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.

One second-order flat reflection on L² and H²

Statement

Assume AC. Let $n \geq 1$ be an integer and fix $K \in {R, C}$ ; all function spaces below are over $K$ . Write $H_{+} = R^{n - 1} \times (0, \infty)$ with coordinates $(y, t)$ . When $n = 1$ , $R^{0}$ is a single point and $∣ y ∣ = 0$ . Define one linear operator on $L^{2} (H_{+})$ by $E f (y, t) = f (y, t) (t > 0), E f (y, t) = 3 f (y, - t) - 2 f (y, - 2 t) (t < 0) .$ It restricts to $f$ on $H_{+}$ , is bounded $L^{2} (H_{+}) \to L^{2} (R^{n})$ and maps $W^{2, 2} (H_{+})$ boundedly into $W^{2, 2} (R^{n})$ . For $f \in W^{2, 2} (H_{+})$ , the same formula matches the value and first normal derivative at $t = 0$ , in the weak trace sense used by the half-space extension theorem. For every $a, b > 0$ , if $f = 0$ almost everywhere on $H_{+} ∖ {(y, t) : ∣ y ∣ < a, 0 < t < b}$ , then $E f = 0$ almost everywhere on $R^{n} ∖ {(y, t) : ∣ y ∣ < a, ∣ t ∣ < b}$ . In particular these smaller bounds leave positive margins inside any larger doubled cylinder ${(y, t) : ∣ y ∣ < A, ∣ t ∣ < B}$ with $A > a$ and $B > b$ .

Facts & Assumptions

Given: The hypotheses and conventions of the Statement.

[F1]

Assume the Axiom of Choice The Axiom of Choice; in particular its countable and dependent instances are available. Its use here is inherited through the weak trace and integration-by-parts interfaces of the half-space supplier in F3; the reflection coefficients are explicit.

[F2]

Integer-order Sobolev spaces and their norms defines the weak-derivative $W^{2, 2}$ space, its component-sum norm and almost-everywhere classes.

[F3]

Integer-order Sobolev extension from a half-space gives the $k = 2, p = 2$ reflection over either scalar field $K \in {R, C}$ with coefficients solving $a_{1} + a_{2} = 1$ and $- a_{1} - 2 a_{2} = 1$ . Its Facts F4–F5 and proof step 3.2 establish the required one-sided weak traces and cancellation of interface terms without assuming smooth-up-to-boundary density.

Proof

Goal: Establish every assertion of the Statement under the stated hypotheses.

Given: The hypotheses and data in the Statement.

1.1F2F3given

The moment equations have the solution $a_{1} = 3, a_{2} = - 2$ , so the formula is precisely the $k = 2$ operator from the half-space theorem. It already defines a map on every $L^{2}$ class: the two reflected coordinate maps preserve null sets. Changing variable $τ = - j t$ gives $∥ f (y, - j t) ∥_{L^{2} (t < 0)} = j^{- 1 / 2} ∥ f ∥_{L^{2} (H_{+})}$ , so the triangle inequality bounds the negative part by $(3 + \sqrt{2}) ∥ f ∥_{2}$ . Adding the unchanged positive part gives a finite common $L^{2}$ bound. Linearity and the restriction identity follow from the formula.

2.1F2F3step 1.1algebra

For a multi-index $α = (β, m)$ with $∣ α ∣ \leq 2$ , the candidate negative-side derivative is $g_{α} = 3 (- 1)^{m} (D^{α} f) (y, - t) - 2 (- 2)^{m} (D^{α} f) (y, - 2 t)$ . Its $L^{2}$ norm is at most $(3 + 2^{m + 1 / 2}) ∥ D^{α} f ∥_{2}$ by the same variable change. The positive-side derivative is $D^{α} f$ . The one-sided traces of order zero and one exist in the half-space theorem’s section sense, and their reflection coefficients are respectively $3 - 2 = 1$ and $- 3 + 4 = 1$ . In the half-space integration-by-parts identity the potential surface terms for second derivatives are multiples of these two mismatches; both vanish. Tangential derivatives remain on the boundary test function as in that theorem, so no derivative of an unconstructed boundary trace is used. Thus the fields $g_{α}$ and their positive-side counterparts are the actual global weak derivatives. Summing their squared bounds proves the $W^{2, 2}$ estimate for the very same $L^{2}$ map.

3.1F3step 1.1step 2.1algebra∎

Let $a, b > 0$ and suppose $f$ vanishes almost everywhere outside $∣ y ∣ < a, 0 < t < b$ in $H_{+}$ . By the null-set preservation in step 1.1, both negative-side summands vanish almost everywhere whenever $∣ y ∣ \geq a$ or $t \leq - b$ : in the latter case $- j t \geq b$ for $j = 1, 2$ . The positive side is unchanged, and the hyperplane $t = 0$ is a null set, so $E f$ vanishes almost everywhere outside $∣ y ∣ < a, ∣ t ∣ < b$ . If $A > a$ and $B > b$ , these bounds retain margins $A - a$ at the lateral edge when $n > 1$ and $B - b$ at the top and bottom of the larger doubled cylinder. For $n = 1$ there is no lateral edge. This proves the localization assertion and completes the endpoint construction without an unproved density passage.

Source notes

The matching is justified by the exact existing weak half-space theorem, not by treating even reflection as an H² extension. No external extension theorem or fresh source-reading claim is used.

Depends on

Used by · 1

A compactly supported common L²/H² extension from a bounded smooth domain

Dependency tree · 2 levels

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