{"id":"be7aa666-c924-43c8-abbb-0898e0b6710d","arxiv_id":"2505.10287","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"Convex viscosity solutions of the Hessian quotient equation are shown to be C^{3,beta} under sharp C^{1,alpha} or W^{2,p} assumptions, via a new Pogorelov type interior C^2 estimate.","lead":"This paper proves a boundary-version second derivative estimate for solutions of a family of Hessian quotient equations, then uses it to establish regularity of convex viscosity solutions under optimal assumptions. A generalist should care because it extends a classical Monge-Ampere tool to a broader class of fully nonlinear equations.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 5.1's two-sided bound 1/C <= G_ii <= C is not justified by Lemma 4.1; the lower bound is equivalent to an upper bound on D^2u, the theorem's target, unless a missing argument using the transformed equation is supplied.","rationale":"The reader correctly identified Lemma 5.1 as the weakest step: the sentence 'By Lemma 4.1, |D^2w| is bounded' is misleading, because Lemma 4.1 bounds only the largest eigenvalue of D^2w, and the lower bound on G_ii is what would make the operator uniformly elliptic. This is load-bearing because Lemma 5.2's local maximum principle and the integration by parts in the proof of Theorem 1.2 require both bounds on G_ii. I do not, however, think this is necessarily fatal. The transformed equation e_{n-k}(mu) = f*, together with mu_1 <= C and the lower bound on f*, may imply a lower bound on each e_{n-k-1}(mu|i), and hence on G_ii. That argument is short but absent, and the paper also omits the proof of Lemma 3.3 and relies on a private-communication reference for Lemma 3.2, though a public preprint is cited alongside. Because the identified gap is localized and plausibly repairable, I would not uphold a flat rejection; instead the paper should be accepted only conditionally on supplying the missing derivation of the lower bound in (5.1) and on clarifying the status of the external concavity lemma. The central idea of the proof appears coherent, and several other steps that initially look suspicious can be justified from already-established facts, so my concern is on the completeness of the written argument rather than on an irreparable mathematical error.","tokens_in":25824,"tokens_out":41776,"duration_ms":367557,"concrete_test":"Write out the missing argument for (5.1): let mu_1 >= ... >= mu_n be the eigenvalues of D^2w in (Sigma_1)^*. From Lemma 4.1, mu_1 <= C; from (4.2), e_{n-k}(mu) = f*(y) with 1/M <= f* <= 1/m. Prove e_{n-k-1}(mu|i) >= c for each i using the elementary inequality e_m(mu|i) <= C(n) mu_1 e_{m-1}(mu|i) and the identity e_m(mu) = mu_i e_{m-1}(mu|i) + e_m(mu|i). If this succeeds, insert the missing step in Lemma 5.1 and the proof proceeds; if the derivation requires assuming D^2u <= C, the argument is circular and Theorem 1.2 fails as written. Separately, verify that Lemma 3.2 is indeed proved as Proposition 2.5 of the cited public preprint arXiv:2501.03386, since the paper's own reference is only a private communication.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"Lemma 5.1 asserts that in (Sigma_1)^*, 1/C <= G_ii <= C, and says this follows from Lemma 4.1. What Lemma 4.1 actually proves is (-v*)^beta * mu_1 <= C, where mu_1 is the largest eigenvalue of D^2w. In (Sigma_1)^*, -v* = 2 - u* >= 1, so mu_1 <= C. This yields the upper bound G_ii <= C, but it does not yield the lower bound G_ii >= 1/C. That lower bound is equivalent to controlling every eigenvalue of D^2w from below, which is equivalent to controlling D^2u from above, precisely the quantity Theorem 1.2 must prove. The paper supplies no argument bridging mu_1 <= C to full ellipticity of the operator sum G_ii d_ii. As written, the local maximum principle in Lemma 5.2 and the subsequent integration by parts rest on an unproved, and apparently circular, premise. The gap is potentially repairable: since w solves e_{n-k}(mu) = f* with f* bounded below, and since e_m(mu|i) <= C mu_1 e_{m-1}(mu|i), one may be able to derive e_{n-k-1}(mu|i) >= c from the equation. But that derivation is absent, and without it the central a priori estimate is not supported as stated.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper claims a Pogorelov-type interior C^2 estimate for convex solutions of the Hessian quotient equation sigma_n/sigma_k(D^2u)=f on sections with homogeneous boundary data (Theorem 1.2), and derives C^{3,beta} regularity for convex viscosity solutions under sharp C^{1,alpha} or W^{2,p} assumptions (Theorem 1.4). The strategy is to use the Legendre transform: Lemma 4.1 gives a lower bound on the smallest eigenvalue of D^2u; Lemma 3.3 (Jacobi inequality) yields an inequality for b = ln lambda_1; Lemma 5.1 claims that in Legendre coordinates b is a subsolution of a uniformly elliptic operator; Lemma 5.2 applies the local maximum principle; and Section 6 bounds the integral of b by integration by parts. The regularity applications follow the work of Urbas and of Collins and Mooney.","tokens_in":26146,"tokens_out":10582,"duration_ms":99569,"significance":"If Theorem 1.2 were established, the paper would be a significant contribution: Pogorelov-type estimates for general Hessian quotient equations have been missing, and the sharp regularity thresholds in Theorem 1.4 would match the example in [23]. The Legendre-transform approach to lower Hessian bounds (Lemma 4.1) and the induction lemma for integration by parts (Lemma 6.1) are interesting and potentially useful. The paper is clearly organized and carefully states the dependencies of constants. However, the central Lemma 5.1 contains an unjustified uniform ellipticity assertion, and the proof relies on a concavity inequality that is only cited to a private communication. As written, the main estimate is not supported.","major_comments":[{"comment":"The two-sided bound 1/C <= G_ii <= C is not a consequence of Lemma 4.1. Lemma 4.1 gives (-u*)^beta * mu_1 <= C, and in (Sigma_1)* we have -u* >= 1, so mu_1 <= C. Since D^2w is positive semidefinite, this yields only the upper bound G_ii <= C. The lower bound G_ii >= 1/C is equivalent to a lower bound on every eigenvalue of D^2w, hence to an upper bound on D^2u, which is precisely the conclusion of Theorem 1.2 that the proof is meant to establish. The chain of inequalities immediately before (5.1) needs -C * sum_i 1/G_ii - C >= -C', which is exactly the missing lower bound. Consequently the uniform ellipticity used in Lemma 5.2 (via the local maximum principle) and in the integration by parts in Section 6 rests on an unproved and effectively circular premise. No argument bridging mu_1 <= C to full control of the coefficients of sum_i G_ii d_ii is supplied.","section":"Lemma 5.1, display (5.1)"},{"comment":"The concavity inequality for the Hessian quotient operator is stated without proof and is referenced only to [14], which is listed in the bibliography as a private communication. Lemma 3.3 and therefore the Jacobi inequality on which the entire proof of Theorem 1.2 depends, rely on this unpublished result. The manuscript should either include a complete proof of Lemma 3.2 or cite a publicly available source that contains its proof (reference [34], if it contains the statement, is not cited in Lemma 3.2). As written, a central ingredient of the main theorem is not verifiable from the manuscript or from a citable reference.","section":"Lemma 3.2"}],"minor_comments":[{"comment":"The displayed conclusion appears as '(-u)^beta * lambda_min <= C', but the proof shows that the quantity to be bounded is (-u*)^beta * mu_1, which equals (-u)^beta / lambda_min. The denominator is missing in the displayed statement.","section":"Lemma 4.1, statement"},{"comment":"The sentence 'By (5.1), sum_i G_ii >= C' uses the same letter C for the constant in (5.2) and for the lower bound; the argument requires a quantitatively chosen constant, for instance writing sum_i G_ii (b* + Lambda |y|^2)_ii >= 0 with Lambda chosen after the constants in (5.1)-(5.2). As written this is a notational shortcut rather than a mathematical error.","section":"Lemma 5.2, proof"},{"comment":"The proof is omitted with the statement 'the rest are exactly the same' as the proof of Lemma 4.1 in [23]. Since [23] is a preprint by the first author, the manuscript would be more self-contained if the argument were included or if the precise dependence on results of [23] were spelled out.","section":"Lemma 3.3, proof"}],"recommendation":"reject","confidential_remarks":"The main issue is not that the result is obviously false, but that the proof as written contains a gap at the central point that appears to assume the conclusion of Theorem 1.2 in the form of the lower bound on G_ii in Lemma 5.1. This is not a local omission that the authors can simply fill with a 'standard argument'; it is equivalent to the desired upper bound on D^2u. The paper could become publishable if the authors prove the lower bound on G_ii from the transformed equation, but that would be a substantial new ingredient. I also note that reliance on a 'private communication' for a key concavity inequality is problematic for a formal submission."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The one thing you should know: this paper probably has the right main theorem, but the proof as written has a real gap at Lemma 5.1. The two-sided bound 1/C ≤ G_ii ≤ C is stated as following from Lemma 4.1, and it does not follow. Without the lower bound, the local maximum principle in Lemma 5.2 and the integration by parts in Section 6 have no uniform ellipticity to stand on.\n\nWhat is genuinely new: the Pogorelov-type interior C^2 estimate for σ_n/σ_k with general k, and the sharp C^{1,α} / W^{2,p} regularity thresholds in Theorem 1.4. The strategy is adapted from Lu's earlier work and from Shankar-Yuan, but the extension to the full quotient range is absent from the literature. The paper uses the Guan-Sroka concavity inequality cleanly and explains the Legendre transform setup well. The exposition is honest about what is new and what is carried over.\n\nThe soft spot is structural. Lemma 4.1 bounds (-v*)^β μ_1 ≤ C, where μ_1 is the largest eigenvalue of D^2w; in (Σ_1)^* that gives μ_1 ≤ C, i.e. an upper bound on the largest eigenvalue of D^2w. Lemma 5.1 needs G_ii ≥ 1/C, which means every eigenvalue of D^2w is bounded below. That is equivalent to an upper bound on D^2u — exactly the quantity Theorem 1.2 must prove. The paper simply asserts (5.1) follows from Lemma 4.1. It does not. The gap may be repairable — the transformed equation σ_{n-k}(D^2w) = f* might force the missing lower bound through elementary symmetric identities — but that argument is not in the paper.\n\nTwo smaller issues: the proof of Lemma 3.3 is omitted entirely (\"same as Lemma 4.1 in [23]\"), and Lemma 3.2 is attributed to a private communication, though an arXiv version [34] appears to cover it. These are minor compared to the Lemma 5.1 gap.\n\nBottom line: this paper deserves a serious referee. The result is important enough that a fix would make it a solid contribution, and the gap is concrete enough that a referee can check whether the missing argument exists. But as it stands, the central a priori estimate is not supported. I would not cite it as a theorem yet.\n\nRecommendation: send to peer review with a referee instruction to focus on Lemma 5.1 first.","headline":"A promising attack on a real regularity problem, but the submitted proof has a load-bearing gap at Lemma 5.1 that the authors do not fix; it deserves peer review, not desk rejection, and not citation yet.","tokens_in":26687,"tokens_out":2837,"would_cite":false,"duration_ms":25444,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35J60","35J15","35J96"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves a Pogorelov-type interior $C^2$ estimate for the Hessian quotient equation $\\sigma_n/\\sigma_k(D^2u)=f$ and uses it to upgrade convex viscosity solutions with sharp Hölder or Sobolev regularity to $C^{3,\\beta}$.","keywords":["Hessian quotient equation","Pogorelov-type estimate","interior C^2 estimate","Legendre transform","viscosity solution","strict convexity","sharp regularity exponents","elementary symmetric functions"],"falsifier":"Take the singular convex solution from the paper's reference [23], where $n-k\\ge3$, and compute the transformed coefficients $G_{ii}=F^{-2}F_{ii}u_{ii}^2$ on a sequence of points where the largest eigenvalue of $D^2u$ diverges; if $G_{ii}$ fails to remain bounded, the uniform ellipticity asserted in (5.1) is false and the mean value inequality in Lemma 5.2 has no basis.","tokens_in":25607,"feed_emoji":"📐","tokens_out":10854,"duration_ms":101217,"temperature":0.7,"pith_summary":"The paper establishes a Pogorelov-type interior $C^2$ estimate for the Hessian quotient equation $\\frac{\\sigma_n}{\\sigma_k}(D^2u)=f$: for a convex solution that vanishes with zero gradient at the origin and equals $2$ on the boundary of its section $\\Sigma_2$, the second derivatives are bounded in a fixed ball $B_\\tau\\subset\\Sigma_1$, with constants depending only on $n,k$, the diameter, the $C^{0,1}$ norm, and the data of $f$. Such an estimate was missing for quotient equations, and it is what upgrades weak convex solutions to classical ones. As an application, the paper proves that convex viscosity solutions with $u\\in C^{1,\\alpha}$ for $\\alpha>1-\\frac{2}{n-k}$, or with $u\\in W^{2,p}$ for $p\\ge\\frac{(n-1)(n-k)}2$, are $C^{3,\\beta}$ in the interior whenever $k\\le n-3$. Both thresholds are sharp, matching the singular example cited as [23] in the paper. A broader consequence is a new route to lower bounds on Hessian eigenvalues, which are usually harder to obtain than upper bounds.","feed_headline":"Hessian quotient viscosity solutions regular under sharp exponents","feed_subtitle":"The method yields C^{3,β} regularity, with Hölder and Sobolev thresholds shown optimal by a known example.","key_machinery":"The load-bearing mechanism is the Legendre transform $w(y)=\\sup_x(x\\cdot y-u(x))$, which converts the quotient equation into the Hessian equation $\\sigma_k(D^2w)=f^{-1}$ on the dual section. The log of the largest Hessian eigenvalue, $b=\\ln\\lambda_1$, is shown to satisfy a Jacobi inequality $\\sum_i F_{ii}b_{ii}\\ge c\\sum_i F_{ii}b_i^2-C$ using a concavity inequality for Hessian quotient operators; after the transform this becomes a subsolution inequality $\\sum_i G_{ii}b^*_{ii}\\ge -C$ for a uniformly elliptic operator whose coefficients are $G_{ii}=F^{-2}F_{ii}u_{ii}^2$. The new strict-positivity estimate of Lemma 4.1, $(-u)^\\beta/\\lambda_{\\min}\\le C$, obtained by a Pogorelov-type argument on the transformed equation, is what supplies the uniform ellipticity and the lower eigenvalue bound used in the mean value inequality and in the final integration by parts with $H_{ij}=\\sigma_kF_{ij}$.","core_discovery":"On its own terms, the central claim is Theorem 1.2: for $n\\ge2$, $1\\le k<n$, a positive $f\\in C^{1,1}(\\Sigma_2\\times\\mathbb{R})$, and a convex $C^4$ solution $u$ of $\\frac{\\sigma_n}{\\sigma_k}(D^2u)=f(x,u)$ with $u(0)=0$, $Du(0)=0$, and $u=2$ on $\\partial\\Sigma_2$, there are constants $\\tau>0$ and $C$, depending only on the stated data, such that $B_\\tau\\subseteq\\Sigma_1$ and $\\max_{B_\\tau}|D^2u|\\le C$. From this a priori bound the paper derives Theorem 1.4: under either of the two sharp regularity assumptions, any convex viscosity solution is in $C^{3,\\beta}(\\Omega)$ for every $\\beta<1$, with quantitative estimates on compact subdomains. The proof is designed so that the same estimate also forces the Hessian of $u$ to be uniformly positive in an interior ball, a strict-convexity property that the regularity argument needs.","pith_inferences":["The same Legendre-transform and strict-positivity strategy may apply to general Hessian quotient equations $\\sigma_k/\\sigma_l=f$ with $k-l\\le2$, where pure interior $C^2$ estimates remain open; the equation treated here is the special case with $l=k$ and $k=n$.","A reader wanting to test the method should check whether the strict positivity estimate can be pushed to the borderline $k=n-2$; if it can, the regularity theorem would extend to that case, which is currently excluded.","The critical Sobolev exponent $p=\\frac{(n-1)(n-k)}2$ suggests an interpolation phenomenon: at exactly this exponent the slicing argument used in the borderline case produces the needed lower bound on the section, while below it the singular example takes over; identifying the analogue for other quotient equations would clarify the boundary between regular and singular regimes.","If the regularity theorem is combined with standard localization arguments, it may extend to equations with $f$ depending on $x$ and $u$ or to nonconstant boundary data, since the a priori estimate has such data in its constants."],"forward_implications":["For $k\\le n-3$, convex viscosity solutions with $u\\in C^{1,\\alpha}$ and $\\alpha>1-\\frac{2}{n-k}$ are $C^{3,\\beta}$ with interior estimates depending on the natural data.","For the same range, convex viscosity solutions with $u\\in W^{2,p}$ and $p\\ge\\frac{(n-1)(n-k)}2$ are also $C^{3,\\beta}$, including the critical borderline case $p=\\frac{(n-1)(n-k)}2$.","The exponents cannot be improved: the singular example cited as [23] shows that the borderline cases are genuinely singular, so the hypotheses of Theorem 1.4 are optimal.","The Pogorelov-type estimate itself holds for all $n\\ge2$ and all $1\\le k<n$ under section boundary conditions, extending earlier results that only covered $n-k\\le2$.","The proof gives a new way to obtain uniform lower bounds on the smallest Hessian eigenvalue, a quantity that is normally much harder to control than upper bounds."],"supporting_citations":[{"why":"Supplies the concavity inequality for positive Hessian quotient operators used in the Jacobi inequality for $b=\\ln\\lambda_1$.","marker":"[14]"},{"why":"Provides the prior interior $C^2$ estimate for $n-k\\le2$ via Legendre transform and also the singular example that makes the exponents sharp.","marker":"[23]"},{"why":"Introduces the $b=\\ln\\lambda_1$ and Legendre-transform method for Hessian estimates that the paper adapts to quotient equations.","marker":"[30]"},{"why":"Regularity theorem for generalized Monge-Ampere solutions whose strict-convexity and approximation argument is followed.","marker":"[37]"},{"why":"Handles the critical borderline case $p=\\frac{(n-1)(n-k)}2$ through slicing and Sobolev embedding.","marker":"[10]"},{"why":"Local maximum principle used to turn the subsolution inequality into a mean value inequality.","marker":"[4]"},{"why":"Equivalence of viscosity and distribution subsolutions needed for the integration by parts step.","marker":"[16]"},{"why":"Existence of smooth convex approximating solutions to the Dirichlet problem for Hessian equations.","marker":"[35]"},{"why":"Earlier non-sharp $W^{2,p}$ regularity result that the paper sharpens.","marker":"[2]"}],"fun_headline_variants":["Optimal thresholds for regularity in Hessian quotient","Sharp interior regularity for Hessian quotient solutions","Pogorelov estimate yields C^3,β regularity for Hessian quotients","Interior C^2 estimate settles regularity for Hessian quotients","Optimal exponents for Hessian quotient regularity"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the lower-bound estimate on the Hessian of $u$ in Lemma 4.1 also gives a uniform upper bound on the transformed coefficients $G_{ii}$; that upper bound is equivalent to the uniform upper bound on $D^2u$ that Theorem 1.2 is trying to prove, and the paper does not supply an independent argument for it.","fun_headline_variants_meta":{"raw":{"variants":["Optimal thresholds for regularity in Hessian quotient","Sharp interior regularity for Hessian quotient solutions","Pogorelov estimate yields C^3,β regularity for Hessian quotients","Interior C^2 estimate settles regularity for Hessian quotients","Optimal exponents for Hessian quotient regularity"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001343,"raw_usage":{"total_tokens":5433,"prompt_tokens":900,"completion_tokens":4533,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":516,"completion_tokens_details":{"reasoning_tokens":4452}},"tokens_in":516,"tokens_out":4533,"duration_ms":30401,"temperature":1.0,"reasoning_tokens":4452,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T21:13:39.134849+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take the singular convex solution from the paper's reference [23], where $n-k\\ge3$, and compute the transformed coefficients $G_{ii}=F^{-2}F_{ii}u_{ii}^2$ on a sequence of points where the largest eigenvalue of $D^2u$ diverges; if $G_{ii}$ fails to remain bounded, the uniform ellipticity asserted in (5.1) is false and the mean value inequality in Lemma 5.2 has no basis.","supporting_citations":[{"cited_title":"Guan and M","cited_arxiv_id":null,"evidence_quote":"Supplies the concavity inequality for positive Hessian quotient operators used in the Jacobi inequality for $b=\\ln\\lambda_1$."},{"cited_title":"Shankar and Y","cited_arxiv_id":null,"evidence_quote":"Introduces the $b=\\ln\\lambda_1$ and Legendre-transform method for Hessian estimates that the paper adapts to quotient equations."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Regularity theorem for generalized Monge-Ampere solutions whose strict-convexity and approximation argument is followed."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Handles the critical borderline case $p=\\frac{(n-1)(n-k)}2$ through slicing and Sobolev embedding."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Local maximum principle used to turn the subsolution inequality into a mean value inequality."},{"cited_title":"Ishii, On the equivalence of two notions of weak solutions, viscosity solutions and distribution solutions, Funkcial","cited_arxiv_id":null,"evidence_quote":"Equivalence of viscosity and distribution subsolutions needed for the integration by parts step."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Existence of smooth convex approximating solutions to the Dirichlet problem for Hessian equations."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Earlier non-sharp $W^{2,p}$ regularity result that the paper sharpens."}],"review_version":1}