{"id":"4f810d45-b5d9-4dab-9879-2b9c1e8fb0ea","arxiv_id":"2607.13543","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"Claims weighted Bergman and Q-space analogues of Gabriel/Frazer inequalities for analytic and harmonic functions, with full p>0 in the harmonic case; internal gaps and factor errors undercut the stated results.","lead":"An analysis paper claims new weighted integral inequalities for analytic and harmonic functions on the disk, extending Gabriel's and Frazer's classical boundary inequalities to Bergman spaces and Möbius invariant spaces. The proofs, however, contain inconsistent constants, an invalid monotonicity step for small p, and an unsupported conjugate estimate, so the highlighted full-range claim is not established.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The full-range harmonic Gabriel inequality (Theorem 1.2) rests on a Riesz-projection estimate for 0<p<∞ that is not established and is false for p≤1; the advertised contrast is therefore unsupported.","rationale":"The reader identified the same load-bearing weakness: the conjugate estimate in Eq. (3.4) is used beyond its known range, and no proof is given for p≤1. I have checked the proof of Theorem 1.2 and found no alternative argument for p≤1; the claimed full-range Theorem 1.2 therefore lacks support. This is an internal gap, not a disagreement with external consensus. I also note the independent flaw in Theorem 1.3's use of monotone integral means for 0<p<1, but the primary reason for rejection is the unsupported p≤1 step in the headline theorem. A verdict of REJECT is appropriate: the paper overclaims, even though portions for p>1 may be repairable.","tokens_in":13846,"tokens_out":7018,"duration_ms":66908,"concrete_test":"Read the statement of [21, Cor. 6]. If its hypotheses require 1<p<∞ (as standard Riesz-projection results for doubling weights do), then Eq. (3.4) is invalid in 0<p≤1 and Theorem 1.2 is unproved in the regime the paper highlights. If the corollary genuinely covers all 0<p<∞, request the precise page/statement and re-check the deduction.","verdict_should_be":"REJECT","load_bearing_attack":"Theorem 1.2 is the central advertised result: it claims Gabriel's inequality for complex-valued harmonic f in a^p_α for every 0<p<∞. The proof of Theorem 1.2 (Section 3, Eq. (3.4)) invokes [21, Cor. 6] to assert ∥F∥_{A^p_α} ≤ c_{p,α}∥Re F∥_{a^p_α} for every 0<p<∞, where F=h+g. This is a boundedness statement for the conjugate/Riesz projection on weighted Bergman spaces. For standard Bergman weights the M. Riesz projection theorem is known only for 1<p<∞; for p≤1 the projection is not bounded, and the manuscript supplies no substitute. In particular, the p=1 case is not a harmless endpoint: the cited theorem cannot be applied. Since Remark 1.1 and the abstract sell exactly the p≤1 regime (contrast with the Hardy-space failure), the proof does not establish the full-range claim. A separate but related p<1 defect is the use in Theorem 1.3 of monotone integral means, which fails for harmonic f when 0<p<1; this does not rescue Theorem 1.2.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes weighted Bergman-space analogues of Gabriel's and Frazer's integral inequalities. Theorem 1.1 states an analytic Gabriel inequality with explicit constant, and Theorem 1.2 claims the harmonic version for every 0<p<∞. Theorem 1.3 gives a circle version, Lemma 1.1 a weighted Kalaj-type estimate, and Theorem 1.4 a two-diameter Frazer inequality. Section 2 transfers these to the Möbius invariant spaces Q(n,p,α) and Q_h(n,p,α) via an embedding lemma from the authors' prior work [16]. The analytic proof is a clean radial-averaging argument. The harmonic proofs, however, rely on a conjugate-function estimate for all 0<p<∞ and on monotonicity of integral means for harmonic functions, both of which are problematic in the stated ranges.","tokens_in":14210,"tokens_out":19906,"duration_ms":181435,"significance":"If Theorem 1.2 were correct, the claimed full-range Bergman-space analogue would be a notable contrast to the known failure of Gabriel's inequality in harmonic Hardy spaces for p≤1. The paper also usefully extends the inequalities to Möbius invariant spaces, and the analytic Theorem 1.1 is an elementary and explicit result. No parameters are fitted and the arguments reduce to classical theorems; the paper is therefore not circular. However, the central harmonic claims are not established by the proofs as written: the conjugate estimate used for p≤1 is not available, and the integral-mean monotonicity used for p<1 is false. These are load-bearing defects in the advertised results.","major_comments":[{"comment":"The proof of Theorem 1.2 invokes [21, Corollary 6] to assert ∥F∥_{A^p_α} ≤ c_{p,α}∥Re F∥_{a^p_α} for every 0<p<∞. This is a Riesz-projection/conjugate-function bound. Standard weighted Bergman theory provides such bounds only for 1<p<∞; no p≤1 version is stated or proved here. Since the abstract and Remark 1.1 explicitly advertise the p≤1 range as the main contrast with harmonic Hardy spaces, Eq. (3.4) does not support the theorem as stated. The authors would need either to prove the p≤1 conjugate estimate or to supply a different argument for the full-range claim.","section":"Section 3, Eq. (3.4)"},{"comment":"The proof asserts that the integral means M_p^p(r,f) are increasing in r for harmonic f. This is false for 0<p<1. For example, with f(z)=1+Re z, the function |f|^p is not subharmonic for p<1, and for p=1/2 a short expansion gives M_p^p(r)=(1/2π)∫_0^{2π}(1+r cosθ)^p dθ = 1 - r^2/16 + O(r^4), which decreases for small r. Thus the proof of the full-range circle inequality Theorem 1.3 is invalid for 0<p<1.","section":"Section 3, proof of Theorem 1.3"},{"comment":"The constant in the displayed theorem misses a factor (α+1). The proof concludes with ∫_C |f|^p(1-|z|^2)^{α+1}|dz| ≤ (4π/(1-R^2)^{α+1}) ∥f∥^p_{A^p_α}. Since ∥f∥^p_{A^p_α} = (α+1)∫_D |f|^p(1-|z|^2)^α dA(z), the stated right-hand side with dA(z) is smaller by the factor (α+1) than what is proved. If the intended measure is dA_α, the notation should be corrected; as written, the theorem is stronger than the proof establishes.","section":"Section 1, Theorem 1.1"},{"comment":"The displayed constant A_p(θ,α) appears to omit a factor of π. The final step of the proof gives L(f) ≤ (2^{1+α}π/(sin θ/2 + cos θ/2)) · (2^{p/2}/(1-|cos(π/p)|)^{p/2}) ∥f∥^p_{a^p_α}. The theorem statement has no π in A_p(θ,α). Unless this is a typographical omission, the stated constant is inconsistent with the proof.","section":"Section 1, Theorem 1.4"},{"comment":"Several displayed lines drop the conjugation on g. The proof obtains an intermediate inequality with |h_r(e^{iθ})+g_r(e^{iθ})| and later ∫_D |h+g|^p dA_α, whereas the lemma is stated for |h+\\bar g|. Since Lemma 1.1 is used in the proof of Theorem 1.4, this is not purely cosmetic. The intended argument is likely to apply [18, Theorem 2.1] to h_r+\\overline{g_r}; the displayed formulas should be corrected consistently.","section":"Section 3, proof of Lemma 1.1"}],"minor_comments":[{"comment":"There are typographical errors such as 'genralization' in Section 1.2, inconsistent use of dA versus dA_α in theorem statements, and missing bars on g in the proof of Lemma 1.1. A careful proofreading pass is needed.","section":"Throughout"},{"comment":"The proof says 'Suppose that f=h+\\bar g with f(0)=g(0)=0'. If f(0)=0, one may choose the canonical decomposition so that h(0)=g(0)=0, but this normalization should be stated explicitly; as written, g(0)=0 looks like an additional assumption.","section":"Theorem 1.4"},{"comment":"The constant contains α+1-np; the condition α>np-1 ensures positivity, but this should be noted, since otherwise the weight exponent could be nonpositive.","section":"Corollary 2.3"},{"comment":"The citation [21, Corollary 6] needs a precise statement of the admissible range of p and α. The current use of it for all 0<p<∞ is the main technical point and should be quoted exactly.","section":"References"}],"recommendation":"reject","confidential_remarks":"The paper has a sound analytic core and a clear organizational structure, but the two main full-range harmonic claims rest on unsupported or false ingredients: a p≤1 conjugate estimate and monotone integral means for p<1. These are central to the advertised contrast with the Hardy-space setting, and the errors are not merely local. I do not see how a revision within the current proof strategy can establish the claims as stated."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nThe paper has a genuine kernel, but the headline result is not proved. The analytic Theorem 1.1 is a clean radial-average argument: apply Gabriel's Hardy-space theorem to dilations, integrate in r, and you get a weighted Bergman Gabriel inequality. That part is basically right. The stated constant is missing a factor (α+1) because the RHS is written with dA rather than dAα, but that is a repairable constant error.\n\nThe real problem is Theorem 1.2, which is the advertised full-range harmonic result. The proof bounds the analytic part F by its real part via [21, Cor. 6] for every 0<p<∞. That is a conjugate-function estimate on weighted Bergman spaces; it is only available for 1<p<∞. The Riesz projection is not bounded for p≤1, and the paper supplies no substitute. Since the abstract and Remark 1.1 specifically sell the p≤1 contrast with Hardy spaces, the central claim is unsupported as written. For p>1 the argument may well go through, but the range needs to be narrowed.\n\nThere are smaller issues in the same direction. Theorem 1.3 uses monotonicity of the integral means M_p^p(r,f) for harmonic f, which is false for 0<p<1; try f=1+Re z. So that theorem is also only proved for p≥1, despite the stated range. Theorem 1.4's explicit constant is missing a π (the proof gives π times the stated constant). In Lemma 1.1's proof, the conjugation on g disappears in several displayed lines when Kalaj's theorem is invoked; that looks like a typo, but it sits exactly at the point where the harmonic decomposition matters.\n\nThe Q-space half is honest: the corollaries follow immediately from the embedding lemma [16, Lemma 2.1]. Low added value, but not circular and not misleading. The citation pattern is fine.\n\nBottom line: this deserves refereeing, not desk rejection, because the analytic result and the p>1 harmonic extension are worth having if the authors fix the constants and rewrite the claims. I would not cite it in its current form, and the abstract needs substantial revision.","headline":"The analytic weighted Bergman Gabriel inequality is a real and mostly correct result; the advertised full-range harmonic version is not proved, and the p≤1 claims rest on unavailable or false estimates.","tokens_in":14635,"tokens_out":12189,"would_cite":false,"duration_ms":110027,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["31A05","30H20","30H05"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper establishes Gabriel's inequality for complex-valued harmonic functions in weighted Bergman spaces for every 0<p<∞, a range where the harmonic Hardy space version fails.","keywords":["weighted Bergman spaces","harmonic mappings","Gabriel inequality","Frazer problem","Riesz–Fejér inequality","Möbius invariant spaces","harmonic conjugate","convex curves"],"falsifier":"For $p=1/2$ and $\\alpha=0$, take $f(z)=1+\\operatorname{Re} z$ and circles $C_r=\\{|z|=r\\}$, and compute the ratio of $\\int_{C_r} |f|^p(1-|z|^2)|dz|$ to $\\int_D |f|^p dA$ for a sequence $r\\to 1^-$; if the ratio grows without bound, the stated broad-range circle version is false. For the headline theorem, check the conjugate estimate on $F(z)=1/(1-z)$ with $p=1/2$: if $\\|F\\|_{A^p_\\alpha}$ is not bounded by a fixed multiple of $\\|\\operatorname{Re} F\\|_{a^p_\\alpha}$, the assumption underlying Theorem 1.2 is refuted.","tokens_in":13718,"feed_emoji":"📐","tokens_out":16013,"duration_ms":137090,"temperature":0.7,"texified_at":"2026-08-05T21:21:51.792219+00:00","pith_summary":"This paper answers a natural question about how far two classical theorems of geometric function theory, Gabriel's and Frazer's arc-length inequalities, extend from Hardy spaces to Bergman spaces. Its main achievement is a full-range result: for every complex-valued harmonic function $f$ in a weighted harmonic Bergman space $a^p_\\alpha$ with $0<p<\\infty$, and for every convex curve $C$ inside the unit disk, the weighted integral of $|f|^p$ along $C$ is bounded by a constant multiple of the weighted area integral of $|f|^p$ over the disk. This is the Bergman-space analogue of Gabriel's theorem, and it holds throughout $0<p<\\infty$, including the small $p$ values for which the harmonic Hardy-space version is known to fail. The authors derive the analytic Bergman version for all $p>0$, then transfer it to harmonic functions through a harmonic-conjugate estimate. They also obtain Frazer's circle and two-diameter formulations, and apply everything to the Möbius invariant spaces $Q(n,p,\\alpha)$ and $Q_h(n,p,\\alpha)$.","texify_model":"deepseek-v4-flash","texify_usage":{"total_tokens":14973,"prompt_tokens":990,"completion_tokens":13983,"prompt_tokens_details":{"cached_tokens":0},"prompt_cache_hit_tokens":0,"prompt_cache_miss_tokens":990,"completion_tokens_details":{"reasoning_tokens":12979}},"feed_headline":"In harmonic Bergman spaces, Gabriel's inequality survives for p ≤ 1","feed_subtitle":"A weighted area integral over the disk controls arc integrals along any convex curve, unlike harmonic Hardy spaces.","key_machinery":"The central mechanism is the transfer from boundary to area: for analytic $f$, dilate to $f_r$, apply the classical Gabriel inequality on the scaled convex curve, multiply by the weight $(1-r^2)^\\alpha r$, and integrate in the dilation parameter $r$. For harmonic $f = h + \\bar{g}$, the paper forms $F = h+g$ and uses a Bergman-space harmonic-conjugate estimate, $\\|F\\|_{A^p_\\alpha} \\leq c_{p,\\alpha} \\|\\operatorname{Re} F\\|_{a^p_\\alpha}$ (adopted for all $p>0$), so that inequalities for the analytic $F$ pull back to inequalities for $f$ through $|f|^p \\leq C_p(|u|^p+|v|^p)$. Gabriel's inequality — the control of an arc integral over a convex curve by an integral over the whole disk or circle — is the named object being extended, and the harmonic-conjugate estim","core_discovery":"The paper's central claim is Theorem 1.2: for every complex-valued harmonic function $f$ in $a^p_\\alpha$ with $0<p<\\infty$, and every convex curve $C$ contained in the disk, $\\int_C |f|^p (1-|z|^2)^{1+\\alpha} |dz| \\leq K_{p,\\alpha,C} \\int_D |f|^p (1-|z|^2)^\\alpha dA(z)$. The proof's key move is to decompose $f = h + \\bar{g}$, form the analytic function $F = h+g$, control the $A^p_\\alpha$ norm of $F$ by the $a^p_\\alpha$ norm of its real part using a Riesz-conjugate estimate (quoted from earlier work, and assumed for every $p>0$), then apply the analytic Gabriel inequality to $F$ and convert the resulting control on $u$ and $v$ into control on $|f|^p$. The same machinery yields the analytic Theorem 1.1 with the explicit constant $4\\pi/(1-R^2)^{1+\\alpha}$, a circle version wit","pith_inferences":["The reduction to the associated analytic function suggests that the harmonic failure in Hardy spaces for p≤1 is best viewed as the absence of a full-range Riesz-conjugate estimate there, not as a fundamental obstruction for harmonic functions; any Bergman-type setting with such an estimate should inherit Gabriel's inequality.","The p≤1 part of the argument relies on a quoted conjugate estimate that the paper does not prove; a direct proof for 0<p≤1, or a counterexample, would settle whether the advertised full range is genuine.","The p-independent constant in the circle version hints that the monotonicity of integral means used in its proof could be bypassed by a measure-distribution argument, which would also cover the delicate p<1 case.","The same transfer device appears adaptable to other weighted settings, such as doubling-weight Bergman spaces or pluriharmonic functions, where the needed conjugate estimates are already available."],"forward_implications":["Weighted harmonic Bergman spaces satisfy Gabriel's inequality for every 0<p<∞, so the p≤1 obstruction found in harmonic Hardy spaces does not appear under the Bergman measure.","The circle version 2πr(α+1)∥f∥^p_{a^p_α} gives an explicit, p-free bound for the weighted arc integral of any harmonic f over every concentric circle.","The Möbius invariant spaces Q(n,p,α) and Q_h(n,p,α) inherit Gabriel and Frazer inequalities through their embedding into weighted Bergman spaces with parameter α-np.","The two-diameter inequality gives an explicit constant, depending on the angle, for the union of two diameters in weighted harmonic Bergman spaces.","These arc-length bounds imply that functions in these spaces cannot concentrate excessively along any convex curve inside the disk relative to their area norm."],"fun_headline_variants":["Gabriel's inequality holds for all p>0 in harmonic Bergman spaces","Harmonic Bergman spaces keep Gabriel's inequality for p ≤ 1","Weighted harmonic Bergman: Gabriel's inequality for every p>0","Gabriel's problem solved for harmonic weighted Bergman spaces"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The full-range harmonic theorem rests on a cited Riesz-conjugate estimate for Bergman spaces that controls an analytic function's weighted norm by the weighted norm of its real part for every $p>0$, whereas the standard theory guarantees this only for $p>1$.","fun_headline_variants_meta":{"raw":{"variants":["Gabriel's inequality holds for all p>0 in harmonic Bergman spaces","Harmonic Bergman spaces keep Gabriel's inequality for p ≤ 1","Weighted harmonic Bergman: Gabriel's inequality for every p>0","Gabriel's problem solved for harmonic weighted Bergman spaces"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00097,"raw_usage":{"total_tokens":4035,"prompt_tokens":888,"completion_tokens":3147,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":632,"completion_tokens_details":{"reasoning_tokens":3082}},"tokens_in":632,"tokens_out":3147,"duration_ms":22678,"temperature":1.0,"reasoning_tokens":3082,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-02T04:52:26.085329+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For $p=1/2$ and $\\alpha=0$, take $f(z)=1+\\operatorname{Re} z$ and circles $C_r=\\{|z|=r\\}$, and compute the ratio of $\\int_{C_r} |f|^p(1-|z|^2)|dz|$ to $\\int_D |f|^p dA$ for a sequence $r\\to 1^-$; if the ratio grows without bound, the stated broad-range circle version is false. For the headline theorem, check the conjugate estimate on $F(z)=1/(1-z)$ with $p=1/2$: if $\\|F\\|_{A^p_\\alpha}$ is not bounded by a fixed multiple of $\\|\\operatorname{Re} F\\|_{a^p_\\alpha}$, the assumption underlying Theorem 1.2 is refuted.","supporting_citations":[],"review_version":1}