CounterexampleConstruction: AI-adaptedVerification: AI-adaptedPipeline-generatedjudge pass (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.

$W^{1, n}$ is not contained in $L^{\infty}$

Statement refuted

Assume the Axiom of Choice (The Axiom of Choice). Let $n \geq 2$ and let $φ \in C_{c}^{\infty} (R^{n})$ satisfy $0 \leq φ \leq 1$ , $φ = 1$ on $B (0, \frac{1}{2})$ and $supp φ \subseteq B (0, 2)$ , and set $f (x) = φ (x) \log \log (1 + \frac{1}{∣ x ∣})$ for $x \neq 0$ , $f (0) = 0$ . Then $f \in W^{1, n} (R^{n})$ but $f$ is unbounded near the origin; therefore this $W^{1, n}$ class admits no bounded (and no continuous, let alone Holder) representative, and the endpoint $p = n$ has no $L^{\infty}$ embedding, while this compactly supported witness belongs to $L^{q}$ for every finite $q$ (the general bounded-domain finite- $q$ embedding is stated separately).

Facts & Assumptions

Given: The Axiom of Choice; $n \geq 2$ ; a cutoff $φ$ as in the statement (A Euclidean bump for a compact set inside an open set); and $f = φ g$ with $g (x) = \log \log (1 + 1 / ∣ x ∣)$ for $x \neq 0$ , $g (0) = 0$ .

[F1]

The witness $g$ lies in $W^{1, n} (B (0, 1))$ and has no bounded representative; consequently no representative of $g$ is bounded on any neighbourhood of the origin (Morrey's inequality has no $p = n$ endpoint).

[F2]

$W^{1, n}$ consists of the $L^{n}$ classes with weak gradient in $L^{n}$ ; multiplication by the smooth compactly supported cutoff $φ$ preserves $W^{1, n}$ classes and $L^{p}$ classes are determined up to null sets (Integer-order Sobolev spaces and their norms, The space $L^{p} (μ)$ as the quotient by null functions).

[F3]

The critical embedding gives $W^{1, n} (Ω) ↪ L^{q} (Ω)$ for every finite $q$ on nonempty bounded domains that are $W^{1, s}$ -extension domains for every $s \in (n / 2, n)$ (The critical Sobolev embedding into every finite $L^{q}$ ).

[F4]

A compact ball inside an open ball admits a smooth cutoff equal to $1$ on the smaller ball and supported in the larger one (A Euclidean bump for a compact set inside an open set); weak derivatives are characterized by the test-function identity (Weak derivative of a locally integrable function).

[F5]

Balls are bounded smooth domains, so under AC they have a bounded extension operator at every Sobolev index (Bounded C^k domains admit integer-order Sobolev extension). Smooth classical derivatives are weak derivatives under CC (Classical derivatives agree with weak derivatives).

Counterexample

technique · direct

1.1F1F2F4F5givenalgebra

Membership in $W^{1, n}$ . On $B (0, 1)$ , $∣ f ∣ \leq ∣ g ∣$ because $0 \leq φ \leq 1$ , so $f \in L^{n} (B (0, 1))$ by [F1]. On $R^{n} ∖ B (0, 1)$ , the function $g$ and its derivatives are smooth and bounded on the compact support of $φ$ , so $f \in L^{n} (R^{n})$ . Choose $χ \in C_{c}^{\infty} (B (0, 1))$ equal to $1$ on $B (0, 1 / 4)$ , and write $a : = χ φ$ and $b : = (1 - χ) φ g$ . Then $a$ is supported in $B (0, 1)$ , while $b$ is smooth and compactly supported away from the origin. For a test function $ψ \in C_{c}^{\infty} (R^{n})$ , $a ψ \in C_{c}^{\infty} (B (0, 1))$ , so the weak derivative identity for $g$ from [F1] gives $\int_{R^{n}} a g \partial_{i} ψ = \int_{B (0, 1)} g \partial_{i} (a ψ) - \int_{B (0, 1)} g (\partial_{i} a) ψ = - \int_{B (0, 1)} (a D_{i} g + g \partial_{i} a) ψ .$ The summand $a g$ therefore has weak derivative $a D_{i} g + g \partial_{i} a$ , which lies in $L^{n}$ by [F1] and boundedness of $a, D a$ ; the summand $b$ has its classical, compactly supported $L^{n}$ weak derivative by [F5]. Thus $f = a g + b \in W^{1, n} (R^{n})$ by [F2].

2.1F1F2step 1.1givenalgebra

No bounded representative. For every $M > 0$ the set ${x \in B (0, \frac{1}{2}) : f (x) > M}$ is a punctured neighbourhood of $0$ and has positive measure, because $f = g \to + \infty$ as $x \to 0$ ; hence every representative of the $W^{1, n}$ class of $f$ is unbounded on every neighbourhood of $0$ . In particular $f$ has neither a bounded, nor a continuous, nor a Holder representative.

3.1F3F5step 1.1step 2.1givenalgebra∎

The endpoint contrast. By [F3], applied on $Ω = B (0, 2)$ , whose all-exponents extension hypothesis is supplied by [F5], the class $f$ lies in $L^{q}$ for every finite $q$ and satisfies $∥ f ∥_{L^{q}} \leq C (n, q) ∥ f ∥_{W^{1, n}}$ . But no finite constant bounds $∥ f ∥_{L^{\infty}}$ , since any representative is unbounded by step 2.1; hence $W^{1, n}$ admits no embedding into $L^{\infty}$ or a Holder class; the finite- $q$ statement here concerns this compactly supported witness and the bounded domains covered by [F3].

Source notes

The witness is Hunter's Example 3.30 and Kinnunen's Remark 3.16(1), printed at p. 66 and p. 73. The membership in $W^{1, n}$ and the absence of a bounded representative are established for the localised logarithm in Morrey's inequality has no $p = n$ endpoint; multiplying by the cutoff $φ$ does not change the behaviour near the origin and makes the function compactly supported, and the finite- $q$ embeddings are the complement recorded in The critical Sobolev embedding into every finite $L^{q}$ .

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

47 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

John K. Hunter, Notes on Partial Differential Equations (UC Davis, revised 18 June 2014, complete 242-page two-quarter notes) (standard reference, not scraped)
Juha Kinnunen, Sobolev Spaces (Aalto University, 2026, complete graduate lecture notes) (standard reference, not scraped)

W1,nW1,n is not contained in L∞L∞