LemmaStatement: AI-adaptedProof: AI-generatedPipeline-generated

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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.

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Bounded coagulation is bilinear and bounded in weighted total variation

Statement

For the coagulation bilinear operator $C$ of Continuous-size coagulation-fragmentation weak model, every $μ, ν \in M_{w} (E)$ satisfies

$∥ C (μ, ν) ∥_{w} \leq \frac{3}{2} K_{*} ∥ μ ∥_{w} ∥ ν ∥_{w} .$

Facts & Assumptions

Given: The weight $w (x) = 1 + x^{2}$ , the bounded kernel $0 \leq K \leq K_{*}$ , and the signed-measure operator.

[F1]

Continuous-size coagulation-fragmentation weak model: $C (μ, ν) = \frac{1}{2} \iint K (x, y) (δ_{x + y} - δ_{x} - δ_{y}) μ (d x) ν (d y)$ .

[F2]

Weighted total-variation space for population measures: $∥ σ ∥_{w} = \int w d ∣ σ ∣$ .

Proof

1.1givenF2algebra

For $x, y \geq a$ , $w (x + y) + w (x) + w (y) = 3 + 2 x^{2} + 2 x y + 2 y^{2} \leq 3 (1 + x^{2}) (1 + y^{2})$ , since the difference is $(x - y)^{2} + 3 x^{2} y^{2} \geq 0$ .

2.1givenF1F2step 1.1algebra∎

Total variation of a signed integral is bounded by the integral against the variation measures. Applying this to [F1] and using $K \leq K_{*}$ gives $∥ C (μ, ν) ∥_{w} \leq \frac{K_{*}}{2} \iint [w (x + y) + w (x) + w (y)] ∣ μ ∣ (d x) ∣ ν ∣ (d y)$ . Step 1.1 factors this bound as $\frac{3}{2} K_{*} (\int w d ∣ μ ∣) (\int w d ∣ ν ∣)$ , which is the claimed estimate by [F2].

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Sources

Philippe Laurençot, Weak compactness techniques and coagulation equations, §§1 and 3 (standard reference, not scraped)
Philippe Laurençot and Stéphane Mischler, From the discrete to the continuous coagulation-fragmentation equations, §§1–2 (standard reference, not scraped)