Alphabeta MathPersonal research
Exterior geometry and function spaces
Specify the obstacle and generalized-ray conventions, Sobolev and time-space notation, Dirichlet realization and spectral scales; connect bounded fractional inputs to exterior localization, traces, compactness and continuation.
1 · Prerequisites
- Integer-order Sobolev spaces and their norms
- Weak derivative of a locally integrable function
- Weak differentiation ignores null-set changes
- Uniqueness of a weak derivative as an almost-everywhere class
- The Sobolev norm descends to equivalence classes
- Distributional derivatives are polynomial Fourier multipliers
- Real-order Bessel-potential completion Hˢ
- The Bessel completion embeds canonically in tempered distributions
- Weighted tempered-distribution characterization of Hˢ
- Plancherel theorem
- Fourier differentiation and multiplication identities on tempered distributions
- Fourier transform agrees with l one and plancherel transforms
- Fourier transform is a topological automorphism of tempered distributions
- Polynomial growth functions define tempered distributions
- Locally integrable functions embed in distributions
- Smooth polynomially bounded multipliers on schwartz space
- Regular distribution from a locally integrable function
- Euclidean Gaussian transform with the 2π normalization
- The exponential dominates every fixed nonnegative integer power at +∞
- Cᵏ maps and multi-index derivative notation in Euclidean space
- Complex Lp classes and Euclidean test-function conventions
- The Axiom of Countable Choice (AC_ω)
- A radial mollifier family in Rn
- Convolution with a mollifier is smooth, and derivatives pass under the integral sign
- Young's convolution inequality under Countable Choice
- Minkowski's integral inequality
- Holder's inequality for integrals, including the endpoint cases
- Translation of a function on ℝⁿ
- An equicontinuous pointwise-bounded family in C(K,ℝ) has a finite net in the supremum metric
- Arzelà--Ascoli for real C(K) under Countable Choice and Dependent Choice: compact closure iff equicontinuous and pointwise bounded
- Finite ε-net and totally bounded metric space
- A totally bounded metric space is bounded, every subspace of a totally bounded space is totally bounded, and the closure of a totally bounded subset is totally bounded
- Closed subspaces of complete metric spaces are complete; the converse under countable choice
- A complete, totally bounded metric space is compact, proved from countable choice used exactly once
- Riesz-Fischer completeness of Lᵖ for 1 ≤ p ≤ ∞
- 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
- Open ball, closed ball and sphere in a metric space
- The space Lᵖ(μ) as the quotient by null functions
- Lebesgue outer measure, Lebesgue measurability and Lebesgue measure are unchanged by translation
- The axiom of dependent choice: a relation in which every element is related to something admits an ℕ-indexed chain
- A finite partition glues the local interior and boundary H² estimates
- Tangential H² estimate near a flat Dirichlet boundary
- The normal second derivative is recovered from the equation
- Interior H² regularity for divergence-form equations
- Weak divergence-form equations are invariant under C² boundary charts
- C² flattening preserves uniform ellipticity quantitatively
- Weak Dirichlet solutions for a divergence-form operator
- Bounded Cᵏ domains and boundary charts
- Uniformly elliptic divergence-form operators and their sesquilinear forms
- Young's inequality for conjugate real exponents
- Maximum principle on a closed strip for bounded holomorphic functions
- The complex exponential is entire and its complex derivative is itself
- The chain rule for complex derivatives
- exp(z+w)=exp z exp w, and the complex exponential extends the real exponential
- Complex logarithms, the principal logarithm, and principal and multivalued complex powers
- The natural logarithm as the inverse of the exponential function
- Order, continuity, range, and the product, quotient, and reciprocal laws for the natural logarithm
- Real powers for positive bases, with the zero-base positive-exponent convention
- Continuity and derivatives of positive-base real powers
- Dedekind completeness: the least-upper-bound property
- Ball mean-value property for harmonic functions under Countable Choice
- Multi-indexed power series in ℂᵐ and their absolute convergence
- Real analytic germs in several variables
- Surface integration on compact C1 hypersurfaces
- The ball Poisson kernel is positive and has unit mass
- Smooth sphere data have a harmonic replacement under Countable Choice
- Sphere and ball measures scale in Rn
- 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 chain rule for total derivatives: D(g∘ f)(a)=Dg(f(a))∘ Df(a)
- Cᵏ Euclidean maps are closed under componentwise algebra and composition
- Continuous ball-mean-value functions are harmonic
- Differentiation under the integral sign
- Continuous Dirichlet problem on a ball
- Dominated convergence
- The multinomial coefficient equals n!/∏_i<m kᵢ!, and (x₀+…+xₘ₋₁)ⁿ = ∑ iotaC(n, k)∏_i<m xᵢ^kᵢ in ℝ
- Multivariable Taylor formula with a Lagrange remainder along a line segment
- For a nonempty subset of ℝⁿ with n≥1, compactness, closedness and boundedness, pseudocompactness, and attainment of extrema by every continuous real-valued function are equivalent
- For n≥1, every Euclidean closed ball and every Euclidean sphere of positive radius is compact
- A continuous real-valued function on a nonempty compact metric space is bounded and attains a greatest and a least value
- Poisson kernel of a Euclidean ball
- The Axiom of Choice
- The notation Hᵏ and the reserved zero-boundary symbol
- Sobolev extension domains and extension operators
- Local Hölder and scaled C-two-alpha norms on balls
- Weak partial derivatives lower the Sobolev order
- The Sobolev conjugate exponent and the scaling identity
- Morrey's inequality for p>n
- The p=1 Gagliardo-Nirenberg-Sobolev inequality
- The Gagliardo-Nirenberg-Sobolev inequality for 1<p<n
- Compactly supported smooth functions are dense in W^k,p(Rⁿ)
- A Euclidean bump for a compact set inside an open set
- Weak Leibniz rule with a smooth factor
- Interior mollification commutes with weak derivatives
- If f : [a,b] → ℝᵐ is differentiable with integrable f' then ∫ₐᵇ f' = f(b)-f(a); and a bounded derivative makes f Lipschitz
- Complex Lp completeness and almost-everywhere subsequences
- Local weak solutions of a divergence-form operator
- Nested-domain induction for interior elliptic derivatives
- The Lᵖ trace operator on a bounded C¹ domain
- The trace agrees with classical restriction for continuous Sobolev functions
- The trace commutes with smooth cutoffs and is chart local
- The Gauss-Green integration-by-parts formula with Sobolev traces
- Zero-boundary Sobolev space as a norm closure
- Ambient smooth restrictions are dense on bounded Cᵏ domains
- Compactly supported Sobolev functions extend by zero in every integer order
- Cᵏ boundary flattening preserves local W^k,p
- Bounded restriction and cutoff localisation in Sobolev spaces
- Finite ambient partitions near compact sets
- Integer-order Sobolev extension from a half-space
- ‖τ_h f - f‖ₚ → 0 in Lᵖ(ℝⁿ) as h → 0, for 1 ≤ p < ∞
- Tonelli and Fubini for the completed product, with only almost-everywhere section measurability
- Fundamental theorem of calculus for absolutely continuous functions
- Complex translation, convolution, approximate identities, and mollification
- Orthogonal decomposition by a closed subspace
- Cauchy–Schwarz: |⟨ x,y⟩|≤‖x‖ ‖y‖, with equality exactly for dependent pairs
- The dual space X^* of a normed space and its dual norm
- A bounded linear operator between normed spaces
- The operator norm as the least bound and as the unit-sphere or unit-ball supremum
- Linear subspace of a vector space
- Orthogonality and the orthogonal complement
- Real and complex inner-product spaces and their induced length
- The half-space trace lies in the fractional Slobodeckij space
- A bounded right inverse of the half-space trace by normal mollification
- The fractional Sobolev space on a compact C¹ boundary
- Chart independence of the fractional boundary norm
- The coordinate-direction form of the Slobodeckij seminorm
- Integral of a measurable function against a projection-valued measure
- The unbounded PVM integral is densely defined, closed and normal
- Cayley correspondence between self-adjoint operators and unitaries
- Cayley transform of a self-adjoint operator
- Spectral theorem for bounded normal operators pvm form
- Support and uniqueness of the spectral measure
- Projection valued measure
- Regular Borel measure on an LCH space
- Spectrum and resolvent of a bounded operator
- Neumann series
- Symmetric, self-adjoint and essentially self-adjoint operators
- Bounded borel pvm integral
- Resolvent and spectrum of an unbounded operator
- Pvm integral is a star homomorphism
- Scalar and complex measures from a pvm
- Monotone convergence for the integral
- ℚ is countably infinite
- Both ℚ and ℝ ∖ ℚ are dense in ℝ, and every nonempty open subset of ℝ is uncountable
- The Axiom of Choice
- The Axiom of Countable Choice (AC_ω)
- Hilbert space
- Infinitesimal generator of a unitary group
- Projection valued measure
- Strongly continuous one-parameter unitary group
- Symmetric, self-adjoint and essentially self-adjoint operators
- Integral of a measurable function against a projection-valued measure
- A self-adjoint operator generates a strongly continuous unitary group
- Bounded borel pvm integral
- Pvm integral is a star homomorphism
- Spectral theorem for unbounded self-adjoint operators (PVM form)
- Unbounded Borel functional calculus: domains, products, spectral mapping
- Closed sectorial form and its associated operator
- Closed semibounded symmetric forms and their associated operators
- Riesz representation for Hilbert spaces
- Cauchy–Schwarz: |⟨ x,y⟩|≤‖x‖ ‖y‖, with equality exactly for dependent pairs
- Cotangent space and cotangent bundle as a disjoint union
- Geometric control condition for the wave equation with a time-dependent observation domain · Section 1.3.1, Definitions 1.3–1.4 and Proposition 1.5, PDF pp. 6–9; Section 1.3.2, Definition 1.6, PDF p. 9
- On nonlinear Schrödinger equations in exterior domains · Introduction, printed pp. 295–296, opening paragraph
- Banach space
- Strongly measurable Banach-valued function
- Bochner-integrable function
- Bochner integrability criterion
- Complex Holder, Minkowski, and the quotient norm
- Minkowski's inequality for integrals, including p = ∞
- Heine-Borel by bisection: every closed bounded interval [a,b] is compact
- A continuous real function on a compact subset of ℝ is bounded
- Lebesgue measurable sets, the family L(ℝⁿ), and the restricted set function λₙ
- 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
- Lebesgue measure is sigma-finite, and every metrically bounded subset of ℝⁿ has finite outer measure
- The completed product measure
- Finite and countable subadditivity of measures
- Arithmetic and lattice operations preserve measurability whenever they are defined
- Real-order Hˢ as weighted Fourier distributions
- The Dirichlet Laplacian generates an analytic heat semigroup
- Sectorial operator with the semigroup sign convention
- Bounded Cᵏ domains admit integer-order Sobolev extension
- Gerd Grubb, Regularity of spectral fractional Dirichlet and Neumann problems, Mathematische Nachrichten 289 (2016), 831–844 · Section 2, PDF pp. 2–3: bijectivity hypothesis before equation (2.2); equation (2.4), discussion and equation (2.5), Definition 2.1 and Theorem 2.2
- Test function space d of an open set
- Test function topology
- Distribution
- Distributional derivative
- A norm on a real vector space, the induced metric, and the dictionary with the metric axioms
- Complex norm and complex normed space
- Realification preserves a complex norm and its metric
- Weak topology on a normed space
- Weak convergence of nets and sequences
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
Theorem
Integer-order W^k,2 and Hᵏ agree with equivalent norms
Theorem
The Fréchet–Kolmogorov compactness criterion in Lᵖ(ℝⁿ)
Theorem
Global H² Dirichlet regularity
Theorem
Hadamard three-lines theorem
Theorem
Harmonic functions are real analytic
Theorem
Higher-order Sobolev embedding
Theorem
Interior Hᵏ⁺² elliptic regularity
Theorem
The kernel of the trace is the closure of the test functions
Theorem
Riesz representation for Hilbert spaces
Theorem
The sharp trace theorem: boundedness and range in the fractional space
Theorem
Spectral theorem for unbounded self-adjoint operators (PVM form)
Theorem
Unbounded Borel functional calculus: domains, products, spectral mapping
Lemma
Flow calculus of a self-adjoint generator: commutation, invariance and differentiability
Theorem
Semibounded closed-form representation and perturbation
Proposition
Independent generalized reflected-ray and GCC convention
Definition
Continuous and mixed time-space function spaces
Definition
Complex integer-order Sobolev notation
Theorem
Bounded smooth fractional Dirichlet domain contract
Definition
Distributional equation convention for complex Schrödinger solutions
Definition