{"id":"758e58d3-4d79-4cb8-b6c6-de351a03d1ea","arxiv_id":"1908.06179","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"Functions with finite non-local energy I_{δ,p}(u,Ω) are exponentially integrable with exponent proportional to δ^{-1}, and the exponent p/d is shown to be sharp.","lead":"The paper proves sharp exponential integrability for functions with finite non-local, non-convex energy, extending Moser-Trudinger-type bounds to a new class of functions. The result matters for Sobolev and Poincaré inequality refinements, where this energy already characterizes Sobolev spaces.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1.1(i) is false as printed: the reduced exponent α(p/d)^β is bounded below by α, while the admissible linear family A_M x_1 has exponential integral growing like e^{cA_M}; an M-independent C cannot exist. The proof's β=ℓ0^{-1} is a multiplier, not an exponent.","rationale":"The reader's weakest_assumption concerns [18, Lemma 17], the radial extension step. That lemma is an external citation and should be checked, but it is not the place where the main argument breaks on its own terms. The loaded issue is inside §2.2: the theorem and proof use different reduced exponents, and as printed the theorem is falsified by linear functions. The linear family is important because it is admissible under the energy constraint for every M and the BMO oscillation can be made large by scaling, so the constraint does not prevent the exponential integral from growing with M. The proof's own definition β=ℓ0^{-1} reinforces the diagnosis: ℓ0 is a threshold level, so the rate obtained by layer-cake integration is α(p/d)/ℓ0 (a multiplier), never α(p/d)^{ℓ0^{-1}} (a power). Therefore the central statement (1.11) must be corrected, or the proof modified. I keep CONDITIONAL rather than REJECT because the proof contains the correct mechanism for a multiplier version of the theorem and part ii/Theorem 1.2 are unaffected; however, the condition is substantive: the exponent in Theorem 1.1(i) must be amended and the omitted proofs supplied.","tokens_in":11835,"tokens_out":18576,"duration_ms":180727,"concrete_test":"Run the layer-cake integration from (2.13) with ℓ0 from (2.12) and record the resulting rate; then evaluate the left side of (1.11) on u_M(x)=A_M x_1 with I_{1,p}(u_M,B1)=M, A_M∼(M/K)^{1/p}. The integrals behave like (2/cA_M)e^{cA_M} with c=α(p/d)^β≥α, so they diverge as M→∞. This shows (i) the proof gives the multiplier rate α(p/d)/ℓ0, not α(p/d)^β, and (ii) the printed statement with an M-independent C is false.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The decisive issue is not [18, Lemma 17] but the statement of Theorem 1.1(i). As printed, (1.11) says that for each M>0 there is β(α,M)∈[0,1] and a constant C depending only on d,p,α such that the exponential integral with exponent α(p/d)^β is uniformly bounded over all u with |B|^{(p-d)/d}δ^{-p}I_{δ,p}(u,B)≤M. This cannot hold. Take B=B1, δ=1, and u_M(x)=A_M x_1 with A_M chosen so I_{1,p}(u_M,B1)=M. By (1.4) and (1.5), A_M≃(M/K_{d,p})^{1/p} is unbounded as M→∞. For every β∈[0,1], the exponent c=α(p/d)^β is at least α, and ∫_{B1}e^{cA_M|x_1|}dx ≃ (2/(cA_M))e^{cA_M}→∞. Hence no M-independent C can bound the supremum in (1.11); as stated, the main theorem is false. The proof in §2.2 independently shows the mismatch: the tail estimate (2.13) gives decay e^{-α(p/d)λ} on the normalized level λ=|u|/ℓ0, so layer-cake integration yields the rate α(p/d)/ℓ0, i.e. a multiplier β=ℓ0^{-1}. The final line claims this is the exponent (p/d)^β, which would require (p/d)^{ℓ0^{-1}}=(p/d)/ℓ0, generically false. Thus the central inequality as stated and the inequality actually proved are different.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies exponential integrability properties of measurable functions with finite non-local, non-convex energy I_{δ,p}(u,O), defined in (1.3) as the double integral of δ^p |x-y|^{-(d+p)} over pairs where |u(x)-u(y)|>δ. The main results are stated as Theorem 1.1 for p>d: for every M>0 and α>0 there is β=β(α,M)∈[0,1] such that the normalized exponential integral with exponent α(p/d)^β over a ball B is bounded by a constant C depending only on d,p,α whenever |B|^{(p-d)/d}δ^{-p}I_{δ,p}(u,B)≤M, and for a small M0 the better exponent α(p/d) is admissible. Theorem 1.2 states an analogous small-energy exponential integrability result in the critical case p=d. Proposition 1.7 gives examples showing that the growth rate p/d cannot be exceeded. The proofs proceed by a radial extension to B_{3/2}, John-Nirenberg estimates, a dyadic level-set iteration, and an external extension lemma from [18].","tokens_in":12244,"tokens_out":14487,"duration_ms":134719,"significance":"If the results were correct, they would fill a natural gap between the John-Nirenberg exponential integrability available for BMO functions and the boundedness or Moser-Trudinger behavior available for Sobolev functions, using only the non-local, non-convex energy I_{δ,p}. The optimality examples in Proposition 1.7 are a useful contribution and appear to be correctly constructed. However, the central theorem, Theorem 1.1(i), is false as stated, and the proof of the other main theorem, Theorem 1.2, is omitted. The paper's advertised main claim is therefore not established, and the current version cannot be accepted.","major_comments":[{"comment":"The statement of Theorem 1.1(i) is false as written. Fix B=B_1 and δ=1, and for M>0 take u_M(x)=A_M x_1 with A_M chosen so that I_{1,p}(u_M,B_1)=M. By (1.4)-(1.5), A_M is comparable to (M/K_{d,p})^{1/p} and hence A_M→∞ as M→∞. For every β∈[0,1] one has (p/d)^β≥1, so the exponent in (1.11) is at least α. A direct estimate gives ∫_{B_1} e^{α(p/d)^β A_M|x_1|} dx ≥ c e^{α A_M}/A_M →∞ as M→∞. Since the constant C in (1.11) is asserted to depend only on d, p, and α and not on M, the supremum in (1.11) is infinite. Thus the theorem cannot hold as stated; the constant would have to depend on M, which would be a substantially weaker statement.","section":"Section 1, Theorem 1.1(i), Eq. (1.11)"},{"comment":"The proof of part i) proves a different inequality from the one stated. The level-set estimate (2.13) reads |{|u|>λℓ0}| ≤ e^{-α(p/d)λ+2}|{|u|>ℓ0}|, which by the layer-cake formula yields exponential integrability of e^{b|u|} with b<α(p/d)/ℓ0, that is, with a multiplier ℓ0^{-1}. The sentence after (2.13), claiming that this implies ∫_{B_1} e^{α(p/d)|u|} dx ≤ C, is not justified unless ℓ0≤1, whereas ℓ0 defined in (2.12) is generally larger than 1 because it is at least c_1 M. Moreover, the final identification β(α,M)=ℓ0^{-1} does not match the exponent (p/d)^β in (1.11): the inequality actually obtained has exponent α(p/d)ℓ0^{-1}, not α(p/d)^{ℓ0^{-1}}. These mismatches confirm that Theorem 1.1(i) is not supported by the proof.","section":"Section 2.2, Eqs. (2.12)-(2.13)"},{"comment":"The proof of Theorem 1.2, which is the main theorem for the critical case p=d, is omitted; the text says 'The proof is similar to the one of part ii) of Theorem 1.1 and is omitted.' The same is true for Propositions 1.5 and 1.6, whose derivations are only sketched via local charts and [18, Lemma 17]. A main theorem cannot rest on an entirely omitted proof, especially when the p=d case requires a separate treatment and the analogous p>d proof already contains the mismatches described above.","section":"Section 2.4"},{"comment":"The passage from B_1 to B_{3/2} is load-bearing. The radial reflection defines ũ, and the two estimates (2.7) and (2.8) are quoted from [18, Lemma 17] without statement or proof. The subsequent John-Nirenberg argument is performed on B_{3/2}, and the conclusion is transferred back to B_1 only through these two estimates. Since the entire level-set iteration depends on this extension step, the manuscript should either state and prove the needed lemma in the present notation or provide a precise reference with all hypotheses. As written, the dependence on an unstated external result makes the proof incomplete.","section":"Section 2.2, Eqs. (2.7)-(2.8)"}],"minor_comments":[{"comment":"There is a typo in the statement: 'let B be a an open ball' should read 'let B be an open ball.'","section":"Section 1, Theorem 1.1"},{"comment":"The two assertions in Proposition 1.7 are both labeled 'i)'; the second should be labeled 'ii)'.","section":"Section 1, Proposition 1.7"},{"comment":"The statement of Lemma 2.2 begins 'Let g ∈ L1_loc(O)' but the integrand is written in terms of u; the lemma should use u consistently.","section":"Section 2.1, Lemma 2.2"},{"comment":"There are numerous typographical errors: 'Propposition' in the introduction, 'founded' for 'found', 'inequaliies' in the abstract, 'Soboleff' in reference [40], and misspelled names in reference [9]. A careful proofreading pass is needed.","section":"Introduction and references"}],"recommendation":"reject","confidential_remarks":"The paper contains a false main theorem, Theorem 1.1(i), as demonstrated by the linear-family counterexample in my report, and the proof of another main theorem, Theorem 1.2, is entirely omitted. These are not fixable by local revision without changing the claims of the paper. Should the authors later submit a corrected version focused on the small-M0 statement with complete proofs, including a full proof of the p=d case and a verification of the extension lemma, it could be worth re-evaluating. I do not see grounds for publication of the current version."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: Theorem 1.1(i) is not just unproven; it is false as printed. The proof gives a bound with exponent α(p/d)/ℓ0 (a multiplier), but the theorem claims α(p/d)^β with β∈[0,1]. Those are different, and the claimed form cannot hold uniformly in M.\n\nThe counterexample is simple. Take u_M(x)=A_M x_1 on B_1 with δ=1 and A_M chosen so that I_{1,p}(u_M,B_1)≈A_M^p≈M. Then for any β∈[0,1], the exponent c=α(p/d)^β is at least α (since p/d>1), and ∫_{B1} e^{c A_M |x1|} dx ~ e^{c A_M}/(c A_M) → ∞ as M→∞. So no C depending only on d,p,α can bound the supremum in (1.11) uniformly in M. The statement would need C to depend on M, or the exponent to be the multiplier form α(p/d)/ℓ0 that the proof actually yields.\n\nWhat is genuinely new and looks right: the small-energy full-exponent result (Theorem 1.1(ii)) and the optimality examples in Proposition 1.7, particularly the p=d construction with energy tending to 0 and divergent exponential integral for any γ>1. Those are careful constructions and they identify the correct critical exponent. If the authors restrict to small M0, the John-Nirenberg plus level-set iteration is a plausible route.\n\nThe soft spots beyond the false theorem: Theorem 1.2's proof is omitted entirely, as are the proofs of Propositions 1.5 and 1.6. That is too much to leave out, even for a short paper. The reliance on [18, Lemma 17] is acceptable as an external standard result; the self-citation is not circular.\n\nWho is this for? Researchers working on non-local characterizations of Sobolev spaces and Moser-Trudinger-type questions. The optimality examples are a useful contribution; the false interpolation statement is a useful caution.\n\nRecommendation: This deserves a serious referee, but the referee should quickly check Theorem 1.1(i) against linear functions on a ball. In current form the main theorem fails; a major revision is needed to either correct the exponent to the one actually proved or allow the constant to depend on M. With that fix, the small-energy result and the optimality examples could stand as a valid paper.","headline":"The main interpolation result is false as stated—a linear counterexample breaks it—but the small-energy bound and the optimality examples are real and should be preserved.","tokens_in":12808,"tokens_out":8153,"would_cite":false,"duration_ms":73856,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["26D10","26A54"],"pacs":[],"model":"deepseek-v4-flash","headline":"Sharp exponential integrability for functions with finite non-local, non-convex energy","keywords":["Moser-Trudinger inequality","exponential integrability","non-local energy","Sobolev inequality","Poincaré inequality","John-Nirenberg inequality","non-convex energy","BMO"],"falsifier":"Construct a sequence of functions u_n on B_1 with normalized energy |B|^((p−d)/d) δ^(−p) I_{δ,p}(u_n,B) → 0 but ∫_B exp(α(p/d)δ^(−1)|u_n − (u_n)_B|) → ∞; such a sequence would directly contradict Theorem 1.1(ii) and Theorem 1.2. Alternatively, compute I_{δ,p} of the radial reflection of oscillating step functions on B_1 and compare with (2.8): an unbounded ratio I_δ(ũ, B_{3/2})/I_δ(u, B_1) would break the extension lemma on which the proof rests.","tokens_in":11616,"feed_emoji":"📈","tokens_out":6177,"duration_ms":55710,"temperature":0.7,"pith_summary":"This paper asks how strongly a function must be integrable when its non-local 'jump energy' is finite—the energy that counts pairs of points whose values differ by more than δ, weighted by the singular kernel |x−y|^(−(d+p)). It proves that a small amount of this energy forces an exponential moment: for p>d, if the normalized energy is at most a small constant M0, then the average of exp(α(p/d)δ^(−1)|u−u_B|) over a ball B is bounded. For larger energy M, the same conclusion holds with a reduced exponent (p/d)^β, where β depends only on M and α. The rate p/d is optimal: for any larger γ, there are admissible functions with divergent exponential integrals. This closes the gap between the standard John-Nirenberg (BMO) exponential integrability and genuine boundedness for this class of non-convex, non-local energies.","feed_headline":"Jump energy forces sharp exponential integrability","feed_subtitle":"For p>d the critical rate is δ⁻¹(p/d); larger exponents fail even with vanishing energy.","key_machinery":"The central object is the non-local, non-convex jump energy I_{δ,p}(u,O). The load-bearing mechanism is a two-step transfer: first a radial reflection extends u from B_1 to B_{3/2} without increasing I_{δ,p} by more than a universal factor, using an extension lemma ([18, Lemma 17]) that also controls level sets, so that John-Nirenberg's inequality applies on the larger ball and gives a starting level-set bound; then two lemmas iterate that bound down to B_1—a geometric counting lemma (Lemma 2.1) that converts level-set volume into the p/d power of the energy, and a dyadic-decay lemma (Lemma 2.2) showing I_{2^k δ,p} ≤ $2^{{−k(p−1)}}$ I_{δ,p}. Iteration of these three ingredients yields the exponential tail estimate $e^{{−α(p/d)λ}}$ and hence the stated integrability.","core_discovery":"Stated on the paper's own terms: for p>d, the sharp exponential integrability threshold for the non-local energy I_{δ,p}(u,B) = ∫∫_{|u(x)−u(y)|>δ} δ^p / |x−y|^{d+p} dx dy is the rate α(p/d)δ^(−1) in the exponent when the normalized energy δ^(−p)|B|^((d−p)/d) I_{δ,p}(u,B) is sufficiently small (Theorem 1.1(ii)), and the sharp threshold is αδ^(−1) when p=d (Theorem 1.2). For an arbitrary bound M on the normalized energy, the same conclusion holds with the exponent scaled by a factor (p/d)^β, β∈[0,1] depending only on M and α. Proposition 1.7 proves optimality: for p>d and any γ>p/d, there exists a function with I_{δ,p}(u,B)≤M yet ∫_B $e^{{αγδ^(−1)|u−u_B|}}$=+∞; for p=d, there exists a sequence with energy tending to 0 whose exponential integrals diverge at any rate γ>1.","pith_inferences":["The dyadic-decay lemma (Lemma 2.2) is a generic mechanism: any energy functional with a comparable split-estimate under midpoint interpolation should inherit exponential integrability, so the result likely extends to kernels other than |x−y|^(−(d+p)) and to weighted or anisotropic variants.","The smallness threshold M0 is not made explicit; a natural next step would quantify M0 in terms of d, p, α, and identify whether the optimal constant in the exponential estimate matches the Moser-Trudinger constant in the limit p→d from above.","Proposition 1.7(i) constructs a function with finite energy but divergent exponential moment at supercritical rate; one might test numerically whether this construction also exhibits a 'truncation' phenomenon in the BMO sense, since the BMO norm stays bounded while the exponential integrability exponent exceeds the threshold.","The p=d sequence with energy tending to 0 suggests a possible non-local analogue of the Moser-Trudinger supremum: normalize the energy to a fixed level and look for the sharp constant multiplying δ^(−1); the paper's result guarantees finiteness but leaves the sharp constant open."],"forward_implications":["For p>d, a small normalized non-local energy implies a strong exponential moment: the average of e^{α(p/d)δ^(−1)|u−u_B|} is bounded by a constant independent of u.","For p=d, the same holds with rate δ^(−1), a Moser-Trudinger-type gain that John-Nirenberg alone cannot deliver because the BMO norm need not vanish as the energy vanishes.","For arbitrary energy level M, a reduced exponential rate (p/d)^β still holds, with β depending only on M and α; the proof makes no use of gradient or boundedness assumptions.","The rate p/d (and rate 1 when p=d) cannot be improved: any excess γ>p/d destroys the exponential integrability even among functions with arbitrarily small energy (p=d) or bounded energy (p>d).","On a smooth bounded domain the same exponential estimates hold for the function itself, up to a factor e^{α(p/d)^β δ^(−1)‖u‖_{L^1}}."],"supporting_citations":[{"why":"Supplies the extension lemma used in Section 2.2: radial reflection from B_1 to B_{3/2} preserves I_{δ,p} up to a universal factor and controls level sets (Lemma 17, inequality (2.8)).","marker":"[18]"},{"why":"John-Nirenberg's inequality provides the starting tail estimate on B_{3/2} that the iteration scheme exploits.","marker":"[24]"},{"why":"Gives the Poincaré-type estimate (1.7) that bounds the BMO norm of u by I_{δ,p}(u,Ω)+δ^d, needed to apply John-Nirenberg.","marker":"[7]"}],"fun_headline_variants":["Sharp exponential rate from nonlocal jump energy","Optimal exponential integrability via a single double integral","Threshold δ^{-1}(p/d) for nonlocal energy functions","Nonconvex energy gives best possible exponential bound","Exact Moser-Trudinger rate for finite jump energy"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof assumes the radial-reflection extension of a function from a ball to a slightly larger ball carries the non-local energy I_{δ,p} with a universal constant and controls the level sets of the extension by those of the original function; if this extension lemma fails uniformly, the John-Nirenberg step on the larger ball cannot be transferred back to the original ball.","fun_headline_variants_meta":{"raw":{"variants":["Sharp exponential rate from nonlocal jump energy","Optimal exponential integrability via a single double integral","Threshold δ^{-1}(p/d) for nonlocal energy functions","Nonconvex energy gives best possible exponential bound","Exact Moser-Trudinger rate for finite jump energy"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000205,"raw_usage":{"total_tokens":1387,"prompt_tokens":931,"completion_tokens":456,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":547,"completion_tokens_details":{"reasoning_tokens":378}},"tokens_in":547,"tokens_out":456,"duration_ms":4951,"temperature":1.0,"reasoning_tokens":378,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T12:54:30.285672+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Construct a sequence of functions u_n on B_1 with normalized energy |B|^((p−d)/d) δ^(−p) I_{δ,p}(u_n,B) → 0 but ∫_B exp(α(p/d)δ^(−1)|u_n − (u_n)_B|) → ∞; such a sequence would directly contradict Theorem 1.1(ii) and Theorem 1.2. Alternatively, compute I_{δ,p} of the radial reflection of oscillating step functions on B_1 and compare with (2.8): an unbounded ratio I_δ(ũ, B_{3/2})/I_δ(u, B_1) would break the extension lemma on which the proof rests.","supporting_citations":[{"cited_title":"Brezis and H.-M","cited_arxiv_id":null,"evidence_quote":"Supplies the extension lemma used in Section 2.2: radial reflection from B_1 to B_{3/2} preserves I_{δ,p} up to a universal factor and controls level sets (Lemma 17, inequality (2.8))."},{"cited_title":"John and L","cited_arxiv_id":null,"evidence_quote":"John-Nirenberg's inequality provides the starting tail estimate on B_{3/2} that the iteration scheme exploits."},{"cited_title":"Bourgain and H.-M","cited_arxiv_id":null,"evidence_quote":"Gives the Poincaré-type estimate (1.7) that bounds the BMO norm of u by I_{δ,p}(u,Ω)+δ^d, needed to apply John-Nirenberg."}],"review_version":1}