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
- Spectral cutoff cores and unitary identifications of completed scales
- Exact scale interpolation and endpoint operator transport
- Spectral theorem for unbounded self-adjoint operators (PVM form)
- Unbounded Borel functional calculus: domains, products, spectral mapping
- Bochner Lp is complete for a Banach value space
- The Axiom of Choice
- Projection valued measure
- Strongly measurable Banach-valued function
- Realification preserves a complex norm and its metric
- Complex Holder, Minkowski, and the quotient norm
- Cauchy–Schwarz: |⟨ u,v⟩|≤‖ u‖‖ v‖, with equality exactly for linearly dependent vectors
- Dominated convergence
- Monotone convergence for the integral
- Every nonnegative measurable function admits an explicit increasing sequence of simple approximations
- Admissible spectral strip profiles and the exact interpolation upper bound
- Riesz representation for Hilbert spaces
- Flow calculus of a self-adjoint generator: commutation, invariance and differentiability
- Closed semibounded symmetric forms and their associated operators
- Semibounded closed-form representation and perturbation
- Integer-order Sobolev spaces and their norms
- Zero-boundary Sobolev space as a norm closure
- The Sobolev space H¹ is a Hilbert space
- Smooth compactly supported functions of an open set are dense in L²
- An explicit common interpolation model for the fractional Strichartz targets
- The essential supremum of a measurable function with respect to a measure
- José C. Bellido and Guillermo García-Sáez, Bessel Potential Spaces and Complex Interpolation: Continuous embeddings, arXiv:2503.04310v6 · Definition 3.6 and Theorem 3.7, PDF p. 17; Theorem 3.21(3), PDF pp. 25–26
- Exterior and truncated obstacle domains
- Complex integer-order Sobolev notation
- Dirichlet Laplacian and positive shift
- Exterior Dirichlet realization, H² domain and form domain
- Weak Leibniz rule with a smooth factor
- Interior mollification commutes with weak derivatives
- Local smooth approximation in integer-order Sobolev spaces
- Compactly supported Sobolev functions extend by zero in every integer order
- Test function cutoffs and euclidean localization
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
Lifted spectral measure and isometric Bochner scale interpolation
Lemma
Positive spectral Hilbert scale: duality and interpolation
Theorem
Independent exterior fractional Sobolev embedding
Definition
Dirichlet spectral Sobolev scale and duality
Lemma