{"id":"b8cc1397-0cf1-4f91-89d4-eb8b54deaf89","arxiv_id":"2607.21144","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For 1≤m<n, direct Skoda-type and strong-openness statements fail, the ball-maximum limit equals the m-Lelong number, and a new capacity scale yields L^p integrability up to (and in some cases beyond) Błocki's predicted exponent.","lead":"This paper shows that for m-subharmonic functions (a middle ground between subharmonic and plurisubharmonic), the exponential-integrability analogues of the Skoda and Guan–Zhou theorems fail, and it answers two open problems about the m-Lelong number. It introduces a Hessian-capacity condition that yields new cases of Błocki's integrability conjecture.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 8.1 rests on unverified radial-tangent uniqueness (8.2); if [6, Thm 3.1] has extra hypotheses or admits non-radial tangents, the equality (8.1) fails.","rationale":"I read the paper carefully, focusing on the proof of Theorem 8.1, which answers Benali–Ghiloufi's Problem 1 and is the main claimed positive result. The proof hinges entirely on (8.2), a strong statement that the rescaled m-subharmonic functions converge in L^1_loc to the radial fundamental solution with coefficient determined by the spherical-mean Lelong number. This is not proved in the paper; it is imported from [6, Thm 3.1]. The stated text does not reproduce the theorem's hypotheses, so the applicability to every m-subharmonic germ is not established. This matches the reader's weakest_assumption exactly. I considered other potential concerns (e.g., the volume-capacity inequality (7.1) and the radial capacity computation in Theorem 9.1), but those are standard or can be verified independently, and the radial case of Theorem 9.1 does not actually need Theorem 8.1. The external tangent-uniqueness theorem, by contrast, is load-bearing for the central claim and is the least secure link. A concrete check—verifying the statement of [6, Thm 3.1] and testing whether all m-subharmonic germs satisfy its hypotheses—would settle whether the concern lands. Since the reader already flagged this and recommended CONDITIONAL, my stress-test agrees and does not change the verdict.","tokens_in":10374,"tokens_out":39988,"duration_ms":307094,"concrete_test":"Retrieve [6, Theorem 3.1] and list its exact hypotheses. Then check whether every u∈SH_m(Ω) satisfies them. If not, exhibit an m-subharmonic function that violates a hypothesis and analytically or numerically compute the L^1_loc limit of u_r(w)=r^{2q}u(rw) to see whether it is radial. Alternatively, construct a non-radial m-subharmonic germ with positive ν (e.g., a small non-radial perturbation of -|z|^{-2q} that preserves m-subharmonicity) and test whether the rescaling converges to -ν/(2q)|w|^{-2q} or to a non-radial homogeneous function; if the latter, Theorem 8.1 is false.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central result, Theorem 8.1, proves the equality ν_u(a)=2lim_{r↓0} M(u,a,r)/φ_m(r) by asserting in (8.2) that the rescalings u_r(w)=r^{2q}u(rw) converge in L^1_loc to the radial function U(w)=-ν_u(0)/(2q)|w|^{-2q}. This convergence is the only mechanism that yields the reverse inequality ℓ_u(a)≥ν_u(a); without it, the squeeze argument (8.3)–(8.4) collapses because limsup of r^{2q}M could be larger than -ν/(2q). The convergence is attributed to the 'strong uniqueness theorem for tangents' of Harvey–Lawson, as stated by Dinew–Kołodziej [6, Thm 3.1], but the paper neither states the theorem nor verifies its hypotheses. If that theorem requires extra assumptions (e.g., finite Hessian mass, local boundedness above by the fundamental solution, or a 'tame' singularity at the point), then arbitrary m-subharmonic germs may not satisfy them. Moreover, for m=n the analogous tangent problem has non-radial homogeneous limits (e.g., log|z_1 z_2| for psh functions); if non-radial tangents can occur for 1≤m<n as well, then sup_{B(0,1)}U would generally exceed -ν/(2q), breaking the claimed equality. The later result for radial functions (Theorem 9.1) also uses the maximum identity, though there it can be replaced by the direct spherical-mean formula, but Theorem 8.1 itself has no such fallback.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies local L^p integrability of m-subharmonic functions for 1≤m<n. It classifies radial power-logarithmic singularities u_{α,β}, determines their m-subharmonicity and exact L^p intervals, and uses them to show that the direct Guan–Zhou strong-openness statement and the direct Skoda criterion in terms of the m-Lelong number both fail in the range m<n. It proves upper/lower semicontinuity properties of the local integrability exponent, including a failure of lower semicontinuity in the function variable on SH_m and a closed endpoint example. It introduces a Hessian-capacity scale (C_{m,δ}) and proves, via the Dinew–Kołodziej volume-capacity inequality and layer-cake formula, that C_{m,δ} implies u∈L^s_loc for every s<(m+δ)n/(n-m); energy classes E_{p,m} are shown to satisfy C_{m,p}. It also answers Benali–Ghiloufi Problem 1 by proving the normalized ball-maximum limit equals the m-Lelong number, using the strong uniqueness theorem for tangents as cited from Dinew–Kołodziej, and shows that radial germs satisfy C_m. The paper is clearly written and the explicit computations are checkable, but the proof of the ball-maximum formula depends on an externally cited tangent-uniqueness theorem whose statement and hypotheses are not reproduced.","tokens_in":10795,"tokens_out":23727,"duration_ms":216169,"significance":"If correct, the paper resolves two open problems of Benali–Ghiloufi and gives a useful capacity-based route toward Błocki's conjecture. The explicit radial counterexamples to direct strong openness and direct Skoda-type criteria are valuable and carefully computed; the endpoint classification of power-logarithmic models is a solid contribution. The capacity criterion (C_{m,δ}) and its application to energy classes are clean and constitute the strongest self-contained part of the paper. However, the central equality in Theorem 8.1 is not self-contained: it rests on the strong uniqueness theorem for tangents, which is neither stated nor verified for arbitrary m-subharmonic germs. The paper is honest about what remains open, and the radial C_m theorem is a nice positive use of the Benali–Ghiloufi mean-value machinery. Overall the manuscript deserves publication after the missing tangent-uniqueness input is made precise.","major_comments":[{"comment":"The proof of Theorem 8.1 rests entirely on the assertion that the rescalings u_r(w)=r^{2q}u(rw) converge in L1_loc to U(w)=-ν_u(0)|w|^{-2q}/(2q) (or 0), attributed to [6, Thm 3.1]. The statement and hypotheses of that theorem are not reproduced. Is convergence for all r>0 or only along subsequences? Does it require u to have finite Hessian mass, a tame singularity, or a special normalization? If any extra hypothesis is needed, it must be verified for arbitrary u∈SH_m with u(0)=-∞. Without (8.2), the Hartogs bound (8.3) fails and the reverse inequality ℓ_u(a)≥ν_u(a) is unproved. Please quote the theorem and explain why non-radial homogeneous tangents (possible for m=n, e.g., log|z1z2| under the logarithmic scaling) are excluded when 1≤m<n.","section":"§8.1, Eq. (8.2)"},{"comment":"The verification of (C_m) for radial germs depends on the ball-capacity formula Cap_m(B(a,ρ),B(a,R0))=c_{n,m}(ρ^{-2q}-R0^{-2q})^{-m}. The text says 'Computing the Hessian mass of this radial extremal function gives...' but gives no computation or reference. This estimate is load-bearing: together with ρ~(A/(qt))^{1/(2q)} it yields the t^{-m} decay that defines C_m. Please include the Hessian-mass computation of h_ρ, check admissibility (-1≤h_ρ≤0), and justify monotonicity of relative capacity in the comparison. A reference to a standard capacitary estimate for balls would also suffice.","section":"§9.1, Theorem 9.1"}],"minor_comments":[{"comment":"Typo: 'partial comfirmation' should be 'partial confirmation'.","section":"Abstract"},{"comment":"Cross-references are inconsistent: 'Theorem 2.1' in the proof of Theorem 3.1 should be 'Lemma 2.1'; 'Theorem 3.2' in Proposition 5.2 and Theorem 8.3 should be 'Proposition 3.2'.","section":"§3 and later"},{"comment":"The sentence 'No counterexample is neither known in those sources' has a double negative and should be rephrased, e.g., 'No counterexample is known in those sources.'","section":"§6"},{"comment":"The phrase 'Part (i) is Theorem 5.1' should refer to 'Proposition 5.1'.","section":"§8.3"},{"comment":"In the convexity computation, F(s)=f(φ_m^{-1}(s)) relies on φ_m^{-1}(s) being positive; the notation is understandable but should be stated explicitly to avoid confusion.","section":"§9.1"},{"comment":"The last row label 'Condition C_{m,δ} below' appears to mean 'above' (since the condition is introduced in Section 7, not below the table). Please correct.","section":"Table 1"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is mathematically careful outside Theorem 8.1. The main risk is the unstated dependence on [6, Thm 3.1]; I would not reject on that basis, since the theorem may well apply verbatim, but the authors must be asked to state it and verify its hypotheses for every m-subharmonic germ. If the theorem only gives subsequential convergence or requires extra assumptions, Theorem 8.1 would need a different proof, making this a substantial revision. The rest of the paper, especially the capacity criterion and the explicit examples, is solid and publishable."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's my take. The paper is a genuine step forward: it answers Benali-Ghiloufi's two problems, gives explicit counterexamples that kill naive Skoda/strong-openness analogues for m-subharmonic functions, and introduces a useful capacity condition C_{m,δ} that yields Błocki's conjectured exponent for radial and energy classes. The radial classification in Section 3 is clean, and the integrability thresholds check out. I have no doubt about the counterexamples or the capacity criterion as a sufficient condition. The soft spot is Theorem 8.1, which is also the headline result. The proof of equality between the normalized maximum limit and the m-Lelong number goes through the assertion in (8.2) that the rescalings u_r(w)=r^{2q}u(rw) converge in L^1_loc to the radial function U(w)= -ν/(2q)|w|^{-2q}, attributed to the Harvey-Lawson strong uniqueness theorem for tangents as stated in [6, Thm 3.1]. But the paper doesn't state that theorem or verify its hypotheses in this setting. If the theorem requires extra conditions, or if non-radial tangents can occur for 1≤m<n (as they do for psh functions), the squeeze argument collapses. This is not a demonstrated error, but it's a load-bearing gap. The author should either reproduce the theorem with hypotheses and check that every m-subharmonic germ satisfies them, or give a direct proof of (8.2). Without that, Theorem 8.1 is conditional. A smaller gap: the ball-capacity estimate in Theorem 9.1 is asserted with 'computing... gives' and no details. That's easily fixable. The rest is solid. The capacity-scale machinery in Section 7 is well-motivated, and the inclusion of E_{p,m} in C_{m,p} is a nice unification. The lower semicontinuity results are straightforward but correctly presented. The citation pattern looks honest; no self-citation games. Who should read it: anyone working on m-subharmonic functions or complex Hessian equations. It deserves a serious referee. My recommendation: send it to review, but make the referee focus on the hypotheses of (8.2). If the author can close that gap, this paper is a solid contribution. If not, the result may still be true but needs a different proof.","headline":"Solid paper with real results, but the proof of the main maximum-identity theorem leans on an unstated tangent-uniqueness theorem that needs verification before publication.","tokens_in":11235,"tokens_out":6234,"would_cite":true,"duration_ms":51379,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["32U05","32U25","32W20","35J60"],"pacs":[],"model":"deepseek-v4-flash","headline":"For every m-subharmonic function, the normalized ball maximum at a point always equals the m-Lelong number, and a new capacity-decay condition yields the conjectured L^p range for a large class of such functions.","keywords":["m-subharmonic functions","m-Lelong number","Hessian capacity","integrability exponents","sharp integrability conjecture","strong openness","radial singularities","power-logarithmic models"],"falsifier":"Take any local m-subharmonic germ with positive m-Lelong number at 0 and compute both 2 lim_{r↓0} M(u,0,r)/φ_m(r) and the spherical-mean limit defining the m-Lelong number. The theorem predicts they are exactly equal; any germ where these differ — for instance one whose rescalings r^{2q}u(rw) converge to a non-radial limit with the same spherical mean but different maximum — would falsify it. Equivalently, checking whether the rescaled sequence always has an L1_loc limit of the form -ν/(2q)|w|^{-2q} settles the issue.","tokens_in":1515,"feed_emoji":"📐","tokens_out":4847,"duration_ms":87654,"temperature":0.7,"pith_summary":"This paper establishes that for every m-subharmonic function — the class between subharmonic and plurisubharmonic in C^n, defined by a Hessian eigenvalue condition — the normalized limit of the maximum value on shrinking balls equals the m-Lelong number, a measure of singularity strength. It also introduces a scale of local Hessian-capacity conditions: if the capacity of deep sublevel sets decays like t^{-(m+δ)}, then the function is locally integrable with every power s < (m+δ)n/(n-m). At the critical case δ=0, this condition is satisfied by compact singularities, local energy classes, and radial germs, giving precisely the strict subcritical range predicted by the long-standing sharp integrability conjecture for these classes. The paper further gives explicit radial examples showing that the direct strong-openness analogue and the direct exponential-integrability analogue from the plurisubharmonic case both fail when m<n, and that the set of integrable powers need not be open. A natural next step toward the full conjecture is to decide whether the critical capacity condition holds for every local m-subharmonic germ.","feed_headline":"Ball-maximum limit equals the m-Lelong number","feed_subtitle":"The equality plus a capacity-decay condition confirms the conjectured L^p range for a wide class of singularities.","key_machinery":"The main device is a scale of local Hessian-capacity conditions C_{m,δ}. For a compact set K inside a bounded m-hyperconvex domain D, the relative Hessian capacity Cap_m(E,D) is defined by a supremum of Hessian masses of comparison functions; the condition asks that the capacity of K∩{u<A-t} decays at least like t^{-(m+δ)}. Combined with a volume-capacity inequality and the layer-cake formula, this decay forces L^s integrability with the stated range. The second load-bearing object is the ball-maximum limit ℓ_u(a)=2 lim M(u,a,r)/φ_m(r), which is shown to equal the m-Lelong number by a tangent-rescaling argument. The radial Hessian test and the classification of power-logarithmic singularitie","core_discovery":"The paper's central identity is ν_u(a) = 2 lim_{r↓0} M(u,a,r)/φ_m(r), valid for every m-subharmonic function u at a point a, where M is the maximum over the ball of radius r and φ_m is the model function (-(r^{2q})/(q))^{-1} with q=(n-m)/m. The proof rescales u by r^{2q}, using a strong uniqueness theorem for tangents to force the rescaled functions to converge in L1_loc to the model singularity -ν/(2q)|w|^{-2q}; then the spherical-mean formula and a maximum principle squeeze the normalized maximum to the same limit. The paper's second main result is the capacity-decay criterion: if u satisfies C_{m,δ}, namely Cap_m(K∩{u<A-t},D) ≤ C t^{-(m+δ)} locally, then u ∈ L^s_loc for every s < (m+δ)n/(","pith_inferences":["The capacity scale may be nearly necessary: it appears plausible that a germ satisfying the sharp L^p range with quantitative control on tails forces sublevel capacity decay of order at least m, so C_m could characterize the conjecture's range (a testable direction).","The failure of lower semicontinuity in the L1_loc topology is tied to the topology; stronger metrics based on capacities or energies might restore semicontinuity for the integrability-exponent functional and could be worth investigating.","Because radial germs satisfy C_m, a promising test toward the full conjecture is whether sums or convex combinations of radial singularities still satisfy C_m; a counterexample there would reveal an intrinsically non-radial obstruction.","The ball-maximum identity may extend to directional refinements — replacing balls with sectors or ellipsoids could yield directional m-Lelong numbers and sharp directional integrability, though the radial rigidity of tangents would be lost."],"forward_implications":["The equality between ball-maximum limit and m-Lelong number gives a practical way to read singularity strength from supremum asymptotics, without computing spherical means.","For radial m-subharmonic germs, the maximum identity turns into a direct verification of C_m, yielding the full strict range p < nm/(n-m) for all such germs.","Energy classes of finite Hessian mass automatically satisfy C_{m,p}, recovering known Sobolev-type integrability exponents and going beyond the critical range.","The counterexamples show that the classical route from bounded exponents to exponential integrability is blocked when m<n; a substitute must involve capacity or tangent profiles rather than only the m-Lelong number.","If the critical capacity condition C_m were proved for every local m-subharmonic germ, the long-standing sharp integrability conjecture would follow for all m between 1 and n."],"fun_headline_variants":["Ball-maximum limit matches m-Lelong number","Capacitary condition nails sharp L^p range","m-subharmonic tangents: max limit equals Lelong","Partial confirmation of Blöcki's L^p conjecture","Max limit and m-Lelong number: exact equality"],"cache_read_input_tokens":12544,"weakest_assumption_plain":"The proof of the ball-maximum identity relies on a strong uniqueness theorem for tangents of m-subharmonic functions whose precise hypotheses are not reproduced; if some local germ admits a non-radial tangent outside that theorem, the equality could fail.","fun_headline_variants_meta":{"raw":{"variants":["Ball-maximum limit matches m-Lelong number","Capacitary condition nails sharp L^p range","m-subharmonic tangents: max limit equals Lelong","Partial confirmation of Blöcki's L^p conjecture","Max limit and m-Lelong number: exact equality"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000204,"raw_usage":{"total_tokens":1332,"prompt_tokens":953,"completion_tokens":379,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":697,"completion_tokens_details":{"reasoning_tokens":302}},"tokens_in":697,"tokens_out":379,"duration_ms":4431,"temperature":1.0,"reasoning_tokens":302,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-01T08:20:11.842164+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take any local m-subharmonic germ with positive m-Lelong number at 0 and compute both 2 lim_{r↓0} M(u,0,r)/φ_m(r) and the spherical-mean limit defining the m-Lelong number. The theorem predicts they are exactly equal; any germ where these differ — for instance one whose rescalings r^{2q}u(rw) converge to a non-radial limit with the same spherical mean but different maximum — would falsify it. Equivalently, checking whether the rescaled sequence always has an L1_loc limit of the form -ν/(2q)|w|^{-2q} settles the issue.","supporting_citations":[],"review_version":1}