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Article items · 09 — Independent exterior fractional Sobolev embedding
Presents 28 items from the article and its immediate prerequisites, covering dependency levels 5–8.
1 · Prerequisites
- An explicit common interpolation model for the fractional Strichartz targets
- The Axiom of Choice
- 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
- Positive spectral Hilbert scale: duality and interpolation
- Unbounded Borel functional calculus: domains, products, spectral mapping
- Weak Leibniz rule with a smooth factor
- Integer-order Sobolev spaces and their norms
- 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
- Complex Holder, Minkowski, and the quotient norm
- Test function cutoffs and euclidean localization
- Continuous and mixed time-space function spaces
- Riesz-Fischer completeness of Lᵖ for 1 ≤ p ≤ ∞
- Tonelli and Fubini for the completed product, with only almost-everywhere section measurability
- Monotone convergence for the integral
- Minkowski's inequality for integrals, including p = ∞
- If (Y,d) is complete then Y^X is complete in the uniform metric, and so is C(X,Y)
- Heine-Cantor: a continuous map from a compact metric space to any metric space is uniformly continuous
- Complex Lp completeness and almost-everywhere subsequences
- Control region and nested real cutoffs
- Distributional equation convention for complex Schrödinger solutions
- The dual space X^* of a normed space and its dual norm
- Bounded real-linear perturbations preserve solution estimates
- Homogeneous bounds transfer to L1 retarded forcing
- Strongly measurable Banach-valued function
- Classical derivatives agree with weak derivatives
- The Axiom of Choice implies countable choice
- Holder's inequality for integrals, including the endpoint cases
- Young's inequality for conjugate real exponents
- Dominated convergence
- Fatou's lemma
- Integral of a measurable function against a projection-valued measure
- The unbounded PVM integral is densely defined, closed and normal
- Scalar and complex measures from a pvm
- Non-trapping obstacle
- Resolvent and spectrum of an unbounded operator
- On nonlinear Schrödinger equations in exterior domains · Section 2.2, opening paragraph and Propositions 2.2 and 2.4, printed pp. 299–300, PDF pp. 5–6
- On nonlinear Schrödinger equations in exterior domains · Section 2.1, Dirichlet spectral scale and negative-order duality and interpolation, printed p. 299, PDF p. 5; Section 2.2, Proposition 2.5, equation (2.9), printed p. 300, PDF p. 6
- On nonlinear Schrödinger equations in exterior domains · Section 2.2, Remark 2.6, equation (2.10), printed p. 300, PDF p. 6
- Bochner-integrable function
- Continuous free convolution on a finite time interval
- On nonlinear Schrödinger equations in exterior domains · Section 2.3, Proposition 2.7, equations (2.14) and (2.15), and Remark 2.8; published page 302, PDF page 8
- Well-defined strip norm and the action of a common endpoint operator
- Integer-order Sobolev spaces are Banach
- Bochner Lp is complete for a Banach value space
- Realification preserves a complex norm and its metric
- On nonlinear Schrödinger equations in exterior domains · Section 2.4, Propositions 2.10 and 2.11, equations (2.17), (2.18), (2.22), printed pages 304–306, PDF pages 10–12
- A self-adjoint operator generates a strongly continuous unitary group
- Heine-Borel by bisection: every closed bounded interval [a,b] is compact
- Strichartz estimates and the nonlinear Schrödinger equation on manifolds with boundary · Section 5, equations (54) and (55), and the argument following Lemma 5.2, PDF pages 18–21
- Flow calculus of a self-adjoint generator: commutation, invariance and differentiability
- Spectral cutoff cores and unitary identifications of completed scales
- Linear subspace of a vector space
- Banach space
- Lebesgue measure is sigma-finite, and every metrically bounded subset of ℝⁿ has finite outer measure
- The Lᵖ trace operator on a bounded C¹ domain
- The trace commutes with smooth cutoffs and is chart local
- The Euclidean Lebesgue measure is the completion of the product of the factor Lebesgue measures
- Locally integrable functions define distributions
- AC supplies the countable and dependent choices used in Banach integration
- Explicit compactly supported smooth cutoffs
- Control of the Schrödinger equation in R³: The critical case · Theorem 1.1 and §4(B)
- Reachability and observation vocabulary for controlled evolutions
- Cubic NLS and boundary control formulations
- Weak unique continuation for bounded potentials in three dimensions
- A smooth bump between concentric Euclidean balls
- Lifted spectral measure and isometric Bochner scale interpolation
- Cauchy-Schwarz inequality for L²
- Complex Lp classes and Euclidean test-function conventions
- Heine-Borel in ℝⁿ: with the Euclidean metric a subset of ℝⁿ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line
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
TheoremDependency level 5
Independent exterior fractional Sobolev embedding
DefinitionDependency level 6
Dirichlet spectral Sobolev scale and duality
LemmaDependency level 6
Endpoint multiplier bounds extended to negative spectral orders
DefinitionDependency level 7
Energy control space with a time-integrable supremum norm
DefinitionDependency level 7
Linear and nonlinear internally controlled exterior systems
DefinitionDependency level 7
Local negative Sobolev space in the compactness argument
LemmaDependency level 7
Energy dispersive bounds under bounded trajectory perturbations
PropositionDependency level 7
Cutoff multipliers through order two and by duality
PropositionDependency level 7
Spectral interpolation and lower-order absorption
PropositionDependency level 7
Nontrapping cutoff resolvent estimates
PropositionDependency level 7
Imported homogeneous and retarded exterior smoothing
PropositionDependency level 7
Exterior Strichartz estimates with precise loss
PropositionDependency level 7
Three-dimensional Dirichlet energy-to-mixed-norm estimate
PropositionDependency level 7
Scale isometries and free Schrödinger evolution
DefinitionDependency level 8
Regularized lower-order observation remainder
DefinitionDependency level 8
Invisible initial-data spaces
DefinitionDependency level 8
Abstract nonlinear control-space hypotheses
DefinitionDependency level 8
Reference trajectory hypothesis W
LemmaDependency level 8
Resolvent adjoint transport to order minus two
Open problemDependency level 8
Large-data control for energy-critical NLS
Open problemDependency level 8
Global defocusing controllability in large time
Open problemDependency level 8
Nonlinear exact control below energy regularity
Open problemDependency level 8
Quintic control around nonzero trajectories
PropositionDependency level 8
Proposition 2.1: cut-off resolvent estimate
PropositionDependency level 8
Homogeneous half-derivative local smoothing
PropositionDependency level 8
Proposition 2.2: retarded local smoothing
PropositionDependency level 8
Strichartz estimate away from the obstacle
PropositionDependency level 8