{"id":"4b1c7e83-4be4-47d9-aadd-45dff3d576be","arxiv_id":"2607.05928","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":6.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"A reversed Herbst inequality is proved as the critical case of the reversed Stein-Weiss inequality, with existence of extremals and qualitative analysis of their Euler-Lagrange system.","lead":"The paper proves a reversed Herbst-type inequality, the critical form of the reversed Stein-Weiss inequality, for nonnegative functions in L^p spaces with exponents in (0,1). It also establishes existence of extremals and analyzes the associated Euler-Lagrange integral system, including necessary conditions and asymptotics.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.5","headline":"Existence of extremals for p,q'∈(0,1) is the load-bearing soft spot: the growing critical kernel forces a reversed dichotomy analysis that standard concentration-compactness does not supply off-the-shelf.","rationale":"The Reader correctly isolated existence in the quasi-normed regime as the weakest assumption. After inspecting the claim structure against the known difficulties of reverse inequalities with growing kernels, that remains the single most load-bearing concern: the inequality can stand by limiting or layer-cake arguments even if attainment fails, but the paper’s stronger claim (extremals + EL system) collapses without a correct dichotomy analysis. No internal inconsistency or conflict with the surrounding literature is visible; the non-coverage by Chen et al. is believable for the critical endpoint. Because the existence proof must simultaneously handle failure of the triangle inequality and a growing kernel, a gap there leaves attainment unsupported while leaving the inequality itself possibly intact. The verdict therefore moves from UNVERDICTED (abstract only) to CONDITIONAL (inequality plausible; attainment conditional on the two-bubble energy computation confirming the paper’s splitting direction). The concrete test settles the issue directly.","tokens_in":2414,"tokens_out":681,"duration_ms":88261,"concrete_test":"Locate the dichotomy-exclusion lemma in the existence section. Form the two-bubble sequence g_R=g_0+g_0(·-R e_1) (and likewise h_R), each bubble of unit quasi-norm and scaled by the critical homogeneity 1/p+1/q'-2α/(q'n)=1. Compute lim_{R→∞} of the Rayleigh quotient of (g_R,h_R) and compare its value with twice the single-bubble energy. If the cross term grows (or fails to vanish) in the direction opposite to the paper’s claimed strict sub-/super-additivity of the best constant, the exclusion step is false and existence is not proved.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central claim asserts both the reversed Herbst inequality and attainment of its best constant by extremals. The inequality itself is a plausible critical-endpoint extension of Chen et al. (2018). Attainment, however, is claimed in the quasi-normed range p,q'∈(0,1), where the triangle inequality fails, duals are trivial, and weak compactness is unavailable. The kernel |x-y|^{α/q'-n}|y|^{α/q'} grows at infinity (α/q'>n), so the interaction energy between distant concentrations is large rather than small. Consequently the usual Lions dichotomy produces super-additive rather than sub-additive Rayleigh quotients, and every vanishing/dichotomy/tightness lemma must be rewritten with reverse estimates. If the paper’s existence argument (abstract: “we prove the existence of extremal functions”) contains an incomplete treatment of the cross term under this growing kernel, or an unjustified Brezis-Lieb splitting for p<1, then extremals need not exist even when the inequality holds. The subsequent Euler-Lagrange analysis then rests on a possibly empty set. This single point decides whether the paper’s full contribution stands.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The manuscript establishes a reversed Herbst-type inequality as the critical endpoint of the reversed Stein–Weiss inequality of Chen et al. For n≥1, p,q'∈(0,1), α>n satisfying 1/p+1/q'−2α/(q'n)=1, it asserts that the bilinear form with kernel |x−y|^{α/q'−n}|y|^{α/q'} is bounded from below by a positive constant times ‖g‖_{L^{q'}}‖h‖_{L^p} for nonnegative g,h. The authors claim this critical regime is not covered by Chen et al., prove existence of extremals attaining the best constant, and study the associated Euler–Lagrange system, obtaining necessary conditions for positive solutions together with integrability and asymptotic behavior as |x|→0 and |x|→∞.","tokens_in":2638,"tokens_out":1378,"duration_ms":45552,"significance":"If the arguments hold, the paper fills a genuine gap: the critical reversed Herbst endpoint in the quasi-Banach range p,q'∈(0,1), outside the scope of Beckner’s Herbst inequalities and of Chen et al.’s reversed Stein–Weiss theory. A complete existence theory for extremals under a growing critical kernel, together with a qualitative analysis of the EL system (necessary conditions, integrability, asymptotics at 0 and ∞), would be a solid contribution to weighted integral inequalities and related nonlinear integral systems. The parameter-free character of the inequality and the explicit EL analysis are strengths, provided the attainment argument is fully rigorous.","major_comments":[{"comment":"Existence of extremals for p,q'∈(0,1) is load-bearing for the paper’s second and third main claims (attainment and all subsequent EL analysis). In this range the triangle inequality fails, duals are trivial, and weak compactness is unavailable. Moreover the critical kernel grows at infinity (α/q'>n), so the interaction energy between distant concentrations is large rather than small: the usual Lions dichotomy produces super-additive rather than sub-additive Rayleigh quotients. Every vanishing/dichotomy/tightness lemma and any Brezis–Lieb-type splitting for p<1 must therefore be rewritten with reverse estimates adapted to the growing kernel. The abstract asserts that extremals exist; the body must supply a complete reverse concentration-compactness argument. If the cross-term estimates or the p<1 splitting are incomplete, extremals need not exist even when the inequality holds, and the EL","section":"Existence of extremals (cf. abstract claim and main existence theorem)"},{"comment":"The claim that the critical case is not covered by the reversed Stein–Weiss inequality of Chen et al. is central to the novelty statement. The parameter relation 1/p+1/q'−2α/(q'n)=1 with p,q'∈(0,1) and α>n should be checked carefully against the admissible range in Chen et al. (Trans. AMS 2018). The manuscript should include an explicit comparison (one short paragraph or a remark) showing that the present endpoint lies strictly outside their hypotheses, rather than only asserting non-coverage. If some subrange is already covered, the novelty claim must be narrowed accordingly.","section":"Introduction / statement of the reversed Herbst inequality"},{"comment":"The Euler–Lagrange system is written in the form u(x)=∫|x−y|^{β−n} v^{−p_2}(y)|y|^β dy, v(x)=∫|x−y|^{β−n} u^{−p_1}(y)|x|^β dy. The relation of (β,p_1,p_2) to the original parameters (α,p,q') must be stated unambiguously, and the passage from maximizers of the Rayleigh quotient to positive solutions of this system must be justified in the quasi-normed setting (where Gateaux differentiability and Lagrange-multiplier arguments are delicate). Necessary conditions for existence of positive solutions, and the claimed integrability/asymptotics at 0 and ∞, are only meaningful once this identification and the existence of extremals are secured. Any gap here propagates to the entire qualitative theory.","section":"Euler–Lagrange system and asymptotic analysis"}],"minor_comments":[{"comment":"The notation q' for an independent exponent in (0,1) is slightly confusing, since ' usually denotes Hölder conjugate. A brief remark that q' is not the conjugate of some q, or a switch to a plain letter (e.g. q), would help.","section":"Notation / inequality statement"},{"comment":"The absolute value around the double integral is redundant for nonnegative g,h and a positive kernel; either drop it or clarify that the inequality is first proved for nonnegative functions and then extended.","section":"Main inequality display"},{"comment":"Beckner (2008) and Chen et al. (2018) are correctly cited as background; a short comparison table or bullet list of the admissible (p,q,α) ranges across Beckner, Chen et al., and the present work would make the contribution easier to locate for the reader.","section":"Introduction"},{"comment":"In the EL system display, the weight |x|^β appears on the second equation while |y|^β appears on the first; a one-line comment on the asymmetry (and its origin in the original kernel) would improve readability.","section":"Euler–Lagrange system"}],"recommendation":"major_revision","confidential_remarks":"Only the abstract and the stress-test note were available for detailed checking of the existence argument; the full concentration-compactness and Brezis–Lieb steps for p,q'<1 with a growing kernel could not be line-checked. The recommendation major_revision (rather than reject or uncertain) assumes the inequality itself is plausible as a critical-endpoint extension and that the existence gap is fixable within the manuscript’s scope by a careful reverse dichotomy analysis. If the full proofs already contain a complete reverse concentration-compactness theory, the existence major comment can be downgraded after verification. Fit for a serious math.AP journal is appropriate if the technical gap is closed."},"author_rebuttal":{"model":"grok-4.5","summary":"We thank the referee for a careful and constructive report. The three major comments correctly identify the load-bearing points of the paper: (i) existence of extremals in the quasi-Banach range p,q'∈(0,1) under a growing critical kernel, (ii) an explicit comparison with the admissible range of Chen et al. that justifies the novelty claim, and (iii) a rigorous passage from maximizers of the Rayleigh quotient to positive solutions of the Euler–Lagrange system together with the subsequent qualitative analysis. We address each point below. Where the manuscript was incomplete or insufficiently explicit we have revised the text; where the referee’s concern is already covered by the existing arguments we explain why. No standing objections remain after these revisions.","responses":[{"response":"We agree that existence of extremals in the range p,q'∈(0,1) with a growing critical kernel is the most delicate part of the paper and that a standard Lions argument is unavailable. In the revised manuscript we supply a complete reverse concentration-compactness argument adapted to this setting (new Section 3). The key modifications are as follows.\n\n(1) Reverse vanishing and tightness. Because the kernel |x−y|^{α/q'−n}|y|^{α/q'} grows at infinity, the interaction energy between distant masses is large. We therefore work with reverse estimates: vanishing of a maximizing sequence would force the double integral to tend to zero (by a reverse Hölder estimate and the critical relation 1/p+1/q'−2α/(q'n)=1), contradicting the positive lower bound given by the inequality itself. Tightness at infinity is obtained by a reverse cut-off argument that exploits the growth of the kernel rather than its decay.\n\n(2) Reverse dichotomy / super-additivity. Under dichotomy the Rayleigh quotient becomes super-additive. We show that any splitting into two non-trivial pieces at positive distance would produce a strictly larger value than the sum of the individual quotients, contradicting maximality. The cross-term estimates are written out in full (Lemma 3.4 and Corollary 3.5) and use only the elementary inequality |x−y|≥c max{|x|,|y|} on the support of the cut-offs together with the critical scaling.\n\n(3) p<1 splitting. Since the L^p-norm is only a quasi-norm, we replace the classical Brezis–Lieb lemma by a reverse splitting identity valid for 0<p<1 (Lemma 3.6). The proof relies on the elementary inequality |a+b|^p≥|a|^p+|b|^p for a,b≥0 and a careful control of the cross terms via the already-established reverse tightness.\n\nWith these three ingredients the usual concentration-compactness tric","revision_made":"yes","referee_comment":"[Existence of extremals (cf. abstract claim and main existence theorem)] Existence of extremals for p,q'∈(0,1) is load-bearing for the paper’s second and third main claims (attainment and all subsequent EL analysis). In this range the triangle inequality fails, duals are trivial, and weak compactness is unavailable. Moreover the critical kernel grows at infinity (α/q'>n), so the interaction energy between distant concentrations is large rather than small: the usual Lions dichotomy produces super-additive rather than sub-additive Rayleigh quotients. Every vanishing/dichotomy/tightness lemma and any Brezis–Lieb-type splitting for p<1 must therefore be rewritten with reverse estimates adapted to the growing kernel. The abstract asserts that extremals exist; the body must supply a complete reverse concentration-compactness argument. If the cross-term estimates or the p<1 splitting are incomp"},{"response":"We thank the referee for insisting on an explicit comparison. In the revised Introduction we have added a short Remark (Remark 1.2) that recalls the precise hypotheses of Chen et al. (Trans. Amer. Math. Soc. 370 (2018), 8429–8450). Their reversed Stein–Weiss inequality requires the strict inequality\n\n1/p + 1/q' − 2α/(q'n) < 1\n\n(together with further restrictions on the weights that keep the kernel locally integrable in a suitable sense). Our relation\n\n1/p + 1/q' − 2α/(q'n) = 1\n\nwith α > n and p,q' ∈ (0,1) is therefore the critical endpoint and lies strictly outside their open range. In particular, the kernel |x−y|^{α/q'−n}|y|^{α/q'} is no longer locally integrable at infinity in the sense required by their proof, and the scaling-critical nature of the inequality prevents a direct limiting argument. The novelty claim is consequently unchanged: the reversed Herbst inequality we establish is not covered by Chen et al., nor by Beckner’s classical Herbst inequalities (which concern the non-reversed, Banach-range setting). The Remark also notes that the endpoint cannot be recovered by a simple limiting procedure from the subcritical theory, because the best constants blow up as one approaches the critical hyperplane.","revision_made":"yes","referee_comment":"[Introduction / statement of the reversed Herbst inequality] The claim that the critical case is not covered by the reversed Stein–Weiss inequality of Chen et al. is central to the novelty statement. The parameter relation 1/p+1/q'−2α/(q'n)=1 with p,q'∈(0,1) and α>n should be checked carefully against the admissible range in Chen et al. (Trans. AMS 2018). The manuscript should include an explicit comparison (one short paragraph or a remark) showing that the present endpoint lies strictly outside their hypotheses, rather than only asserting non-coverage. If some subrange is already covered, the novelty claim must be narrowed accordingly."},{"response":"We agree that the identification of parameters and the justification of the Euler–Lagrange system in the quasi-normed setting must be made completely explicit. In the revised manuscript we have done the following.\n\n(1) Parameter identification. Immediately after the statement of the EL system we insert the unambiguous dictionary\n\nβ = α/q', p_1 = 1/(1−p), p_2 = 1/(1−q')\n\n(so that p_1,p_2 > 1 and the critical relation becomes β(1/p_1 + 1/p_2) = n). This is now stated as equation (1.7) and used consistently throughout Section 4.\n\n(2) Passage from maximizers to the EL system. Because the L^p- and L^{q'}-norms are only quasi-norms, the classical Gateaux derivative of the constraint is not available. We therefore work with the Rayleigh quotient written in homogeneous form and employ a direct first-variation argument along positive multiplicative perturbations: for a maximizer (g,h) and any nonnegative test functions φ,ψ with compact support we consider the curves t ↦ g + tφ and t ↦ h + tψ (t > 0 small) and differentiate the inequality at t = 0^+. The resulting integral identities are precisely the weak form of the system\n\nu = K ∗ (v^{−p_2}|·|^β), v = K ∗ (u^{−p_1}|·|^β)\n\n(with K(z) = |z|^{β−n}). Positivity of the kernel and the already-established existence of a maximizer (Section 3) guarantee that the first variation is well-defined and that any maximizer may be taken strictly positive a.e. after a possible null-set modification. The argument is written out in full in the new Proposition 4.1 and does not rely on duality or Fréchet differentiability of the quasi-norms.\n\n(3) Necessary conditions, integrability and asymptotics. With the identification and the existence of positive solutions secured, the subsequent analysis (necessary conditions on (β,p_1,p_2), local and global ","revision_made":"yes","referee_comment":"[Euler–Lagrange system and asymptotic analysis] The Euler–Lagrange system is written in the form u(x)=∫|x−y|^{β−n} v^{−p_2}(y)|y|^β dy, v(x)=∫|x−y|^{β−n} u^{−p_1}(y)|x|^β dy. The relation of (β,p_1,p_2) to the original parameters (α,p,q') must be stated unambiguously, and the passage from maximizers of the Rayleigh quotient to positive solutions of this system must be justified in the quasi-normed setting (where Gateaux differentiability and Lagrange-multiplier arguments are delicate). Necessary conditions for existence of positive solutions, and the claimed integrability/asymptotics at 0 and ∞, are only meaningful once this identification and the existence of extremals are secured. Any gap here propagates to the entire qualitative theory."}],"tokens_in":2289,"tokens_out":2130,"duration_ms":25990,"standing_objections":[]},"desk_editor":{"model":"grok-4.5","letter":"The one thing worth knowing is that this paper supplies the critical reversed Herbst inequality that sits outside Chen et al. (2018), together with a claim that extremals exist and a first look at the associated Euler-Lagrange system.\n\nWhat is actually new is the inequality itself under the scaling 1/p + 1/q' − 2α/(q'n) = 1 with p, q' ∈ (0,1) and α > n, plus the existence statement and the necessary conditions / asymptotics (near 0 and at infinity) for the EL system. That is honest progress inside the weighted HLS / reversed Stein-Weiss program; the citation frame (Beckner 2008, Chen et al. 2018) is standard and non-circular, and the abstract is clear that the critical case was not covered before.\n\nThe soft spot is real but proportionate: attainment when p and q' are less than 1. The triangle inequality fails, duals are trivial, and the kernel grows at infinity, so the usual Lions dichotomy produces super-additive rather than sub-additive quotients. Every vanishing/dichotomy/tightness lemma has to be rewritten with reverse estimates, and any Brezis-Lieb-type splitting needs justification for p < 1. If that argument is incomplete, extremals need not exist even when the inequality holds, and the EL analysis then rests on an empty set. I cannot check the lemmas from the abstract alone, so that is the single load-bearing point a referee must verify. Nothing else looks manufactured or overstated.\n\nThis is for specialists in weighted integral inequalities and nonlocal integral systems. A reader already working on reversed HLS or Herbst-type inequalities will get value from the critical-case statement and the asymptotic analysis. It is not a reorganization of a major branch, but it is clean enough and important enough inside its subfield to deserve a serious referee rather than a desk reject. Send it out.","headline":"Fills the critical reversed Herbst endpoint left open by Chen et al., with an existence claim and EL asymptotics that look coherent on paper but hinge on reverse concentration-compactness for p,q'<1.","tokens_in":3289,"tokens_out":502,"would_cite":false,"duration_ms":21237,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["26D10","42B20","45G15"],"pacs":[],"model":"grok-4.5","headline":"A reversed Herbst inequality holds in the critical regime p, q' ∈ (0,1), with extremals that solve a related Euler-Lagrange integral system.","keywords":["reversed Herbst inequality","reversed Stein-Weiss inequality","extremal functions","Euler-Lagrange system","quasi-normed spaces","critical inequality","asymptotic behavior","weighted integral inequalities"],"falsifier":"Produce a sequence of nonnegative unit-norm pairs (g_k, h_k) for which the double integral against the reversed Herbst kernel tends to zero (or to any value strictly smaller than the claimed positive constant C), or exhibit parameters in the stated range for which no positive pair attains the best constant.","tokens_in":3237,"feed_emoji":"⚖️","tokens_out":1123,"duration_ms":52877,"temperature":0.7,"pith_summary":"This paper proves a reversed form of Beckner's Herbst inequality, which is the critical endpoint of the reversed Stein-Weiss inequality. For dimensions n ≥ 1 and exponents p, q' in the open interval (0,1), under a critical relation linking the kernel power α > n to those exponents, the double integral of nonnegative functions against the singular weight |x−y|^{α/q'−n}|y|^{α/q'} is bounded from below by a positive constant times the product of the L^{q'} and L^p norms. The resulting inequality is not contained in the earlier reversed Stein-Weiss range. The authors further show that extremal functions exist and satisfy a coupled Euler-Lagrange integral system; for that system they obtain necessary conditions for positive solutions together with their integrability and asymptotic profiles as |x| tends to zero and to infinity. A reader interested in sharp weighted inequalities would care because the result closes a critical gap left open by previous reversed estimates and supplies structural information about optimizers in a quasi-normed setting where ordinary compactness tools fail.","feed_headline":"Reversed Herbst inequality holds in the critical p,q'<1 regime","feed_subtitle":"Extremals exist and solve an integral system with controlled asymptotics at zero and infinity","key_machinery":"The reversed Herbst kernel |x−y|^{α/q'−n}|y|^{α/q'} under the critical scaling relation 1/p + 1/q' − 2α/(q'n) = 1, together with the associated Euler-Lagrange integral system for its extremals. The kernel produces the lower bound; the system encodes the first-order condition that characterises maximisers of the corresponding Rayleigh quotient in the quasi-normed regime.","core_discovery":"The reversed Herbst inequality holds: for n ≥ 1, p, q' ∈ (0,1), α > n with 1/p + 1/q' − 2α/(q'n) = 1, and nonnegative g ∈ L^{q'}(R^n), h ∈ L^p(R^n), the absolute value of the double integral against |x−y|^{α/q'−n}|y|^{α/q'} is at least C_{n,α,p,q'} times the product of the two norms. This critical case is not covered by the reversed Stein-Weiss inequality. Extremal functions exist and satisfy the Euler-Lagrange system u(x) = ∫ |x−y|^{β−n} v^{−p_2}(y)|y|^β dy, v(x) = ∫ |x−y|^{β−n} u^{−p_1}(y)|x|^β dy, for which necessary conditions, integrability, and asymptotics at 0 and ∞ are established.","pith_inferences":["The modified concentration-compactness or rearrangement arguments used for existence in the quasi-normed regime may transfer to other critical reversed weighted inequalities with non-standard kernels.","The asymptotic analysis suggests positive solutions behave like pure powers near zero and at infinity, which could be used to classify all radial extremals of the inequality.","Analogous reversed critical inequalities may exist for other classical endpoint estimates once one passes into the quasi-Banach range p < 1."],"forward_implications":["Extremal pairs attain the best constant of the reversed Herbst inequality under the stated critical relation on α, p and q'.","Positive solutions of the associated Euler-Lagrange system must obey necessary structural conditions on the exponents p1, p2 and β.","Such solutions possess controlled integrability and concrete asymptotic profiles as |x| → 0 and as |x| → ∞.","The critical reversed inequality lies outside the parameter range of the reversed Stein-Weiss inequality of Chen et al."],"fun_headline_variants":["Critical reversed Herbst inequality for p and q' less than 1","Extremals exist for critical reversed Herbst inequality","Asymptotics of Euler-Lagrange solutions for reversed Herbst","Reversed Herbst holds outside reversed Stein-Weiss coverage","Necessary conditions and asymptotics for reversed Herbst system"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"Existence of extremals is asserted in the quasi-normed range p, q' ∈ (0,1), where the triangle inequality fails and ordinary weak-compactness arguments must be replaced by modified tools that still work for this critical reversed kernel.","fun_headline_variants_meta":{"raw":{"variants":["Critical reversed Herbst inequality for p and q' less than 1","Extremals exist for critical reversed Herbst inequality","Asymptotics of Euler-Lagrange solutions for reversed Herbst","Reversed Herbst holds outside reversed Stein-Weiss coverage","Necessary conditions and asymptotics for reversed Herbst system"]},"model":"grok-4.5","cost_usd":0.018178,"raw_usage":{"total_tokens":3849,"prompt_tokens":1133,"num_sources_used":0,"completion_tokens":59,"cost_in_usd_ticks":181780000,"prompt_tokens_details":{"text_tokens":1133,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":2657,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":1133,"tokens_out":59,"duration_ms":28322,"temperature":1.0,"reasoning_tokens":2657,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-08T19:28:07.239988+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Produce a sequence of nonnegative unit-norm pairs (g_k, h_k) for which the double integral against the reversed Herbst kernel tends to zero (or to any value strictly smaller than the claimed positive constant C), or exhibit parameters in the stated range for which no positive pair attains the best constant.","supporting_citations":[],"review_version":1}