{"id":"1958253f-c822-4f3f-afda-abb1c30feca5","arxiv_id":"2507.20069","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The logarithmic convolution energy sup over E of the double integral of log(1/|x-y|) G(u(x))G(u(y)) is finite exactly when G is subcritical with exponent gamma >= 1, and maximizers exist in the strictly subcritical case.","lead":"This paper proves fractional Trudinger-Moser type inequalities with logarithmic convolution potentials in one dimension, including existence of extremal functions. It also proves radial symmetry of positive solutions to the associated Euler-Lagrange system.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Corollary 2.8's radial convolution identity is false even for even functions; the constant-function test gives LHS 6−4log2 vs RHS 1/2, so Proposition 3.2 does not actually bound Φ(u).","rationale":"The reader identified the radial-rearrangement reduction as the weak point, which is related to my concern, but the actual failure is more severe: the identity in Corollary 2.8 is false even for even functions. A single constant-function check disproves it. Since Proposition 3.2's proof of finiteness, Proposition 3.4's convergence argument, and Proposition 4.2's existence argument all invoke this identity, the submitted proof does not establish the main finiteness and attainment claims. The main theorems may still be true and a corrected proof may be possible, but the current manuscript contains a demonstrably false central identity rather than a merely missing justification. I therefore cannot regard the paper as conditionally acceptable without knowing whether the corrected one-dimensional potential estimates close the gap.","tokens_in":33775,"tokens_out":23280,"duration_ms":224965,"concrete_test":"Evaluate Corollary 2.8(ii) with v = 1_{(−1,1)} on both sides. The left side equals 6 − 4 log 2 by a direct one-dimensional computation; the right side equals 1/2. If this is confirmed, replace Lemma 2.7 by the correct one-dimensional potential formula w(r) = ∫_0^∞ [log(1/|r−s|) + log(1/(r+s))] v(s) ds, and rerun the estimates of Lemma 2.10 and Proposition 3.2 with the corresponding expression for Φ. The theorem is salvageable only if the corrected radial bound remains finite for all u ∈ E.","verdict_should_be":"UNVERDICTED","load_bearing_attack":"The load-bearing defect is in Corollary 2.8, the radial convolution identity used in Proposition 3.2 and again in Propositions 3.4 and 4.2. For even v on R, (2.14) claims that ∫∫ log(1/|x−y|) v(x)v(y) dxdy = 2∫_0^∞ v(r) log(1/r) ∫_0^r v(ρ) dρ dr. This identity is false. Take v = 1_{(−1,1)}. The left side is ∫_{−1}^1 ∫_{−1}^1 log(1/|x−y|) dx dy = 6 − 4 log 2 ≈ 3.23, while the right side is 2∫_0^1 r log(1/r) dr = 1/2. The source of the error is Lemma 2.7: in one dimension the logarithmic potential of an even radial density does not reduce to the two-dimensional Newton-shell formula. The correct expression contains an additional log(1/(r+s)) term and the proper even-integration factors. Consequently Proposition 3.2 bounds a different radial expression, not the functional Φ(u). The same incorrect identity underwrites Remark 2.9, the decomposition in Proposition 3.4, and the Ψ2 estimates in Proposition 4.2. This is not merely an unjustified reduction to radial functions: even after assuming u is even and decreasing, the central estimate is based on a numerically false equality.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript studies one-dimensional fractional Trudinger-Moser type inequalities with logarithmic convolution potentials. For a nonnegative even, increasing G satisfying the exponential bound (A), it claims in Theorem 1.6 that the supremum m(G) over the unit ball E of H^{1/2}_0(I) is finite when G has at most γ-critical growth with γ ≥ 1, that m(G) is attained with even decreasing maximizers when γ > 1 and (1.5) holds, and that m(G) = ∞ when G has at least γ-critical growth with γ < 1. Theorem 1.7 states the analogous dichotomy on the whole line for Ψ. The proofs use a radial representation of the logarithmic energy (Lemma 2.7 and Corollary 2.8), a one-dimensional Moser sequence, concentration-compactness arguments, and a moving-plane argument for the corresponding Euler-Lagrange system.","tokens_in":34056,"tokens_out":6767,"duration_ms":61304,"significance":"The problem is natural and the stated threshold γ = 1 is plausible; if established, the results would extend the Cingolani–Weth theory to the fractional one-dimensional setting, and the symmetry theorem for the nonlocal system is of independent interest. The paper contains substantial technical work, including detailed asymptotic estimates for Moser sequences and a careful decay analysis for the fractional Laplacian. However, the central radial convolution identity used throughout the paper is false, even for even functions. Since this identity underlies the bounds in Propositions 3.2, 3.4, and 4.2, the proofs of the main finiteness and attainment theorems do not go through. The manuscript therefore does not establish its headline claims in the submitted form.","major_comments":[{"comment":"Corollary 2.8(ii), Eq. (2.14), is false even for even functions. Taking v = 1_{(-1,1)}, the left side equals ∫_{-1}^1∫_{-1}^1 log(1/|x-y|) dx dy = 6 - 4 log 2 ≈ 3.23, whereas the right side equals 2∫_0^1 r log(1/r) dr = 1/2. The error originates in Lemma 2.7: in one dimension the logarithmic potential of an even density is not constant on the two-point set {|y| = r}, so the Newton-shell formula (2.12) does not hold. Since Proposition 3.2 uses (2.14) to reduce Φ(u) to the radial expression bounded in Lemma 2.10, Theorem 1.6(i) is not proved.","section":"Section 2, Eq. (2.14)"},{"comment":"The mixed identity (2.13) in Corollary 2.8(i), also based on Lemma 2.7, is invalid for the same reason. For v = w = 1_{(-1,1)}, the right-hand side of (2.13) gives 1/2 instead of the true value 6 - 4 log 2. This invalidates the decomposition in Proposition 3.4(ii), where the cross term IIn is identified as G(0)∫_0^1 v_n(r)(1-r) dr; the function f(r) = 1-r is not the correct one-dimensional weight. The proof of (3.10) therefore does not establish the claimed continuity at 0.","section":"Section 2, Lemma 2.7 and Eq. (2.13)"},{"comment":"The proof assumes that the sequence {u_n} is radial and non-increasing after asserting u_n → 0 in L^p: the sentence 'And since u_n are radial and non-increasing in the radial coordinate' introduces an hypothesis that is not derived. A general maximizing sequence in H^{1/2}_0(I) is not radial, and the paper does not justify a symmetric-decreasing rearrangement that preserves both the energy Φ and the weak convergence property used in the argument. The uniform convergence on [δ,1] used in (3.7) and (3.11) depends on this missing reduction. The same unjustified radiality appears in the proof of Proposition 4.2.","section":"Section 3, Proposition 3.4"},{"comment":"The decomposition Ψ(u_n) = 2(Ψ_2(u_n) - Ψ_1(u_n)) at the start of the proof invokes Corollary 2.8 and therefore inherits the false radial identity. Even if the sequence were radial, the representation is wrong; consequently the upper semicontinuity claim (4.8) is not supported. Since attainment of m∞(G) in Theorem 1.7(ii) rests on this claim, Theorem 1.7 is also not established by the submitted proof.","section":"Section 4, Proposition 4.2"}],"minor_comments":[{"comment":"The phrase 'even and radially decreasing' should be defined for functions on an interval; for I = (-1,1), radial decreasing should be stated explicitly as depending only on |x| and being non-increasing in |x|.","section":"Abstract and Theorem 1.6"},{"comment":"The notation I_{1/n}(0) is used without definition; presumably it denotes the interval (-1/n, 1/n), but this should be stated.","section":"Proposition 3.3"},{"comment":"The equality case in Lemma 2.3(iii) is asserted without proof or precise reference; please provide a citation or a short argument.","section":"Lemma 2.3"}],"recommendation":"reject","confidential_remarks":"The central identity is not a minor gap but a numerically false equality that invalidates the main proof. The statements may be true, but the submitted argument would need a fundamentally corrected one-dimensional potential representation and a re-derivation of the subsequent bounds. I therefore recommend rejection rather than minor revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThis paper tackles a natural 1D fractional analogue of the Cingolani–Weth logarithmic-potential Trudinger–Moser inequality, proving finiteness, extremal existence, and symmetry at the critical growth exponent gamma = 1. The topic is timely, and there is real work here: the Moser sequence asymptotics in Proposition 3.3 are intricate, and the moving-plane section contains some original devices, such as the explicit Fourier computation of (-Delta)^{1/2}(1/(1+x^2)) and the estimate log(1+|x|) <= sqrt(|x|). If the main inequality were established, Theorems 1.6 and 1.7 would be solid contributions.\n\nUnfortunately, the central estimates rest on an identity that is false even for even functions. Corollary 2.8(ii) claims that, for even v,\n\n  int int log(1/|x-y|) v(x)v(y) dxdy = 2 int_0^infty v(r) log(1/r) int_0^r v(rho) drho dr.\n\nTake v = 1_{(-1,1)}. The left side is 6 - 4 log 2 ≈ 3.23, while the right side equals 2 int_0^1 r log(1/r) dr = 1/2. The source of the error is Lemma 2.7: Newton's shell theorem applies to the logarithmic kernel in R^2, not in R. For a 1D even density, the correct reduction includes log(1/(r+s)) contributions, and the simple min(log|x|, log|y|) formula does not hold.\n\nThis is a load-bearing flaw, not a typo. Proposition 3.2 uses the false identity to bound Phi(u) by a much smaller radial integral, so the finiteness of Phi on E is not established. Proposition 3.4's decomposition and the cross-term computation with the weight 1-r depend on the same false formula, so the convergence Phi(u_n) -> Phi(0) is unproven. The same incorrect identity underpins the Psi estimates in Section 4 and the lower bound in Proposition 3.3. The true double integral might still behave as the authors need, but the written proofs do not show it.\n\nThe paper is not incoherent or deceptive; the authors followed the 2D framework of [9] and likely believed the shell theorem transferred. But it does not transfer, and the derived bounds do not bear on the stated functional. The results may be true and repairable with a correct 1D representation, but that requires substantial rewriting of the core proofs, not localized fixes.\n\nMy bottom line: this manuscript should not be accepted in its current form, and a referee would quickly find the counterexample. I would still send it to a serious referee once, because the question is natural and parts of the machinery are valuable, but the expected recommendation is rejection or major revision. I would not cite it as is.","headline":"The paper's main estimates rely on a one-dimensional radial convolution identity that is numerically false even for even functions, so Theorems 1.6 and 1.7 are currently unproven despite substantial technical work.","tokens_in":34585,"tokens_out":6455,"would_cite":false,"duration_ms":58452,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35J50","35Q40","31A10"],"pacs":[],"model":"deepseek-v4-flash","headline":"At decay exponent γ ≥ 1 the fractional Trudinger–Moser supremum with logarithmic convolution potential is finite, with extremals for γ > 1; at γ < 1 it is infinite.","keywords":["fractional Trudinger-Moser inequality","logarithmic convolution potential","extremal functions","critical growth exponent","fractional Sobolev space","radial symmetry","moving plane method","Moser sequence"],"falsifier":"Compute the supremum of $\\Phi$ on $I=(-1,1)$ for the exact $\\gamma=1$ nonlinearity $G(s)=e^{\\pi s^2}/(1+|s|)$; the theorem predicts $m(G)<\\infty$, so any sequence $u_n$ in the unit ball with $\\Phi(u_n)\\to\\infty$ would refute part (i) of Theorem 1.6.","tokens_in":33562,"feed_emoji":"🧮","tokens_out":9430,"duration_ms":80696,"temperature":0.7,"pith_summary":"This paper pins down the exact decay rate at which a one-dimensional fractional Trudinger–Moser inequality with a logarithmic convolution potential stays finite. The authors prove that on a bounded interval $I$, the supremum of $\\Phi(u) = \\int_I\\int_I \\log(1/|x-y|) G(u(x))G(u(y))\\,dx\\,dy$ over the fractional Dirichlet unit ball is finite whenever $G$ has at most $\\gamma$-critical growth with $\\gamma \\ge 1$, and is infinite when $G$ has at least $\\gamma$-critical growth with $\\gamma < 1$. Under the additional subcritical growth condition $G(s)-G(0)\\le c(e^{\\pi s^2}-1)$, they show for $\\gamma>1$ that the supremum is attained and every maximizer is, up to sign, even and radially decreasing. The same dichotomy is established in $H^{1/2}(\\mathbb{R})$ for the whole-space functional $\\Psi$. Since the borderline value $\\gamma=1$ separates finiteness from blow-up, the result completes the one-dimensional counterpart of the planar logarithmic-convolution inequalities and identifies the sharp threshold for existence of extremals.","feed_headline":"γ ≥ 1: sharp cutoff for a fractional energy","feed_subtitle":"A fractional Trudinger–Moser energy with logarithmic convolution is finite and attained for γ>1, and blows up for γ<1.","key_machinery":"The load-bearing identity is the radial representation $\\Phi(u) = 2\\int_0^1 G(u(r))\\log(1/r)\\int_0^r G(u(\\rho))\\,d\\rho\\,dr$, valid for even functions via Newton's theorem for the logarithmic kernel. Lemma 2.10 bounds the two-weight functional $\\Phi_{\\gamma_1,\\gamma_2}$ whenever $\\gamma_1+\\gamma_2\\ge 2$, converting the exponential growth of $G$ in the $\\gamma\\ge 1$ regime into an integrable singularity at $r=0$. For the blow-up case $\\gamma<1$, a Moser-type sequence with explicitly computed fractional norm asymptotics is shown to make $\\Phi(w_n)\\to\\infty$. Attainability rests on Proposition 3.4, where the additional growth condition (1.5) and a Chebyshev-truncation argument force $\\Phi(u_n)\\to\\Phi(0)$ whenever $u_n\\rightharpoonup 0$; the moving-plane part uses the decay bounds $u(x)=O(|x|^{-2})$ and $w(x)=O(\\log|x|)$ and a Hardy–Littlewood–Sobolev estimate with the logarithmic kernel.","core_discovery":"The central discovery is a sharp exponent threshold. For the functional $\\Phi(u) = \\int_I\\int_I \\log(1/|x-y|) G(u(x))G(u(y))\\,dx\\,dy$ on the unit ball of $W^{1/2,2}_0(I)$ determined by $\\|(-\\Delta)^{1/4}u\\|_{L^2}\\le 1$, the supremum $m(G)$ is finite exactly when $G$ has at most $\\gamma$-critical growth for some $\\gamma\\ge 1$, and infinite when $G$ has at least $\\gamma$-critical growth for some $\\gamma<1$. When $\\gamma>1$ and $G(s)-G(0)\\le c(e^{\\pi s^2}-1)$, the supremum is attained, and every maximizer is, up to sign, an even and radially decreasing function in the constraint set. The corresponding statement on $H^{1/2}(\\mathbb{R})$ holds up to translation. The positivity of maximizers and the moving-plane analysis further show that positive classical solutions of the associated Euler–Lagrange system decay like $|x|^{-2}$ and are radially symmetric.","pith_inferences":["The $\\gamma=1$ endpoint under only at-most-critical growth remains open for attainment; the present machinery yields maximizers only for $\\gamma>1$.","In higher dimensions or for other fractional exponents $s$, the critical decay exponent may shift because the radial weight multiplying the convolution changes with dimension.","A numerical test on non-even peaked functions could reveal whether the radial-rearrangement reduction is essential, by comparing $\\Phi(u)$ and $\\Phi(u^*)$ for unit-norm functions."],"forward_implications":["For any bounded interval and any $G$ of at most $\\gamma$-critical growth with $\\gamma\\ge 1$, the supremum $m(G)$ is finite; the inequality holds with a constant independent of $u$.","For $\\gamma>1$ together with condition (1.5), the supremum is attained, so the constrained maximization problem has a genuine solution rather than only a maximizing sequence.","Every maximizer in the attaining regime is, up to sign, even and radially decreasing; on the whole line it is so up to translation as well.","Positive classical solutions of the Euler–Lagrange system decay polynomially and are radially symmetric, complementing the one-dimensional fractional landscape.","The sharp threshold at $\\gamma=1$ means that adding a polynomial decay factor to the critical exponential growth is exactly the right amount to restore compactness."],"supporting_citations":[{"why":"Supplies the one-dimensional fractional Trudinger–Moser inequality (1.2) used to bound exponential integrals of $u^2$.","marker":"[21]"},{"why":"Gives the improved fractional Trudinger–Moser inequality (1.3) used to control $e^{\\pi u^2}$ uniformly on the constraint sets.","marker":"[5]"},{"why":"Provides the identification of the fractional Sobolev norm with the $L^2$ norm of $(-\\Delta)^{1/4}u$, defining the constraint set $E$.","marker":"[17]"},{"why":"Establishes the planar logarithmic-convolution inequality and the rearrangement lemmas (two-sided estimates under symmetric rearrangement) that the one-dimensional proof adapts.","marker":"[9]"},{"why":"Supplies the sharpened planar result and the extremal-function framework that the authors transfer to $\\gamma>1$.","marker":"[10]"},{"why":"Concentration-compactness principle used in Propositions 3.5 and 4.2 to split maximizing sequences into vanishing and non-vanishing cases.","marker":"[23]"},{"why":"Provides Newton's theorem and the Hardy–Littlewood–Sobolev inequality used in the radial representation and the moving-plane estimate.","marker":"[22]"},{"why":"Comparison-function decay argument for fractional Schrödinger equations that motivates the polynomial decay estimates in Lemmas 5.1 and 5.2.","marker":"[18]"}],"fun_headline_variants":["Sharp γ≥1 for Trudinger–Moser with log kernel","Fractional energy finite exactly at γ≥1","Log-convolution energy: cutoff at γ=1","Maximizers radial for γ>1 critical growth","Exponent threshold governs energy finiteness"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof assumes that every function in the constrained set may be replaced by its symmetric decreasing rearrangement, so that the radial representation and the uniform convergence on $[\\delta,1]$ apply to arbitrary maximizing sequences; if rearrangement changes the weak-limit behaviour, the attainment argument collapses.","fun_headline_variants_meta":{"raw":{"variants":["Sharp γ≥1 for Trudinger–Moser with log kernel","Fractional energy finite exactly at γ≥1","Log-convolution energy: cutoff at γ=1","Maximizers radial for γ>1 critical growth","Exponent threshold governs energy finiteness"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000213,"raw_usage":{"total_tokens":1434,"prompt_tokens":968,"completion_tokens":466,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":584,"completion_tokens_details":{"reasoning_tokens":390}},"tokens_in":584,"tokens_out":466,"duration_ms":4358,"temperature":1.0,"reasoning_tokens":390,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T17:51:48.010989+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the supremum of $\\Phi$ on $I=(-1,1)$ for the exact $\\gamma=1$ nonlinearity $G(s)=e^{\\pi s^2}/(1+|s|)$; the theorem predicts $m(G)<\\infty$, so any sequence $u_n$ in the unit ball with $\\Phi(u_n)\\to\\infty$ would refute part (i) of Theorem 1.6.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the one-dimensional fractional Trudinger–Moser inequality (1.2) used to bound exponential integrals of $u^2$."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the improved fractional Trudinger–Moser inequality (1.3) used to control $e^{\\pi u^2}$ uniformly on the constraint sets."},{"cited_title":"Di Nezza, G","cited_arxiv_id":null,"evidence_quote":"Provides the identification of the fractional Sobolev norm with the $L^2$ norm of $(-\\Delta)^{1/4}u$, defining the constraint set $E$."},{"cited_title":"Cingolani and T","cited_arxiv_id":null,"evidence_quote":"Establishes the planar logarithmic-convolution inequality and the rearrangement lemmas (two-sided estimates under symmetric rearrangement) that the one-dimensional proof adapts."},{"cited_title":"Cingolani, T","cited_arxiv_id":null,"evidence_quote":"Supplies the sharpened planar result and the extremal-function framework that the authors transfer to $\\gamma>1$."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Concentration-compactness principle used in Propositions 3.5 and 4.2 to split maximizing sequences into vanishing and non-vanishing cases."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides Newton's theorem and the Hardy–Littlewood–Sobolev inequality used in the radial representation and the moving-plane estimate."},{"cited_title":"Felmer, A","cited_arxiv_id":null,"evidence_quote":"Comparison-function decay argument for fractional Schrödinger equations that motivates the polynomial decay estimates in Lemmas 5.1 and 5.2."}],"review_version":1}