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Article items · 11 — Partial extension across an artificial sphere and annular cutoff gluing
Presents 28 items from the article and its immediate prerequisites, covering dependency levels 11–15.
1 · Prerequisites
- The Axiom of Choice
- Simultaneous Sobolev extension and localization
- Localized smooth coordinate changes and cutoffs on Fourier Hʳ
- Test function cutoffs and euclidean localization
- Canonical higher traces on a bounded boundary or compact interior hypersurface
- The kernel of the trace is the closure of the test functions
- Zero-boundary Sobolev space as a norm closure
- Compactly supported smooth functions are dense in W^k,p(Rⁿ)
- Zero extension of W₀^1,p has no boundary derivative
- Integer-order W^k,2 and Hᵏ agree with equivalent norms
- Reachability and observation vocabulary for controlled evolutions
- Continuous and mixed time-space function spaces
- Bessel potentials shift Sobolev order
- Restriction Bessel-potential spaces on polyhedral domains
- The Axiom of Countable Choice (AC_ω)
- Classical normal derivative
- The Laplacian of a C² function and of a C² vector field
- Exterior and truncated obstacle domains
- Reference trajectory hypothesis W
- Conditional energy-level unique continuation hypothesis
- Non-trapping obstacle
- Unique continuation for Schrödinger operators with partially Gevrey coefficients · Author manuscript dated 31 March 2025: abstract, PDF p. 1; Section 1.2, equation (1.18), Theorem 1.7 and equation (1.19), PDF pp. 6–7.
- Control region and nested real cutoffs
- Dirichlet Laplacian and positive shift
- Dirichlet spectral Sobolev scale and duality
- Local negative Sobolev space in the compactness argument
- Weak convergence notation in normed state spaces
- Distributional equation convention for complex Schrödinger solutions
- Distributional time derivatives with values in a Banach space
- Scale isometries and free Schrödinger evolution
- Homogeneous half-derivative local smoothing
- Positive spectral Hilbert scale: duality and interpolation
- Exterior spectral domains and the critical trace index
- Fractional compactness on bounded collar domains
- General Banach-triple Aubin–Lions–Simon compactness
- Weak Leibniz rule with a smooth factor
- Complex completeness, density, and inner product: the consumer interface
- Bochner integrability criterion
- Bounded linear maps commute with Bochner integration
- Exterior Dirichlet realization, H² domain and form domain
- Complex integer-order Sobolev notation
- Bounded Cᵏ domains admit integer-order Sobolev extension
- Bounded Cᵏ domains and boundary charts
- Bounded restriction and cutoff localisation in Sobolev spaces
- Real-order Bessel-potential completion Hˢ
- The Bessel completion embeds canonically in tempered distributions
- Weighted tempered-distribution characterization of Hˢ
- Regularized lower-order observation remainder
- Endpoint multiplier bounds extended to negative spectral orders
- Spectral interpolation and lower-order absorption
- Weakly convergent sequences are norm bounded
- Hahn-Banach dominated extension theorem for real vector spaces
- Atlas-independent hypersurface Sobolev norms and H^3/2 tangential derivatives
- Real-order Hˢ as weighted Fourier distributions
- Linear and nonlinear internally controlled exterior systems
- Uniform localized H2 algebra on a smooth exterior domain
- The Lᵖ trace operator on a bounded C¹ domain
- The trace commutes with smooth cutoffs and is chart local
- Dominated convergence
- For an open subset of ℝⁿ, connectedness, path-connectedness and polygonal connectedness are equivalent
- Integer-order Sobolev spaces are Banach
- Linearized equation and its real-duality adjoint
- Weak product rule for bounded Sobolev functions
- Linearity, locality, and commutation of weak derivatives
- Quadratic-cubic remainder after subtracting a trajectory
- Energy control space with a time-integrable supremum norm
- Tame bounds for the quadratic-cubic trajectory remainder
- Integer-order Sobolev spaces and their norms
- An explicit common interpolation model for the fractional Strichartz targets
- Global Strichartz estimate with derivative loss
- Independent exterior fractional Sobolev embedding
- Three-dimensional L4 time Linfinity space estimate
- Tonelli and Fubini for the completed product, with only almost-everywhere section measurability
- Holder's inequality for integrals, including the endpoint cases
- Fatou's lemma
- Heine-Cantor: a continuous map from a compact metric space to any metric space is uniformly continuous
- Invisible initial-data spaces
- Lemma 3.3: observability modulo a compact remainder
- Hilbert spaces are reflexive
- Reflexivity is equivalent to weak subsequential compactness of bounded sequences
- For a metric space, compact, countably compact, limit point compact, sequentially compact, and complete together with totally bounded are all equivalent, given countable choice and dependent choice
- In an infinite-dimensional normed space the closed unit ball is not compact
- Open cover, subcover, compact metric space, and compact subset of a metric space
- Finite-dimensional vector space, and its dimension dim_F V; infinite-dimensional means having no finite basis
- Bounded real-linear perturbations preserve solution estimates
- Scale Riesz map and free HUM operator
- Cutoff multipliers through order two and by duality
- Realification of Hilbert pairings and conjugate-linear coefficients
- Bochner-integrable function
- Cubic NLS and boundary control formulations
- A ray trapped between two convex obstacles
- Locally integrable functions as regular distributions
- Homogeneous bounds transfer to L1 retarded forcing
- Lemma 4.1: retarded smoothing down to negative order two
- Spectral cutoff cores and unitary identifications of completed scales
- A finite-dimensional normed subspace is closed
- Linearity of the Bochner integral
- Strongly measurable Banach-valued function
- A bounded linear operator between normed spaces
- Vanishing of an eigenfunction invisible outside a ball
- Every endomorphism of a nonzero finite-dimensional vector space over an algebraically closed field has an eigenvalue
- Fundamental theorem of algebra: every nonconstant complex polynomial has a complex root
- An algebraically closed field: every nonconstant polynomial has a root in the field
- Compatible Banach pairs and their integrals
- Time translations and mollification in Bochner Lp
- Bochner Lp is complete for a Banach value space
- Realification preserves a complex norm and its metric
- Bochner integral norm inequality
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
LemmaDependency level 11
Partial extension across an artificial sphere and annular cutoff gluing
Open problemDependency level 11
Question B: controls on part of the boundary
Open problemDependency level 11
Unique continuation for general non-trapping obstacles and trajectories
PropositionDependency level 11
Localized free-flow bounds and compactness mechanism
PropositionDependency level 11
Low-order identifications of the Dirichlet scale
LemmaDependency level 12
Lemma 3.2: compactness of the remainder
LemmaDependency level 12
Collar traces and extension across an artificial boundary
PropositionDependency level 12
Exterior H2 algebra and polynomial nonlinear estimates
PropositionDependency level 12
Time-dependent potential and conjugation bounds on the scale
PropositionDependency level 12
Quadratic-cubic estimates near a reference trajectory
PropositionDependency level 12
Two-dimensional exponents removing the Strichartz loss
DefinitionDependency level 13
States compatible with the uncontrolled obstacle boundary
LemmaDependency level 13
Lemma 3.8: energy-data time-integrable supremum bound
PropositionDependency level 13
Finite dimensionality of invisible spaces
PropositionDependency level 13
Forward and backward real-linear propagators
DefinitionDependency level 14
Annular extension and cutoff gluing construction
DefinitionDependency level 14
Real HUM operator for the trajectory linearization
DefinitionDependency level 14
Resolvent-regularized adjoint and its forcing
Open problemDependency level 14
Nonlinear control for trapping obstacle configurations
PropositionDependency level 14
Free and retarded bounds in the energy control space
PropositionDependency level 14
Time cutoff bootstrap for invisible adjoint solutions
PropositionDependency level 14
Locally constant invisible spaces are Laplacian invariant
PropositionDependency level 14
Duhamel formula for a real-linear propagator
RemarkDependency level 14
Necessary endpoint compatibility for boundary controls
LemmaDependency level 15
Lemma 3.11: extension preserving the obstacle Dirichlet condition
LemmaDependency level 15
Lemma 3.4: free unique continuation on the scale
LemmaDependency level 15
Lemma 4.2: smoothing of the regularized adjoint
LemmaDependency level 15