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Dispersive estimates and evolution
Organize the independent exterior resolvent, smoothing and Strichartz inputs with their dimensional, exponent and derivative-loss limits, then the negative-scale, causal L1 and real-linear perturbation transfers used by the article.
1 · Prerequisites
- Compatible Banach pairs and their integrals
- Strongly continuous shifted orbits in finite Bochner norms
- Bochner dominated convergence theorem
- Holder's inequality for integrals, including the endpoint cases
- Bochner integrability criterion
- Bochner integral norm inequality
- Bounded linear maps commute with Bochner integration
- Absolute continuity of the integral
- Scalar Banach-function retarded estimates by Fatou approximation
- Riesz-Fischer completeness of Lᵖ for 1 ≤ p ≤ ∞
- Minkowski's integral inequality
- Fatou's lemma
- Tonelli's theorem for nonnegative measurable functions on a sigma-finite product
- The essential supremum is attained as the least essential bound
- The Axiom of Countable Choice (AC_ω)
- Finite, sigma-finite, and semifinite measures
- The space Lᵖ(μ) as the quotient by null functions
- Banach space
- The essential supremum of a measurable function with respect to a measure
- Continuous free convolution on a finite time interval
- A contraction of a nonempty complete metric space into itself has exactly one fixed point, the limit of the iterates from any starting point
- The spaces B(X,Y) and B(X) of bounded linear operators
- The operator norm as the least bound and as the unit-sphere or unit-ball supremum
- Exterior and truncated obstacle domains
- Independent generalized reflected-ray and GCC convention
- Strongly continuous one-parameter unitary group
- Hilbert space
- Strongly measurable Banach-valued function
- Bochner-integrable function
- Gronwall's integral inequality with variable and constant coefficients
- The Axiom of Choice
- Weak shifted Dirichlet form and energy bound on an open set
- Collar and tail assembly of the exterior graph domain
- Integer-order Sobolev spaces and their norms
- Zero-boundary Sobolev space as a norm closure
- Symmetric, self-adjoint and essentially self-adjoint operators
- Fubini's theorem for L¹ functions on a sigma-finite product
- Bochner dominated convergence theorem
- Bounded linear maps commute with Bochner integration
- Linearity of the Bochner integral
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
Abstract retarded integration in compatible time spaces
Lemma
Spatial Lq classes satisfy the scalar mixed-norm hypotheses
Lemma
Short-interval real Volterra contraction
Definition
Non-trapping obstacle
Lemma
Homogeneous bounds transfer to L1 retarded forcing
Lemma
Global real evolution, cocycle and inverse
Lemma
Exterior Dirichlet realization, H² domain and form domain
Lemma