LemmaStatement: AI-generatedProof: AI-generated

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

An explicit common interpolation model for the fractional Strichartz targets

Statement

Assume AC. For $1 < q < \infty$ , $0 < s < 1$ and a smooth exterior domain $Ω$ , fix the convention $W^{s, q} (Ω) = [L^{q} (Ω), W^{1, q} (Ω)]_{s}$ by complex strip interpolation. For a compatible couple $E_{0}, E_{1}$ embedded in a common Hausdorff vector space, its sum has norm $\inf_{x = x_{0} + x_{1}} (∥ x_{0} ∥_{E_{0}} + ∥ x_{1} ∥_{E_{1}})$ ; the strip functions are bounded and continuous in this sum, norm-holomorphic inside, and have continuous $E_{j}$ boundary values tending to zero at imaginary infinity. The interpolation norm is the infimum of their maximum boundary supremum norm over functions taking value $x$ at $s$ . A single collar-reflection/free-tail extension, bounded at both $L^{q}$ and $W^{1, q}$ endpoints and right-inverse to restriction, identifies this space with the restriction quotient of whole-space $H^{s, q}$ , with equivalent norms. For each $0 < T < \infty$ and $1 \leq p < \infty$ , let $L^{p} (0, T; X)$ denote the Bochner space of strongly measurable almost-everywhere classes with norm $(\int_{0}^{T} ∥ f (t) ∥_{X}^{p} d t)^{1 / p}$ . Then the corresponding target interpolation is $[L^{p} (0, T; L^{q} (Ω)), L^{p} (0, T; W^{1, q} (Ω))]_{s} = L^{p} (0, T; W^{s, q} (Ω))$ . This is an explicit model chosen for the proposed applications; it is not an assertion that BGT specified this unique model, nor an identification with Slobodeckij spaces for $q \neq 2$ .

External input

The abstract strip-space Banach property and exact operator theorem are the independent contract in reference 0, §2.2, PDF pp.9–10. Its whole-space Bessel identity is Definition 3.6 and Theorem 3.7, PDF p.17, using Definitions 3.1 and 3.3 on pp.15–16. The finite-exponent Bochner identity is Example 2.7(i), PDF p.11. These imported inputs have the hypotheses and reading limits in the source record. The exterior extension, its simultaneous endpoint bounds and the restriction quotient application are local arguments in the five named intermediate items.

Facts & Assumptions

Given: The hypotheses of the Statement and the independently cited external contracts.

[F1]

Assume the Axiom of Choice The Axiom of Choice; in particular its countable and dependent instances are available.

[F2]

Banach space and A bounded linear operator between normed spaces define complex Banach spaces and common bounded linear endpoint operators; Integer-order Sobolev spaces and their norms fixes the $L^{q}, W^{1, q}$ weak-derivative endpoints. Under AC, Integer-order Sobolev spaces are Banach proves their completeness on the open set $Ω$ , using $k = 0, 1$ , exponent $q$ and scalar field $C$ .

[F3]

Well-defined strip norm and the action of a common endpoint operator verifies the stated sum/strip norms, their Banach well-definedness and exact interpolation of a common endpoint map.

[F4]

A single collar-reflection and free-tail exterior extension constructs the one collar/reflection plus tail map, and Restriction quotient and Bochner interpolation for the exterior extension proves the quotient-norm comparison and finite-time Bochner identity for that map.

[F5]

[F6]

Strongly measurable Banach-valued function defines strong measurability by almost-everywhere norm limits of measurable simple functions, for complex as well as real Banach values. The Bochner norm and null-set class convention in the Statement are those of Restriction quotient and Bochner interpolation for the exterior extension.

Proof

Given: The hypotheses and data in the Statement.

Goal: Prove the conclusions in the statement under its stated hypotheses.

1.1F1F2F3given

By the Sobolev completeness theorem with $k = 0, 1$ , exponent $q$ and scalar field $C$ , the endpoints $L^{q} (Ω)$ and $W^{1, q} (Ω)$ are complex Banach spaces under AC. Both inject into $L^{q} (Ω)$ ; indeed their sum is $L^{q} (Ω)$ and the displayed sum norm is its $L^{q}$ norm because $∥ u ∥_{q} \leq ∥ u ∥_{W^{1, q}}$ . The strip-model helper therefore makes the stated interpolation convention a definite complete normed space, with precisely the boundary continuity and decay specified in the Statement. All spaces are spaces of almost-everywhere classes, so there is no representative-dependent boundary value of a spatial function in this definition.

2.1F3F4F5step 1.1

The exterior extension helper supplies one map at both endpoints. The quotient/retraction helper applies the strip theorem to this map and restriction, identifies the whole-space target using the cited Bessel convention, and proves both inequalities comparing the interpolation and restriction-quotient norms. Thus $W^{s, q} (Ω)$ has exactly the asserted quotient realization without importing a bounded-domain extension as an exterior theorem.

3.1F4F5F6step 2.1∎

Let $0 < T < \infty$ and $1 \leq p < \infty$ , the finite Bochner range in the Statement. Example2.7(i) and the quotient/retraction helper give $[L^{p} (0, T; L^{q} (Ω)), L^{p} (0, T; W^{1, q} (Ω))]_{s} = L^{p} (0, T; W^{s, q} (Ω))$ . The maps are consistent on sums and on strongly measurable classes as proved there. This supplies exactly the chosen fractional target model; it adds no identification with a different fractional convention and no claim about how BGT named its spaces.

Source notes

The prior planned citation’s p.23 Bochner locator has been replaced by the independently inspected Example2.7(i), PDF p.11. Local proof provenance records the extension and retraction argument; the abstract and whole-space inputs remain explicit external premises. No proof or scaffold review is asserted.

Depends on

Used by · 5

Dependency tree · 2 levels

Each result is drawn at its shortest distance from this item. Arrows run from prerequisites to the items that use them; this item has a heavier outline. External source nodes link directly to the cited text. The view shows up to two dependency steps; open a prerequisite to continue.

External dependencies

J. C. Bellido and G. García-Sáez, Bessel Potential Spaces and Complex Interpolation: Continuous embeddings, arXiv:2503.04310v6 · Compatible couples and sums: PDF pp.4–6; §2.2, PDF pp.9–10 (strip definition, Banach property and exact operator bound); Example 2.7(i), PDF p.11 (Bochner interpolation); Definitions 3.1, 3.3, 3.6 and Theorem 3.7, PDF pp.15–17 (whole-space Bessel convention and identity).

For compatible complex Banach couples the strip class F with the maximum boundary norm is Banach, evaluation at an interior strip point is continuous into the sum, and complex interpolation is exact of exponent theta. For 1<q<infinity and 0<s<1, [Lq(Rn),W1q(Rn)]_s is the Bessel-potential space {f in S prime: F inverse((1+4 pi squared |xi| squared)^(s/2) Ff) in Lq}, with equivalent norms. Example 2.7(i) states [Lp0(E0),Lp1(E1)]_theta=Lp(theta)([E0,E1]_theta) for 1<=p0,p1<infinity.

Hypotheses: Complex Banach spaces continuously embedded into a common Hausdorff topological vector space; common bounded endpoint operator on their sum; 0<theta<1. The whole-space Sobolev assertion requires 1<q<infinity. Bochner identity uses the same Lebesgue measure space at both endpoints and finite 1<=p0,p1<infinity; here use (0,T) and p0=p1=p.

Reading limits: This adjudication inspected the compatible-couple and common operator passage on v6 PDF p.4, the strip definitions and exactness on pp.9–10, Example 2.7(i) on p.11, and Definitions 3.1, 3.3, 3.6 and the statement of Theorem 3.7 on pp.15–17. Theorem 3.7 and Mihlin proofs were not reconstructed; the original interpolation books were not read. No exterior extension theorem is imported; the extension is supplied by the existing local helpers.

Sources

J. C. Bellido and G. García-Sáez, Bessel Potential Spaces and Complex Interpolation: Continuous embeddings, arXiv:2503.04310v6 · Compatible couples and sums: PDF pp.4–6; §2.2, PDF pp.9–10 (strip definition, Banach property and exact operator bound); Example 2.7(i), PDF p.11 (Bochner interpolation); Definitions 3.1, 3.3, 3.6 and Theorem 3.7, PDF pp.15–17 (whole-space Bessel convention and identity).