LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-6-luna)verified 2026-10-22 (gpt-6.1-sol)

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.

Full-space BV Sobolev inequality for L² distributional TV

Statement

Assume Countable Choice. Every $u \in L^{2} (R^{2}; R)$ with $T V_{2} (u; R^{2}) < \infty$ satisfies $∥ u ∥_{L^{2} (R^{2})} \leq T V_{2} (u; R^{2}) .$

Statement references

The full-space L² TV is the extended distributional test-field supremum.

For globally integrable BV functions in two dimensions, $∥ v ∥_{2} \leq T V (v)$ .

Compactly supported BV cutoffs approximate every finite-variation L² function.

Facts & Assumptions

Given: Countable Choice and $u \in L^{2} (R^{2}; R)$ with finite $T V_{2} (u; R^{2})$ .

[A1]

Countable Choice supplies the finite-Radon representation used in the cutoff approximation. (The Axiom of Countable Choice ( ${A C}_{ω}$ ))

[F1]

Compactly supported BV cutoffs for finite-variation L² functions: there are $v_{k} \in B V (R^{2}) \cap L^{1} (R^{2})$ with $v_{k} \to u$ in $L^{2}$ and ${lim sup}_{k} T V (v_{k}; R^{2}) \leq T V_{2} (u; R^{2})$ .

[F2]

BV Sobolev inequality in full space: every globally integrable $v \in B V (R^{2})$ satisfies $∥ v ∥_{2} \leq T V (v; R^{2})$ .

Proof

1.1A1F1F2givenchoose

By [F1], choose $v_{k} \in B V (R^{2}) \cap L^{1} (R^{2})$ with $v_{k} \to u$ in $L^{2}$ and the stated variation upper bound. Apply [F2] to each $v_{k}$ to obtain $∥ v_{k} ∥_{2} \leq T V (v_{k}; R^{2})$ .

2.1step 1.1algebra∎

Norm continuity gives $∥ v_{k} ∥_{2} \to ∥ u ∥_{2}$ , so $∥ u ∥_{2} = \lim_{k} ∥ v_{k} ∥_{2} \leq {lim sup}_{k} T V (v_{k}; R^{2}) \leq T V_{2} (u; R^{2}) .$ This is the required inequality.

Depends on

Used by

Primal compactness and selection of a minimum-TV limit Lemma

Dependency tree · two levels

18 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

José A. Iglesias, Gwenael Mercier, and Otmar Scherzer, A note on convergence of solutions of total variation regularized linear inverse problems, §1.1 (standard reference, not scraped)
Gláucio Terra, Introduction to Geometric Measure Theory — Lecture Notes, version 1.6, §7.5 (standard reference, not scraped)