{"id":"457561dd-c2fa-44cc-b077-86a30df96eaa","arxiv_id":"2607.28221","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Hg is bounded on Hp iff g' multiplies the Hilbert range into Hp; this equals the mixed-norm space only for p≤2 and is strictly smaller for p>2, refuting the endpoint conjecture.","lead":"Generalized Hilbert operators on Hardy spaces are bounded exactly when the symbol derivative is a Hadamard multiplier from the Hilbert range. For p>2 the natural endpoint Lipschitz condition fails, disproving a 2014 conjecture.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.5","headline":"No significant objection identified","rationale":"The manuscript’s strongest claim is a clean, explicit refutation of a published conjecture by means of a lacunary series that lies in the endpoint mixed-norm space but fails to multiply the Hilbert range. Every step after the cited integral estimate is elementary Hardy-space theory (Fejér–Riesz, dyadic blocks, Paley). The external lemma is a routine estimate, not an unproved hypothesis special to the paper; its failure would be surprising and would be immediately visible by direct computation. Consequently the reader’s ACCEPT / high-confidence assessment needs no adjustment. The concrete test above is merely a one-line sanity check that any referee can perform in a few minutes.","tokens_in":18596,"tokens_out":566,"duration_ms":10982,"concrete_test":"Independently verify the single integral appearing in the proof of Theorem 3.5: confirm that ∫-π^π |1-rei\theta|-1 |log(e/(1-rei\theta))|-p/2 d\theta remains bounded as r\to1- for every fixed p>2 (e.g., by the change of variables t=(1-r)/|1-rei\theta| or by consulting the cited Lemma 1 of Avetisyan). If the integral is O(1), the membership fp∈Hp and the subsequent counter-example stand.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central strict-inclusion claim (Theorem 3.5 / Corollary 3.6) rests on three classical ingredients that hold: (i) the dyadic characterization of H(p,∞,1/p'), (ii) the lower moment bound (7) obtained by restricting the integral of fp to the interval IN=[1-1/N,1-1/(2N)], which uses only elementary comparisons tN≳1, (1-t)-1/p≃ N1/p and (log e/(1-t))-1/2≃(log N)-1/2, and (iii) Paley’s theorem for Hadamard-gap series. The sole external citation is Avetisyan’s integral estimate guaranteeing fp∈Hp; that estimate is standard (the weight |log|-p/2 with p/2>1 is integrable against the Poisson kernel of 1/|1-z|). No hidden boundedness assumption, circular appeal, or gap in the lacunary construction appears. The reader’s weakest-assumption note is therefore a presentational dependence rather than a load-bearing correctness risk.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The paper studies the generalized Hilbert operator Hg(f)(z)=\\int0^1 f(t)g'(tz)dt on Hardy spaces Hp. It introduces the range space Rp=H(Hp) with the pullback norm and the Hadamard multiplier space Mp=(Rp,Hp). The central results are the equivalences Hg:Hp\\to Hp bounded \\Leftrightarrow g'\\in Mp \\Leftrightarrow Bp(g)<\\infty (Theorem 3.2), the identification Mp=H(p,\\infty,1/p') for 1<p\\le2 (Theorem 3.4), and the strict inclusion Mp\\subsetneq H(p,\\infty,1/p') for p>2 (Theorem 3.5), which yields an explicit counter-example gp\\in\\Lambda(p,1/p) for which Hg is unbounded and thereby answers the conjecture of Galanopoulos–Girela–Peláez–Siskakis in the negative. The paper further embeds two known sufficient classes Ap and Ep strictly inside Mp, gives a complete coefficient characterization for symbols with nonnegative decreasing Taylor coefficients, and proves that the family {Mp} is strictly increasing in p.","tokens_in":18814,"tokens_out":753,"duration_ms":11973,"significance":"The work cleanly separates the operator-theoretic question from the symbol by reducing boundedness of Hg to membership of g' in a concrete multiplier space. The negative resolution of the Galanopoulos–Girela–Peláez–Siskakis conjecture for p>2 is definitive: the lacunary series \\Phi p and the integrated symbol gp are explicit, and the argument relies only on standard tools (dyadic characterizations, Fejér–Riesz, Paley’s theorem for Hadamard gaps). The structural results—strict increase of Mp in p, proper containment of Ap and Ep, and the monotone-coefficient characterization—give a coherent picture of the multiplier scale and recover earlier sufficient conditions as special cases. The contribution is solid and of clear interest to the Hardy-space operator community.","major_comments":[],"minor_comments":[{"comment":"In the proof of Theorem 3.4 the sentence “Since 2<p\\le2, the embedding (1) shows…” is a typographical contradiction; the intended range is 1<p\\le2 and the embedding D(p,p,1)\\subset Hp should be invoked.","section":"Theorem 3.4 proof"},{"comment":"Membership fp\\in Hp is cited from Avetisyan [1, Lemma 1] rather than proved. A one-line sketch that the weight |log|^{-p/2} (p/2>1) is integrable against the Poisson kernel of 1/|1-z| would make the counter-example self-contained.","section":"Theorem 3.5 proof"},{"comment":"Several minor typos appear: “folliwing” (p. 2), “imples” (p. 7), “The folliwing embeddings”, and inconsistent spacing around operators (Hg vs H_g). A careful copy-edit is needed.","section":"Throughout"},{"comment":"The abstract and introduction mention that Mp contains Cauchy transforms; this is proved only for the kernels \\kappa\\zeta in Theorem 4.7. A brief clarifying sentence would align the claims.","section":"Abstract / §5"}],"recommendation":"accept","confidential_remarks":"The manuscript is ready for acceptance after routine copy-editing. The external citation for fp\\in Hp is standard and does not affect correctness. No novelty or scope concerns."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"The one thing worth knowing is that this paper settles the Galanopoulos–Girela–Peláez–Siskakis conjecture from 2014 in the negative: for p>2, g in Λ(p,1/p) does not force Hg bounded on Hp. They do it by rewriting boundedness of Hg as membership of g' in the Hadamard multiplier space Mp=(Rp,Hp), identifying Mp exactly with H(p,∞,1/p') when p≤2, and exhibiting a concrete gap series that sits in the growth space but not in Mp when p>2.\n\nWhat is new is the reformulation itself (Theorem 3.2: Hg bounded ⇔ g'∈Mp ⇔ Bp(g)<∞), the sharp identification for p≤2, the strict inclusion plus the explicit Φp/fp counter-example for p>2, the placement of the two earlier sufficient classes Ap and Ep strictly inside Mp, the monotone-coefficient characterization, and the fact that the family {Mp} is strictly increasing in p. The arguments stay inside classical Hp theory—dyadic blocks, Stieltjes–Mellin, duality pairings, Paley gaps, standard embeddings—and the logic is transparent. The stress-test is right: the Avetisyan integral that puts fp in Hp is a standard Poisson-kernel estimate, not a hidden assumption, so the counter-example holds.\n\nSoft spots are minor and presentational. There is a typographical slip (“Since 2<p≤2”) in the p≤2 proof, the fp membership is cited rather than reproved, and the paper does not give an intrinsic description of Mp for p>2 beyond the sandwich Ap ⊊ Mp ⊊ H(p,∞,1/p') and Ep ⊊ Mp. None of that touches the load-bearing claims. Citations look appropriate; self-citations are to earlier operator papers used as background.\n\nThis is for people who work on Hilbert-type operators and coefficient multipliers on Hardy spaces. It is a correct, self-contained resolution of a stated open question inside its subfield. I would send it to a serious referee and I would cite the counter-example and the multiplier identification.","headline":"Clean negative answer to the 2014 endpoint conjecture on generalized Hilbert operators, via an exact range-multiplier reformulation and an explicit lacunary counter-example.","tokens_in":19487,"tokens_out":550,"would_cite":true,"duration_ms":9249,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["47B35","30H10","30H20"],"pacs":[],"model":"grok-4.5","headline":"Generalized Hilbert operators on Hardy spaces Hp are bounded exactly when the symbol derivative multiplies the classical Hilbert range into Hp; for p>2 that multiplier space is strictly smaller than the usual growth space, so the mean-Lipsc","keywords":["Generalized Hilbert operator","Hardy space","Hadamard multiplier","Mixed norm space","Mean Lipschitz space","Hilbert matrix","Coefficient multipliers"],"falsifier":"Compute or disprove the lower bound on the moments of fp(z)=(1-z)-1/p (log e/(1-z))1/2 against the lacunary coefficients of Φp; if those moments decay fast enough that Φp*H(fp) lands in Hp, the claimed strict inclusion fails.","tokens_in":19428,"feed_emoji":"∞","tokens_out":992,"duration_ms":24160,"temperature":0.7,"pith_summary":"This paper pins down when a generalized Hilbert operator, built from an analytic symbol g, acts boundedly on the Hardy space Hp. The decisive move is to rewrite the operator as a Hadamard product of g' against the range of the classical Hilbert matrix; boundedness is then exactly membership of g' in that multiplier space. For 1<p≤2 the multiplier space coincides with the familiar mixed-norm growth space H(p,∞,1/p'), recovering the known characterization in terms of the mean Lipschitz class. For p>2 the same growth space is too large: an explicit lacunary series lies in the growth space yet fails to multiply the Hilbert range back into Hp, producing a concrete symbol in Λ(p,1/p) whose operator is unbounded. The work also embeds two earlier sufficient classes inside the true multiplier space, gives a complete coefficient criterion for symbols with nonnegative decreasing Taylor coefficients, and shows that the multiplier spaces themselves form a strictly increasing scale in p.","feed_headline":"Mean-Lipschitz symbols fail for Hilbert operators when p>2","feed_subtitle":"The true multiplier space is strictly smaller than the usual growth space, killing a prior conjecture","key_machinery":"The elementary identity Hg(f)=g'*H(f), which converts operator boundedness into the concrete coefficient-multiplier problem of mapping the pullback range Rp of the classical Hilbert operator into Hp.","core_discovery":"For every 1<p<∞ the operator Hg is bounded on Hp if and only if g' belongs to the Hadamard multiplier space (Rp,Hp), equivalently if a certain Hilbert-matrix bilinear form is finite. When 1<p≤2 this space equals H(p,∞,1/p'). When p>2 the inclusion is strict, so there exist symbols g in the mean Lipschitz class Λ(p,1/p) for which Hg fails to be bounded on Hp, answering a prior conjecture in the negative.","pith_inferences":["An intrinsic description of (Rp,Hp) for p>2—perhaps via vector-valued Carleson measures or tent spaces—would finish the classification the paper begins.","The same range-multiplier dictionary should adapt to weighted Bergman or Dirichlet-type settings where generalized Hilbert operators are already studied.","Compactness and essential-norm formulae for Hg are natural next targets once the bounded multipliers are understood."],"forward_implications":["For p>2, membership of g in Λ(p,1/p) is necessary but no longer sufficient for Hg to be bounded on Hp.","Any symbol whose derivative lies in the union over q<p of H(q,∞,1/q') or in the mixed space H(p,tp,1/p') automatically yields a bounded Hg.","When the Taylor coefficients of g are nonnegative and decreasing, Hg is bounded on Hp if and only if n bn stays bounded.","The multiplier spaces (Rp,Hp) increase strictly with p, so a multiplier for a smaller exponent remains a multiplier for every larger exponent."],"fun_headline_variants":["Multiplier space strictly smaller than mean-Lipschitz for p>2","Hilbert operators: Λ(p,1/p) fails to bound Hp when p>2","Exact multipliers for generalized Hilbert operators on Hp","Conjecture dies: mean-Lipschitz symbols do not suffice for p>2","(Rp,Hp) equals H(p,∞,1/p') only up to p=2"],"cache_read_input_tokens":128,"weakest_assumption_plain":"The counter-example for p>2 rests on an external integral estimate guaranteeing that a particular test function with a logarithmic factor still lies in Hp; if that estimate failed, the strict-inclusion argument would collapse.","fun_headline_variants_meta":{"raw":{"variants":["Multiplier space strictly smaller than mean-Lipschitz for p>2","Hilbert operators: Λ(p,1/p) fails to bound Hp when p>2","Exact multipliers for generalized Hilbert operators on Hp","Conjecture dies: mean-Lipschitz symbols do not suffice for p>2","(Rp,Hp) equals H(p,∞,1/p') only up to p=2"]},"model":"grok-4.5","effort":"low","cost_usd":0.00473,"raw_usage":{"total_tokens":1502,"prompt_tokens":1016,"num_sources_used":0,"completion_tokens":88,"cost_in_usd_ticks":47304000,"prompt_tokens_details":{"text_tokens":1016,"audio_tokens":0,"image_tokens":0,"cached_tokens":128},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":398,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":1016,"tokens_out":88,"duration_ms":6617,"temperature":1.0,"reasoning_tokens":398,"cache_read_input_tokens":128,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-31T14:13:31.708516+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Compute or disprove the lower bound on the moments of fp(z)=(1-z)-1/p (log e/(1-z))1/2 against the lacunary coefficients of Φp; if those moments decay fast enough that Φp*H(fp) lands in Hp, the claimed strict inclusion fails.","supporting_citations":[],"review_version":1}