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Article items · 10 — Three-dimensional L4 time Linfinity space estimate
Presents 29 items from the article and its immediate prerequisites, covering dependency levels 8–10.
1 · Prerequisites
- The Axiom of Choice
- Non-trapping obstacle
- Dirichlet Laplacian and positive shift
- Scale isometries and free Schrödinger evolution
- Three-dimensional Dirichlet energy-to-mixed-norm estimate
- Complex integer-order Sobolev notation
- Continuous and mixed time-space function spaces
- Exterior and truncated obstacle domains
- Control region and nested real cutoffs
- Dirichlet spectral Sobolev scale and duality
- Spectral cutoff cores and unitary identifications of completed scales
- Helmholtz continuation through a harmonic lift
- Test function space d of an open set
- Locally integrable functions embed in distributions
- Endpoint multiplier bounds extended to negative spectral orders
- Linear and nonlinear internally controlled exterior systems
- Infinitesimal generator of a C0-semigroup
- Classical, strong and mild abstract Cauchy solutions
- Variation of constants for the inhomogeneous abstract Cauchy problem
- Bochner integral norm inequality
- AC supplies the countable and dependent choices used in Banach integration
- Classical derivatives agree with weak derivatives
- 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
- The image of a compact metric space under a continuous map is compact, and so is the image of any compact subset
- For a nonzero real c, dilation by c multiplies Lebesgue outer measure by |c|ⁿ, and reflection in the origin preserves it
- Lebesgue outer measure, Lebesgue measurability and Lebesgue measure are unchanged by translation
- Integral invariance under measure-preserving maps
- Banach-valued simple function and integral
- Bochner-integrable function
- Positive spectral Hilbert scale: duality and interpolation
- Cutoff multipliers through order two and by duality
- Distributional derivative
- Weak Leibniz rule with a smooth factor
- Locally integrable functions as regular distributions
- Multiplication of a distribution by a smooth function
- Leibniz rule for distributions
- Linearity of the Bochner integral
- Bochner integrability criterion
- Exterior Dirichlet realization, H² domain and form domain
- Complex polynomial weak derivatives and zero boundary preservation
- The Axiom of Countable Choice (AC_ω)
- The chain rule for total derivatives: D(g∘ f)(a)=Dg(f(a))∘ Df(a)
- A C¹ diffeomorphism satisfies the change-of-variables formula for nonnegative Lebesgue measurable functions
- Complex Lp classes and Euclidean test-function conventions
- The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case
- Reachability and observation vocabulary for controlled evolutions
- Bounded real-linear perturbations preserve solution estimates
- Adjoint of a densely defined operator
- Positive square root
- César A. Bortot and Marcelo M. Cavalcanti, Asymptotic Stability for the Damped Schrödinger Equation on Noncompact Riemannian Manifolds and Exterior Domains, Communications in PDE 39 (2014), 1791–1820 · Theorem 1.1, printed p.1793; exterior equation (3.16) and Assumption 3.1, p.1797; Assumption 3.2 and energy (3.18), p.1798; dissipation identity (4.19), p.1800. PDF pages 4,8,9,11 including publisher cover.
- Emmanuel Trélat, Control in finite and infinite dimension, author lecture notes · §5.3.4, Theorem 31, printed/PDF pp.119–120; Remark 59 on p.122. Equations (5.27)–(5.28).
- Reference trajectory hypothesis W
- Distributional equation convention for complex Schrödinger solutions
- Realification of Hilbert pairings and conjugate-linear coefficients
- Real and imaginary parts, complex conjugation, and modulus
- Conjugation is an involutive real-field automorphism, zz̄=|z|², and modulus is definite, multiplicative, and subadditive
- Chain sum product and composition rules for Banach derivatives
- Cauchy–Schwarz: |⟨ u,v⟩|≤‖ u‖‖ v‖, with equality exactly for linearly dependent vectors
- 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
- A common exterior extension and explicit collar–tail retraction
- Well-defined strip norm and the action of a common endpoint operator
- Abstract nonlinear control-space hypotheses
- Homogeneous half-derivative local smoothing
- Unbounded Borel functional calculus: domains, products, spectral mapping
- Bochner Lp is complete for a Banach value space
- Dominated convergence
- Proposition 2.1: cut-off resolvent estimate
- Nontrapping cutoff resolvent estimates
- Resolvent adjoint transport to order minus two
- A smooth bump between concentric Euclidean balls
- Plancherel theorem
- Continuous free convolution on a finite time interval
- Holder's inequality for integrals, including the endpoint cases
- Laplace resolvents of a unitary group
- Flow calculus of a self-adjoint generator: commutation, invariance and differentiability
- Tonelli's theorem for nonnegative measurable functions on a sigma-finite product
- Fubini's theorem for L¹ functions on a sigma-finite product
- Bounded linear maps commute with Bochner integration
- Agreement of the integral and L2 transforms
- Spectral interpolation and lower-order absorption
- Global Strichartz estimate with derivative loss
- Bounded C1 domains and their outward normals
- Classical normal derivative
- On nonlinear Schrödinger equations in exterior domains · Proposition 2.11 and its proof; Remarks 1.2, 2.3 and 2.9
- Strichartz estimates and the nonlinear Schrödinger equation on manifolds with boundary · Introduction, p. 3, final application paragraph; Theorem 1.1, p. 3; section 5, pp. 18–21, especially Theorem 5.1, estimate (55) and the spectral decomposition on pp. 20–21
- Integer-order Sobolev spaces and their norms
- The Lᵖ trace operator on a bounded C¹ domain
- The trace agrees with classical restriction for continuous Sobolev functions
- Zero-boundary Sobolev space as a norm closure
- Heine-Cantor: a continuous map from a compact metric space to any metric space is uniformly continuous
- Lebesgue measure is sigma-finite, and every metrically bounded subset of ℝⁿ has finite outer measure
- The Euclidean Lebesgue measure is the completion of the product of the factor Lebesgue measures
- Complex Lp completeness and almost-everywhere subsequences
- Tonelli and Fubini for the completed product, with only almost-everywhere section measurability
- Complex Holder, Minkowski, and the quotient norm
- The Sobolev space H¹ is a Hilbert space
- Hilbert space
- Regularized lower-order observation remainder
- Proposition 2.2: retarded local smoothing
- The spaces C_c(ℝⁿ) and C_c^∞(ℝⁿ)
- Localized smooth coordinate changes and cutoffs on Fourier Hʳ
- Atlas-independent hypersurface Sobolev norms and H^3/2 tangential derivatives
- Fourier trace on a hyperplane and one-sided H¹ uniqueness
- The sharp trace theorem: boundedness and range in the fractional space
- The trace commutes with smooth cutoffs and is chart local
- Real-order Hˢ as weighted Fourier distributions
- Integer-order W^k,2 and Hᵏ agree with equivalent norms
- Bounded Cᵏ domains and boundary charts
- Finite ambient partitions near compact sets
- Smooth Euclidean hypersurface graphs and compact localization
- The Euclidean tubular neighbourhood theorem
- The normal addition map for a Euclidean submanifold
- Restriction Bessel-potential spaces on polyhedral domains
- Embedded submanifolds and slice charts
- Codimension and hypersurfaces
- Slice-chart restrictions form a smooth atlas
- Young's convolution inequality under Countable Choice
- Fourier transform of a product with one integrable transform
- Fourier inversion on Schwartz space
- Schwartz functions are dense in the Fourier Bessel-potential norm
- Distributional derivatives are polynomial Fourier multipliers
- Integer-order Sobolev spaces are Banach
- Schwartz space and its seminorms
- Bounded smooth fractional Dirichlet domain contract
- The kernel of the trace is the closure of the test functions
- Compactly supported smooth functions are dense in W^k,p(Rⁿ)
- Translation, modulation, linear dilation and reflection laws
- The Fréchet–Kolmogorov compactness criterion in Lᵖ(ℝⁿ)
- Uniformly supported families have vanishing tails
- 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
- Energy control space with a time-integrable supremum norm
- Cubic and quintic time bounds from L4 Linfinity control
- The inner product is jointly continuous
- The Axiom of Choice implies countable choice
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
PropositionDependency level 8
Three-dimensional L4 time Linfinity space estimate
PropositionDependency level 8
Vanishing of an eigenfunction invisible outside a ball
PropositionDependency level 8
Linear forced well-posedness by Duhamel formula
PropositionDependency level 8
Integer resolvent commutator and localized first-order bounds
PropositionDependency level 8
Mild formulation of the controlled power NLS
PropositionDependency level 8
Power nonlinearity preserves the zero-boundary energy space
RemarkDependency level 8
Critical nonlinearities named in the article
RemarkDependency level 8
Linear damping decay and observability comparison
DefinitionDependency level 9
Scale Riesz map and free HUM operator
DefinitionDependency level 9
Linearized equation and its real-duality adjoint
LemmaDependency level 9
Simultaneous Sobolev extension and localization
Open problemDependency level 9
Three-dimensional intermediate Sobolev control
PropositionDependency level 9
Local observation has a lower initial regularity cost
PropositionDependency level 9
Negative resolvent bounds and the retarded Fourier argument
PropositionDependency level 9
Lifting zero-order observability to the order-two endpoint
PropositionDependency level 9
Pointwise and energy difference estimates for odd powers
RemarkDependency level 9
Remark 3: loss, Littlewood–Paley input and Neumann estimates
RemarkDependency level 9
Prior energy-critical null control on Euclidean space
RemarkDependency level 9
Resolvent frequencies, cutoff signs and higher compatibility
DefinitionDependency level 10
Quadratic-cubic remainder after subtracting a trajectory
DefinitionDependency level 10
Conditional energy-level unique continuation hypothesis
LemmaDependency level 10
Lemma 3.3: observability modulo a compact remainder
LemmaDependency level 10
Lemma 4.1: retarded smoothing down to negative order two
LemmaDependency level 10
Canonical higher traces on a bounded boundary or compact interior hypersurface
LemmaDependency level 10
Fractional Hilbert Sobolev algebra on a smooth exterior domain
LemmaDependency level 10
Exterior spectral domains and the critical trace index
LemmaDependency level 10
Fractional compactness on bounded collar domains
PropositionDependency level 10
Cubic and quintic energy-space nonlinear time estimates
PropositionDependency level 10