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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
- Choice function
- Hilbert space
- A bounded linear operator between normed spaces
- The operator norm as the least bound and as the unit-sphere or unit-ball supremum
- Orthogonality and the orthogonal complement
- Continuity of a map between metric spaces, at a point and globally, in the ε-δ form
- Bochner integrability criterion
- The Banach-valued simple integral is well defined
- Additivity of the nonnegative Lebesgue integral
- Monotonicity and nonnegative homogeneity of the nonnegative integral
- Bochner-integrable function
- Vector addition and scalar multiplication are continuous in a normed space
- Lipschitz map, α-Hölder map for rational 0 < α ≤ 1, and contraction
- Complete metric space: every Cauchy sequence converges in the space
- Cauchy sequence in a metric space
- The recursion theorem
- For |r| < 1 the sequence rᵏ is null, and for |r| > 1 the sequence |r|ᵏ diverges to +∞
- Integer powers aᵐ
- Convergence of a sequence in a metric space: xₖ → x iff d(xₖ, x) → 0 in ℝ
- A sequence in a metric space has at most one limit
- Contraction implies Lipschitz implies uniformly continuous implies continuous; every Hölder map is uniformly continuous, and a Lipschitz map on a bounded space is Hölder for every exponent
- Metric continuity characterisations, with countable choice for the sequential converse
- Factorisation of bⁿ - aⁿ, and the resulting Lipschitz estimate
- Finite sums and finite products, by recursion
- Laws of finite sums and finite products
- The principle of mathematical induction
- Metric space: d(x,y) = 0 iff x = y, symmetry, and the triangle inequality; pseudometric and ultrametric
- Inverses of positives are positive, and reciprocation reverses order
- Algebra of limits: sums, scalar multiples, products and quotients
- Laws of integer exponents
- Nonnegativity of a metric is a consequence of the other axioms, not an axiom
- Limits and Cauchy sequences of reals
- The rationals embed densely in the reals
- The exponential function is smooth and (exp)'=exp
- The exponential is positive and satisfies exp(-x)=1/exp(x)
- The chain rule, in one line from Carathéodory: if g is differentiable at c and f is differentiable at g(c), then f ∘ g is differentiable at c with (f ∘ g)'(c) = f'(g(c)) g'(c)
- Sums, scalar multiples, products and quotients: (f+g)'(c) = f'(c) + g'(c), (α f)'(c) = α f'(c), (fg)'(c) = f'(c)g(c) + f(c)g'(c), and (f/g)'(c) = (f'(c)g(c) - f(c)g'(c))/g(c)² when g(c) ≠ 0
- The first fundamental theorem: if f is integrable on [a,b] and continuous at c, then F'(c) = f(c); in particular a continuous f has F as a primitive
- The second fundamental theorem: if G is differentiable on [a,b] with G' = f and f is integrable, then ∫ₐᵇ f = G(b)-G(a)
- If f ≤ g on [a,b] and both are integrable then ∫ₐᵇ f ≤ ∫ₐᵇ g; and m(b-a) ≤ ∫ₐᵇ f ≤ M(b-a)
- For 1 < p < ∞, the Lᵖ norm of a nonnegative function is the supremum of its pairings with L^q unit vectors
- Tonelli's theorem for nonnegative measurable functions on a sigma-finite product
- Holder's inequality for integrals, including the endpoint cases
- Monotone convergence for the integral
- Spectral theorem for unbounded self-adjoint operators (PVM form)
- The unbounded PVM integral is densely defined, closed and normal
- Integral of a measurable function against a projection-valued measure
- Resolvent and spectrum of an unbounded operator
- Symmetric, self-adjoint and essentially self-adjoint operators
- Projection valued measure
- Dominated convergence
- Bounded borel pvm integral
- Pvm integral is a star homomorphism
- Scalar and complex measures from a pvm
- ℚ is countably infinite
- Both ℚ and ℝ ∖ ℚ are dense in ℝ, and every nonempty open subset of ℝ is uncountable
- Banach space
- A closed subspace of a Banach space is Banach
- Bounded linear maps commute with Bochner integration
- Measure spaces
- Topological vector spaces over the real and complex fields
- Strongly measurable Banach-valued function
- Finite products of Banach spaces are Banach
- Continuity of a map of topological spaces at a point and globally
- Hausdorff space: distinct points have disjoint open neighbourhoods; every metrizable space is Hausdorff and the indiscrete topology on two points is not
- The metric topology: a set is open when every one of its points has a ball around it inside the set; closed means open complement
- The Axiom of Countable Choice (AC_ω)
- 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 dominated convergence theorem
- Lebesgue outer measure, Lebesgue measurability and Lebesgue measure are unchanged by translation
- The integral of a nonnegative simple function
- The nonnegative Lebesgue integral
- The space Lᵖ(μ) as the quotient by null functions
- Fubini's theorem for L¹ functions on a sigma-finite product
- Fatou's lemma
- Strongly continuous semigroup
- Bochner dominated convergence theorem
- Bounded linear maps commute with Bochner integration
- Assuming countable choice, L(ℝⁿ) is a sigma-algebra containing every elementary set and λₙ is a complete measure extending elementary volume
- A box in ℝⁿ with parameters aᵢ≤ bᵢ is Lebesgue measurable of measure ∏_i<n(bᵢ-aᵢ), whichever of its faces are included
- For every ε > 0 in a complete ordered field there is a natural n ≥ 1 with 1/n < ε
- Open cover, subcover, and compact topological space; a compact subset is a subspace that is compact in its own right
- Separation of a topological space, connected and disconnected spaces, clopen sets, and connected subsets
- Interior, closure, boundary, exterior, derived set and isolated point in a topological space
- Open ball, closed ball and sphere in a metric space
- Bounded Cᵏ domains and boundary charts
- In a Hausdorff space a point and a disjoint compact set, and two disjoint compact sets, have disjoint open neighbourhoods; hence every compact subset is closed, and in a compact Hausdorff space the compact subsets are exactly the closed ones
- Distinct points of a metric space have disjoint balls around them
- Independent generalized reflected-ray and GCC convention
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
Definition
The Axiom of Choice
Definition
Strongly continuous one-parameter unitary group
Lemma
Bochner integral norm inequality
Lemma
Linearity of the Bochner integral
Theorem
A contraction of a nonempty complete metric space into itself has exactly one fixed point, the limit of the iterates from any starting point
Theorem
Gronwall's integral inequality with variable and constant coefficients
Theorem
Minkowski's integral inequality
Theorem
Unbounded Borel functional calculus: domains, products, spectral mapping
Lemma
Compatible Banach pairs and their integrals
Lemma
Strongly continuous shifted orbits in finite Bochner norms
Lemma
Scalar Banach-function retarded estimates by Fatou approximation
Lemma
Continuous free convolution on a finite time interval
Definition