{"id":"c803fd73-703f-4ebf-a427-5c33a95bf55e","arxiv_id":"2411.14843","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For BMOA symbols g1 and g2, the meromorphic optimal domains (T_{g1},H^p) and (T_{g2},H^p) coincide exactly when each symbol is the other symbol integrated against a bounded analytic function, with the two functions reciprocals.","lead":"This paper introduces a space of meromorphic functions that a generalized Volterra integral operator sends into Hardy spaces, and gives a clean condition for when two symbols produce the same space. The result advances the optimal domain program for integral operators on H^p.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified: Eq. (3) is an immediate consequence of the definition, so the reader's weakest assumption does not threaten Theorem 2.3.","rationale":"The reader identified Eq. (3) as the weakest assumption, but Eq. (3) is not an independent unproved result: it is exactly the definition of (T_g,H^p) rewritten. For every f in the meromorphic optimal domain, h=T_g(f) belongs to H^p and h′=fg′, so f=h′/g′. Conversely, any h∈H^p produces f=h′/g′ with fg′=h′∈Hol(D) and T_g(f)=h−h(0)∈H^p. Hence the isometric isomorphism in Theorem 2.1(e), the W_g characterization in Theorem 2.2, and the equality criterion in Theorem 2.3 all rest on a direct definitional identity. The proof of Theorem 2.3 is sound: equality of meromorphic optimal domains is equivalent to W_{g1}=W_{g2} by definition, and Theorem 2.2 gives W_g=T_g(H^∞)+C, yielding exactly g1=T_{g2}(k1)+g1(0) and g2=T_{g1}(k2)+g2(0); differentiating and using that g1′ is not identically zero gives k2=1/k1. The converse direction is equally direct from the same inclusion. I also checked the surrounding arguments for circularity. Theorem 2.2's converse uses Theorem 2.1(c), whose multiplier proof is correct modulo an unstated closed-graph argument to ensure an algebraic multiplier is bounded; this is standard and does not affect the central claim. The remaining gaps are in peripheral statements: Theorem 2.4(b) does not prove the strict containment for q1=(1+z)^{1/p}, although the check above shows it is true; Corollary 5.4 contains a sign typo; and Theorem 2.4(a) writes W_g=T_g(H^∞) instead of W_g=T_g(H^∞)+C. These are worth fixing but do not undermine Theorem 2.3. Therefore no significant objection to the central claim is identified, and the verdict should remain unchanged.","tokens_in":15010,"tokens_out":31944,"duration_ms":291035,"concrete_test":"Verify the omitted strict-containment step in Theorem 2.4(b) for q1(z)=(1+z)^{1/p}: define F=1/(g′(z)(1+z)^{1/p+1}). Then T_g(F)=p((1+z)^{−1/p}−1)∉H^p, while T_{T_g(q1)}(F)=log(1+z)∈H^p, confirming (T_{T_g(q1)},H^p)⊋(T_g,H^p). If this fails, the intersection proof needs repair.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Equation (3) is definitional, not a load-bearing unproved lemma. If f∈(T_g,H^p), then h=T_g(f)∈H^p and h′=fg′, so f=h′/g′; conversely, f=h′/g′ with h∈H^p gives fg′=h′∈Hol(D) and T_g(f)=h−h(0)∈H^p. Thus Theorem 2.1(e), Theorem 2.2, and Theorem 2.3 rest on a tautology, and the reader's stated concern does not land. The central chain Theorem 2.2, relying on Theorem 2.1(c), whose proof invokes the boundedness of the companion operator from [4], is internally coherent. The only substantive omissions I see are peripheral: Theorem 2.4(b) asserts strict containment for T_g(q1) without proof, and Corollary 5.4 has a sign error (−g_i(0) instead of +g_i(0)). Neither affects the equality criterion in Theorem 2.3.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces, for g in BMOA and 1 <= p < infinity, the meromorphic optimal domain (T_g,H^p) of the generalized Volterra operator T_g, consisting of meromorphic functions f such that f g' is holomorphic in the unit disc and T_g(f) lies in H^p. It identifies this space with the set of quotients h'/g' with h in H^p (Eq. (3)), proves Banach-space properties of these domains (Theorem 2.1), characterizes the space W_g of symbols h whose meromorphic optimal domain contains (T_g,H^p) as W_g = T_g(H^infinity) + C (Theorem 2.2), and uses this to prove the main structural result (Theorem 2.3): two meromorphic optimal domains coincide exactly when the symbols can be written as g_j = T_{g_l}(k_j) + g_j(0) for reciprocal bounded analytic functions k_1,k_2. The paper then studies the Banach-space structure and intersection properties of W_g (Theorem 2.4) and gives partial results for equality of the holomorphic optimal domains [T_g,H^p] for locally univalent symbols, polynomial symbols, and a weighted Bergman-space reformulation in the Hilbert case.","tokens_in":15210,"tokens_out":20399,"duration_ms":185950,"significance":"If the local gaps are patched, the central Theorem 2.3 is a clean, parameter-free structural characterization of equality of meromorphic optimal domains. The proof is genuinely from definitions plus published multiplier and integral-operator facts, and Theorem 2.2 is a useful p-independent description of W_g. The paper is also honest about the parts of the holomorphic-optimal-domain problem that remain open. The main chain of reasoning for the central claim is sound; the issues I found are confined to secondary results and to statement-level typos.","major_comments":[],"minor_comments":[{"comment":"The assertion that (T_{T_g(q1)},H^p) and (T_{T_g(q2)},H^p) strictly contain (T_g,H^p) for q1=(1+z)^{1/p} and q2=(1-z)^{1/p} is not actually derived. The first part of the proof constructs a strict extension only for a special q_h2; it does not apply directly to q1 and q2. The authors should supply explicit functions showing strictness, for example F1 = 1/(g'(1+z)^{1+1/p}) for q1 and F2 = 1/(g'(1-z)^{1+1/p}) for q2.","section":"Section 4, proof of Theorem 2.4(b)"},{"comment":"The sign in condition (b) is wrong: the constants should be +g1(0) and +g2(0), as in Theorem 2.3. The displayed version with minus signs would force g1(0)=g2(0)=0.","section":"Section 5.1, Corollary 5.4"},{"comment":"The identification (T_g,H^p) = {h'/g' : h in H^p} is stated without proof, with a citation to [11, Proposition 3.2]. Since it follows in one line from Definition 1.1, it would be cleaner to include that line instead of omitting the proof.","section":"Section 2, Eq. (3)"},{"comment":"The sentence 'the graph of T_g, which is precisely W_g' is imprecise: W_g is T_g(H^infinity) + C, not the graph itself. The subsequent argument still works because the sum is direct, but the wording should be corrected.","section":"Section 4, proof of Theorem 2.4(a)"},{"comment":"The displayed formula for T_g(phi) has a sign error: the factor should be +p2, not -p2, since the derivative of p2((1-z)^{-1/p2}-1) is (1-z)^{-1/p2-1}.","section":"Section 3, proof of Theorem 2.1(b)"},{"comment":"The argument that point evaluation at z0 in Z(g') is unbounded because 1/g' has a pole at z0 is not fully formal, since point evaluation is not defined on functions with a pole. A cleaner argument would exhibit a sequence of functions in (T_g,H^p) whose point evaluations at z0 tend to infinity.","section":"Section 3, proof of Lemma 3.3"}],"recommendation":"minor_revision","confidential_remarks":"The paper is within scope and the central characterization theorem is sound. The main corrections needed are the sign in Corollary 5.4, an explicit verification of strict containment in Theorem 2.4(b), and several proof-presentation issues. Once those are fixed, the manuscript should be acceptable."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Hi,\n\nI read the Bellavita–Belli–Nikolaidis–Stylogiannis paper on meromorphic optimal domains for Volterra operators. Bottom line: the paper is in good shape, and the main characterization in Theorem 2.3 is correct as far as I can tell. The report's principal worry — that Eq. (3) is a load-bearing unproved lemma — does not survive contact with the paper. Eq. (3) is just the definition of (T_g,H^p) written in derivative form: for f in the domain, h = T_g(f) is in H^p and h' = f g', so f = h'/g'; conversely that formula gives T_g(f) = h − h(0) in H^p. So the foundation is tautological, not an omitted proof.\n\nWhat's genuinely new: the meromorphic optimal domain itself, and the description W_g = T_g(H∞) + C, which yields the clean equality criterion for two symbols. The structural results — Banach, multipliers H∞, separability, duality with H^q, interpolation — are useful and mostly proved tightly. The paper credits prior work appropriately: the companion-operator boundedness from [4] and the optimal-domain program from [5] and [11] are cited where used.\n\nSoft spots, in proportion: Theorem 2.4(b) is the weakest section. The proof asserts strict containment for the specific functions T_g(q_i) without really deriving it; the earlier construction was for a different purpose and doesn't directly transfer. The sign error in Corollary 5.4 (−g_i(0) instead of +g_i(0)) is a typo but should be fixed. A few OCR-level typos elsewhere. None of these touch the central argument.\n\nWho this is for: researchers working on Volterra operators, Hardy spaces, and optimal domains. It extends an active line and gives a complete answer to a natural question. I'd send it to a competent referee; it's a solid contribution that needs minor revision, not a desk reject.\n\nRegards","headline":"A solid, mostly correct paper introducing meromorphic optimal domains; the main concern about Eq. (3) is a red herring, and the remaining issues are minor and fixable.","tokens_in":15744,"tokens_out":4311,"would_cite":true,"duration_ms":38209,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["30H10","47G10"],"pacs":[],"model":"deepseek-v4-flash","headline":"Meromorphic optimal domains of Volterra operators coincide exactly when the symbols' derivatives differ by reciprocal bounded analytic factors.","keywords":["Optimal domain","Volterra type operators","Integral operators","Hardy spaces","Analytic bounded mean oscillation","Meromorphic functions","BMOA","Cesàro operator"],"falsifier":"A concrete test of Theorem 2.3: with $g_1(z)=z$ and $g_2(z)=z^2$, the criterion predicts $(T_z,H^p)\\ne(T_{z^2},H^p)$, since $g_2'/g_1'=2z$ is not bounded with bounded reciprocal. Direct computation confirms it: $f(z)=1/(2z)$ satisfies $f g_2'=1\\in\\mathrm{Hol}(\\mathbb{D})$ and $T_{g_2}(f)=z\\in H^p$, but $f\\notin(T_z,H^p)$ because $f$ is not holomorphic. A pair of BMOA symbols achieving equality of domains without reciprocal bounded derivative factors would refute the characterization.","tokens_in":14825,"feed_emoji":"","tokens_out":12713,"duration_ms":106348,"temperature":0.7,"pith_summary":"For a BMOA symbol $g$, the paper studies the meromorphic optimal domain $(T_g,H^p)$: all meromorphic functions $f$ on the unit disc for which $f g'$ is holomorphic and the generalized Volterra integral $T_g(f)(z)=\\int_0^z f(\\zeta)g'(\\zeta)\\,d\\zeta$ lies in the Hardy space $H^p$. The central result is a complete equality criterion: for nonconstant $g_1,g_2\\in BMOA$, the two meromorphic optimal domains coincide if and only if $g_1' = k g_2'$ and $g_2' = (1/k)g_1'$ for some $k,1/k\\in H^\\infty$, equivalently $g_1=T_{g_2}(k)+g_1(0)$ and $g_2=T_{g_1}(1/k)+g_2(0)$. The engine is the space $W_g=T_g(H^\\infty)+\\mathbb{C}$ of symbols whose domain contains $(T_g,H^p)$, which turns domain equality into a check on bounded analytic factors of derivatives. The same comparison technique also yields information about the harder holomorphic optimal domains $[T_g,H^p]$ for locally univalent, polynomial, and Blaschke-type symbols.","feed_headline":"Optimal Volterra domains agree iff derivatives match by reciprocal factors","feed_subtitle":"For BMOA symbols, equality of the function spaces reduces to a bounded-invertible-factor condition on derivatives.","key_machinery":"The load-bearing object is the representation $(T_g,H^p)=\\{f\\in\\mathrm{Mer}(\\mathbb{D}): f=h'/g'\\text{ for some }h\\in H^p\\}$, taken over without proof from the Cesàro-operator template. This makes $T_g$ an isometric isomorphism from $(T_g,H^p)$ onto $H^p_0$, transferring separability, duality, and interpolation from the Hardy spaces. On top of it, the space $W_g=T_g(H^\\infty)+\\mathbb{C}$ characterizes exactly the symbols whose meromorphic optimal domain contains $(T_g,H^p)$, and the equality criterion follows by comparing $W_{g_1}$ and $W_{g_2}$: the two derivatives must differ by a factor $k$ with $k,1/k\\in H^\\infty$.","core_discovery":"The paper introduces the meromorphic optimal domain as the natural recipient of the generalized Volterra operator $T_g$ when $g\\in BMOA$ and $1\\le p<\\infty$: a meromorphic $f$ belongs to $(T_g,H^p)$ exactly when $f g'$ is holomorphic and $T_g(f)\\in H^p$. Because $g'$ can vanish, this is strictly larger than the holomorphic optimal domain $[T_g,H^p]$ studied earlier. The paper's main theorem, Theorem 2.3, states that $(T_{g_1},H^p)=(T_{g_2},H^p)$ for nonconstant BMOA symbols if and only if there exist $k_1,k_2\\in H^\\infty$ with $g_1=T_{g_2}(k_1)+g_1(0)$, $g_2=T_{g_1}(k_2)+g_2(0)$, and then necessarily $k_2=1/k_1$. In derivative form this says the two derivatives differ by reciprocal bounded analytic factors. The proof routes through Theorem 2.2, which identifies $W_g=\\{h\\in BMOA: (T_h,H^p)\\supseteq(T_g,H^p)\\}$ as $T_g(H^\\infty)+\\mathbb{C}$; equality of domains is equivalent to $W_{g_1}=W_{g_2}$.","pith_inferences":["The criterion effectively defines an equivalence relation on BMOA by $g_1'\\sim g_2'$ when $g_1'=k g_2'$ with $k,1/k\\in H^\\infty$; because $k_2=1/k_1$ is forced, the relation is simply 'derivatives differ by a bounded invertible analytic factor'.","The example $g(z)=z^2$ suggests that zeros of $g'$ are exactly what separate the meromorphic from the holomorphic theory; a natural next step is to ask whether the reciprocal-factor condition survives for holomorphic domains when $g'$ has zeros but can be factored through an $H^\\infty$ unit.","It would be interesting to test whether an analogous characterization holds for other Hardy-based spaces, such as weighted $H^p$ or Bergman spaces, where the same derivative representation and companion-operator estimates are available."],"forward_implications":["Since $W_g=T_g(H^\\infty)+\\mathbb{C}$ does not depend on $p$, equality of meromorphic optimal domains is $p$-independent: if it holds for one $1\\le p<\\infty$, it holds for all such $p$.","For $g\\in BMOA\\cap U_{loc}(\\mathbb{D})$, the meromorphic and holomorphic optimal domains coincide, so the reciprocal-factor criterion of Corollary 5.4 gives a complete answer to when $[T_{g_1},H^p]=[T_{g_2},H^p]$ in the locally univalent case.","Each $(T_g,H^p)$ is isometrically isomorphic to $H^p_0$ via $T_g$, so Hardy-space duality and interpolation transfer verbatim; in particular $((T_g,H^1),(T_g,H^\\infty))_{1-1/p,p}=(T_g,H^p)$.","Separability of $(T_g,H^p)$ forces separability of the holomorphic optimal domains $[T_g,H^p]$ and of the non-radial weighted Bergman spaces $A^2(|g'|^2(1-|z|^2))$, even in cases where polynomials are not dense.","For analytic polynomial symbols, equality with the classical Volterra domain $[T_z,H^p]$ is decided by zeros of $g'$: all zeros inside the disc give equality, while a zero on the boundary yields a strictly larger domain."],"supporting_citations":[{"why":"Supplies the Cesàro-operator representation whose line of proof is transferred to equation (3), the identifying description of $(T_g,H^p)$.","marker":"[11]"},{"why":"Defines the holomorphic optimal domain, proves its closed embedding in the meromorphic domain, and supplies the multiplier theorem used in the proof of Theorem 2.2.","marker":"[5]"},{"why":"Establishes that $T_g$ is bounded on $H^p$ exactly for BMOA symbols and that $T_g(H^\\infty)\\subset BMOA$, the ambient fact behind $W_g$.","marker":"[2]"},{"why":"Proves the companion-operator estimate used to show every $k\\in H^\\infty$ multiplies $(T_g,H^p)$ into itself.","marker":"[4]"},{"why":"Supplies density of polynomials in $H^p$ for separability and the integral estimate used in the polynomial-symbol lemma.","marker":"[13]"},{"why":"Establishes BMOA as the symbol class for which $T_g$ is bounded on $H^2$, setting the ambient space for the paper.","marker":"[18]"}],"fun_headline_variants":["Volterra domains equal iff derivatives are reciprocal factors","Meromorphic optimal domains: equality via reciprocal derivatives","When do Volterra domains match? When derivatives are reciprocal","Optimal domains agree iff derivatives differ by reciprocal factors","BMOA symbols: same optimal domain iff derivative reciprocity"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The entire framework rests on the unproved identification that a meromorphic $f$ belongs to $(T_g,H^p)$ exactly when $f=h'/g'$ for some $h\\in H^p$; if that identification failed, the isometric isomorphism, duality, interpolation, and equality results built on it would collapse.","fun_headline_variants_meta":{"raw":{"variants":["Volterra domains equal iff derivatives are reciprocal factors","Meromorphic optimal domains: equality via reciprocal derivatives","When do Volterra domains match? When derivatives are reciprocal","Optimal domains agree iff derivatives differ by reciprocal factors","BMOA symbols: same optimal domain iff derivative reciprocity"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000818,"raw_usage":{"total_tokens":3570,"prompt_tokens":923,"completion_tokens":2647,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":539,"completion_tokens_details":{"reasoning_tokens":2581}},"tokens_in":539,"tokens_out":2647,"duration_ms":23692,"temperature":1.0,"reasoning_tokens":2581,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T14:50:00.663369+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A concrete test of Theorem 2.3: with $g_1(z)=z$ and $g_2(z)=z^2$, the criterion predicts $(T_z,H^p)\\ne(T_{z^2},H^p)$, since $g_2'/g_1'=2z$ is not bounded with bounded reciprocal. Direct computation confirms it: $f(z)=1/(2z)$ satisfies $f g_2'=1\\in\\mathrm{Hol}(\\mathbb{D})$ and $T_{g_2}(f)=z\\in H^p$, but $f\\notin(T_z,H^p)$ because $f$ is not holomorphic. A pair of BMOA symbols achieving equality of domains without reciprocal bounded derivative factors would refute the characterization.","supporting_citations":[{"cited_title":"Extension of th e classical Ces´ aro operator on Hardy spaces","cited_arxiv_id":null,"evidence_quote":"Supplies the Cesàro-operator representation whose line of proof is transferred to equation (3), the identifying description of $(T_g,H^p)$."},{"cited_title":"Optimal domain of generalized Volterra operators","cited_arxiv_id":null,"evidence_quote":"Defines the holomorphic optimal domain, proves its closed embedding in the meromorphic domain, and supplies the multiplier theorem used in the proof of Theorem 2.2."},{"cited_title":"An integra l operator on H p","cited_arxiv_id":null,"evidence_quote":"Establishes that $T_g$ is bounded on $H^p$ exactly for BMOA symbols and that $T_g(H^\\infty)\\subset BMOA$, the ambient fact behind $W_g$."},{"cited_title":"Some closed range integral operators o n spaces of analytic functions","cited_arxiv_id":null,"evidence_quote":"Proves the companion-operator estimate used to show every $k\\in H^\\infty$ multiplies $(T_g,H^p)$ into itself."},{"cited_title":"Theory of H p Spaces","cited_arxiv_id":null,"evidence_quote":"Supplies density of polynomials in $H^p$ for separability and the integral estimate used in the polynomial-symbol lemma."},{"cited_title":"Schlichte Funktionen und analy tische Funktionen von beschr¨ ankter mittlerer Os- zillation","cited_arxiv_id":null,"evidence_quote":"Establishes BMOA as the symbol class for which $T_g$ is bounded on $H^2$, setting the ambient space for the paper."}],"review_version":1}