503.04310v6 · Compatible couples and sums: PDF pp.4–6; §2.2, PDF pp.9–10 (strip definition, Banach property and exact operator bound); Example 2.7(i), PDF p.11 (Bochner interpolation); Definitions 3.1, 3.3, 3.6 and Theorem 3.7, PDF pp.15–17 (whole-space Bessel convention and identity).
- Continuous and mixed time-space function spaces
- Flow calculus of a self-adjoint generator: commutation, invariance and differentiability
- The Gauss-Green integration-by-parts formula with Sobolev traces
- The sharp trace theorem: boundedness and range in the fractional space
- Ambient smooth restrictions are dense on bounded Cᵏ domains
- Finite ambient partitions near compact sets
- Bounded C1 domains and their outward normals
- Surface integration on compact C1 hypersurfaces
- Chart and partition independence of surface measure
- The completed product measure
- Tonelli and Fubini for the completed product, with only almost-everywhere section measurability
- Complex Holder, Minkowski, and the quotient norm
- Complex Lp completeness and almost-everywhere subsequences
- Heine-Cantor: a continuous map from a compact metric space to any metric space is uniformly continuous
- Sequential suprema, infima, limsup, liminf, and pointwise limits of measurable functions are measurable
- Every section of a product-measurable function is measurable
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
Exact scale interpolation and endpoint operator transport
Lemma
Restriction quotient and Bochner interpolation for the exterior extension
Definition
Dirichlet Laplacian and positive shift
Lemma
An explicit common interpolation model for the fractional Strichartz targets
Lemma