{"id":"39983c23-2f1a-4e69-8f6a-cf1589bf1207","arxiv_id":"2411.15379","paper_version":2,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Holomorphic functions in the new mixed-Fourier-norm spaces on G×Y are exactly those whose Fourier transforms factor as e^{-2π⟨ξ,y⟩} times a distribution on Ĝ.","lead":"This paper develops a general framework for spaces of functions on a product G×Y where G is an Abelian Lie group acting freely, using Fourier analysis in the G-direction. Within this framework it characterizes the holomorphic functions by a simple factorization of their Fourier transform, unifying three standard models of the disc.","discovery_kind":"unification","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified: the central Fourier characterization is internally consistent, and the surjectivity and trivialization hypotheses are stated explicitly rather than hidden.","rationale":"The reader's verdict of ACCEPT with moderate confidence is appropriate. I scrutinized the main structural results: the Cauchy-Riemann distributional characterization (Lemma 7 and Proposition 4), the local-integrability lemma (Lemma 9), and the isometry/surjectivity statement (Proposition 6). The apparent concern about measure-dependence of weak derivatives resolves upon checking that the transpose operator ∂_y^µ acts on distributions identified via ν as the standard coordinate derivative; hence the kernel of ∂_y^µ is y-independence relative to ν, exactly as the proof requires. The only genuinely conditional part of the central description is the surjectivity of the isometry in Proposition 6, but the condition is stated explicitly and is satisfied in the worked examples. The global trivialization point raised by the reader is a genuine limitation of scope, not an internal inconsistency: the paper's theorems are formulated on G×Y, and the passage to a general domain is handled through explicit biholomorphisms in Section 7 and the examples. The hyperbolic example contains a likely typo (0,π vs 0,2π) and the ρ formula there would benefit from independent verification, but these are peripheral to the central claim. I therefore see no reason to change the verdict.","tokens_in":3,"tokens_out":52103,"duration_ms":1269714,"concrete_test":"Independently re-derive the final equivalence in Proposition 4 using the parabolic measure νλ=(λ+1)(2y)^λ dy with λ≠0: take u(x,y)=e^{-2πy}e^{2πix}, compute its half-Fourier transform, and verify that the distribution ∂_y^µ(e^{2πξy}û(ξ,y)) is zero iff û factorizes as e^{-2πξy}u0(ξ) with u0 independent of y. If a zero-order density term survives, the measure-independence claim fails; otherwise the central factorization step is confirmed.","verdict_should_be":"UNCHANGED","load_bearing_attack":"I read the central claim as Proposition 4 plus Proposition 6: a G-tempered distribution on G×Y satisfying the Cauchy-Riemann equations has a half-Fourier transform of the form e^{-2π⟨ξ,y⟩}u0(ξ), and for holomorphic elements of the mixed-Fourier-norm space the map u ↦ u0 is an isometry onto the weighted space Ξ(Ĝ,ρ). I checked the load-bearing steps. The transpose bookkeeping in Proposition 4 is measure-consistent: for a G-invariant measure dµ=dx dν with density a(y), the operator ∂_y^µ (defined by transposition) satisfies ∂_y^µ(a w)=a ∂_y w, so the kernel of ∂_y^µ is exactly y-independent distributions with respect to ν. Thus the factorization conclusion does not depend on the chosen smooth measure. Lemma 9 correctly extracts local integrability of u0 from Bochner membership of e^{-2π⟨·,·⟩}u0, and the norm identity in Proposition 6 follows by homogeneity of the Y-norm. The converse direction of Proposition 6 is explicitly conditional on the Bochner-measurability of ξ ↦ e^{-2π⟨·,ξ⟩}/ρ(ξ); this is a stated hypothesis, not a hidden gap. The global trivialization restriction is likewise explicit: the theory is developed on G×Y, and Section 7 explains how the examples embed G×Y into the target domain. The examples in Sections 8.2 and 8.3 are announced as sketches with details deferred, so the central theorem does not rest on their unproved parts. I do not find a load-bearing flaw.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops a general functional-analytic framework for mixed-Fourier-norm spaces on trivial principal bundles G×Y, where G is a connected Abelian Lie group and Y⊂R^n. The main objects are the G-tempered distributions D(G×Y)'=S(G)'\\hat⊗C_c^∞(Y)', the half-Fourier transform, and the spaces Ξ(Ĝ,Y(Y)) and X(G×Y) defined by requiring the Fourier image to take values in a Banach space Y(Y) with norm in Ξ(Ĝ). The central results are Proposition 4/Corollary 4, which characterize holomorphic G-tempered distributions by the factorization \\hat u(ξ,y)=e^{-2π⟨y,ξ⟩}\\hat u_0(ξ), and Proposition 6, which identifies the holomorphic subspace A_X(G×Y) isometrically with the weighted space Ξ(Ĝ,ρ), with an explicit Bochner-measurability hypothesis for surjectivity. Section 7 derives conditional Paley-Wiener support and boundedness properties, and Section 8 sketches the elliptic, parabolic, and hyperbolic models of the unit disc and half-plane.","tokens_in":33243,"tokens_out":29204,"duration_ms":280040,"significance":"The framework is substantial and mostly self-contained, and the central derivation is sound: Proposition 4 cleanly converts the Cauchy-Riemann equations, after half-Fourier transform and the appropriate measure transpose, into ∂_y(e^{2π⟨y,ξ⟩}\\hat u)=0, yielding the factorization. I checked the transpose bookkeeping and found it measure-consistent, and the norm identity in Proposition 6 is not circular because the holomorphic characterization is independent of the weighted norm definition. The paper is honest about its technical hypotheses: the trivialization of the domain as G×Y, the lattice/uniform-embedding assumptions, the open completeness questions, and the conditional surjectivity in Proposition 6 are all stated explicitly. The main limitation is scope: the theory is developed only on the product G×Y with a global complex slice, and Section 7's transfer to a general domain Ω is a collection of examples rather than a general theorem. This is a stated restriction, not a hidden error.","major_comments":[],"minor_comments":[{"comment":"The displayed formula for ρ in the parabolic example with X=L^p(R_+,ν_λ) appears to have an incorrect constant: for λ=0 the weight reduces to Lebesgue measure and ρ(ξ)=‖e^{-2πξ y}‖_{L^p(R_+)}=(2πξ p)^{-1/p} for ξ>0, whereas the formula as written gives ξ^{-1/p}. The membership conclusions are unaffected, but the explicit constant should be corrected.","section":"§8.2"},{"comment":"The strip is first described as Γ=R×(0,π), but a few lines later the text says y∈(0,2π)=Y, while the measure and all subsequent formulas use Y=(0,π). Please resolve the inconsistency.","section":"§8.3"},{"comment":"There are several typographical errors in this section, including 'comleteness' for 'completeness' and 'Bargman' for 'Bergman'; these should be corrected in the final version.","section":"§8.3"},{"comment":"The paper would benefit from stating more prominently in the introduction that the analysis requires a global trivialization G×Y and that nontrivial bundles, or domains without a global slice for the G-action, are not covered by the general framework; Section 7 only illustrates how the product case embeds into examples.","section":"§7 and Introduction"},{"comment":"The proof of Proposition 3 invokes a 'slight modification' of Theorem 2.30 in [1] for the completeness of L^0(Ĝ,Y(Y)); since this is a partial result, this is acceptable, but a precise reference or a brief indication of the modification would improve readability.","section":"§4, Proposition 3"}],"recommendation":"minor_revision","confidential_remarks":"The paper is a framework paper whose main deliverable is the general Fourier characterization; the examples in Sections 8.2 and 8.3 are announced as sketches with details deferred to future work. If the journal expects full development of the motivating applications, that is a scope issue, but the central theorems are sound as far as I can determine. I recommend minor revision rather than acceptance as-is because of the small errors in the examples and the need for clearer scope statements."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nRead this one. The core is a genuine generalization: instead of the disc with Fourier coefficients, the authors set up mixed-Fourier-norm spaces on G×Y for any connected Abelian Lie group G, with half-Fourier transform, and characterize the holomorphic subspace via the Cauchy–Riemann equations. Proposition 4 is the clean result: a G-tempered distribution satisfies CR iff its half-Fourier transform factorizes as e^{-2π⟨y,ξ⟩} û_0(ξ). The proof is self-contained, and the transpose bookkeeping is measure-invariant; I checked the step where ∂_y^μ(a w)=a ∂_y w, and it holds. Proposition 6 turns that into an isometry onto a weighted space on the dual group, with surjectivity conditional on an explicit Bochner-measurability hypothesis. The paper is honest about what is not settled: completeness of the general spaces is left open, and the global-trivialization assumption (G×Y rather than a non-trivial bundle) is stated plainly.\n\nWhat it does well: it gives a unified language for the elliptic, parabolic, and hyperbolic Bergman-type spaces; the elliptic example reproduces known results (a feature, not a bug); and the technical assumptions (uniform embeddings, lattice properties) are explicit rather than hidden. The examples for parabolic and hyperbolic are, however, sketches—the authors say the details will appear elsewhere. So the claimed unification is real at the level of framework, but the two non-elliptic cases are not yet fully worked out. That is the main soft spot, and it is a small one relative to the central theorem.\n\nMinor points: a couple of typos ('comleteness', 'Bargman') suggest a final proofread, and the norm on the Fourier side is to some extent chosen to make the isometry work, but the holomorphic characterization is substantive.\n\nVerdict: worth a serious referee. The central result is correct and the limitations are acknowledged. I would send it to review, and ask the authors to either complete or more clearly delimit the parabolic and hyperbolic examples in the next version.","headline":"A clean and honest framework paper whose central Fourier-characterization theorem is correct; the promised unification of the three disc geometries is real at the framework level but the parabolic and hyperbolic examples remain sketches.","tokens_in":33730,"tokens_out":5140,"would_cite":true,"duration_ms":48029,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["30H20","46E30","46E15"],"pacs":[],"model":"deepseek-v4-flash","headline":"A single Fourier profile decides holomorphic function spaces on domains with free Abelian group actions.","keywords":["mixed-Fourier-norm spaces","half-Fourier transform","holomorphic functions","Paley-Wiener theorem","Abelian Lie groups","tempered distributions","Bergman spaces","unit disc geometries"],"falsifier":"In the parabolic half-plane model with $\\mathcal Y(Y)=L^p(\\mathbb R_+,\\nu_\\lambda)$ and $\\Xi(\\hat G)=L^q(\\mathbb R)$, take $\\hat u_0(\\xi)=e^{-\\pi(\\xi-\\xi_0)^2}$ for some $\\xi_0>0$ and compute both sides of the claimed isometry $\\|F^{-1}(e^{-2\\pi\\langle\\xi,y\\rangle}\\hat u_0(\\xi))\\|_{L_{q;X}(\\Pi)}=\\bigl(\\int|\\hat u_0(\\xi)|^q\\rho(\\xi)^q\\,d\\xi\\bigr)^{1/q}$ using the paper's explicit formula for $\\rho$; any mismatch for some $p,q,\\lambda$ would refute Proposition 6.","tokens_in":32728,"feed_emoji":"🧮","tokens_out":11663,"duration_ms":107002,"temperature":0.7,"pith_summary":"This paper establishes a single Fourier-side description for function spaces on domains $G\\times Y$ carrying a free action of an Abelian Lie group $G$. Its central claim is that a distribution is holomorphic exactly when its half-Fourier transform factorizes as $\\hat{u}(\\xi,y)=e^{-2\\pi\\langle\\xi,y\\rangle}\\hat{u}_0(\\xi)$, so membership of a holomorphic function in a mixed-Fourier-norm space reduces to one weighted condition on the profile $\\hat{u}_0$ on the dual group. The authors prove this as an isometric isomorphism from the holomorphic subspace to a weighted Banach-type space $\\Xi(\\hat G,\\rho)$, and derive Paley-Wiener support and boundedness properties from the weight. In general the mixed-Fourier-norm space is only a topological cone, becoming a normed space under lattice-type assumptions, with completeness settled only partially. The payoff is a unified treatment of the elliptic, parabolic, and hyperbolic models of the unit disc, previously handled by separate arguments.","feed_headline":"Holomorphic functions reduce to one weighted Fourier profile","feed_subtitle":"A factorization identity turns holomorphic membership in mixed-Fourier-norm spaces into one weighted condition on u0.","key_machinery":"The carrying mechanism is the half-Fourier transform $F$ in the $G$-variable, defined on $D(G\\times Y)=\\mathcal S(G)\\hat\\otimes C_c^\\infty(Y)$ and extended by duality to $G$-tempered distributions. Under the product complex structure, the Cauchy-Riemann operator becomes $\\partial_x+i\\partial_y$, and Proposition 4 shows that the CR equations are equivalent to $\\partial_{y_i}(e^{2\\pi\\langle\\xi,y\\rangle}\\hat u)=0$, forcing the factorization $\\hat u(\\xi,y)=e^{-2\\pi\\langle\\xi,y\\rangle}\\hat u_0(\\xi)$. The mixed-Fourier-norm space $X(G\\times Y)$ is the Fourier preimage of $\\Xi(\\hat G,\\mathcal Y(Y))$, the space of maps $\\xi\\mapsto\\hat u(\\xi,\\cdot)$ whose $\\mathcal Y(Y)$-norm lies in $\\Xi(\\hat G)$; the weight $\\rho(\\xi)=\\|e^{-2\\pi\\langle\\cdot,\\xi\\rangle}\\|_{\\mathcal Y(Y)}$ turns the factorization into the membership condition $|\\hat u_0|\\rho\\in\\Xi(\\hat G)$. Proposition 6 asserts that $F_0:AX(G\\times Y)\\to\\Xi(\\hat G,\\rho)$ is an isometry, and Propositions 7 and 8 convert the growth of the weight into Paley-Wiener support and boundedness.","core_discovery":"On $G\\times Y$, with $Y\\subset\\mathbb R^n$ supplying the imaginary directions, the half-Fourier transform converts the Cauchy-Riemann equations $\\bar\\partial_{z_i}u=0$ into $e^{2\\pi\\langle\\xi,y\\rangle}\\hat u$ being independent of $y$. Consequently every $G$-tempered holomorphic distribution has Fourier transform $\\hat u(\\xi,y)=e^{-2\\pi\\langle\\xi,y\\rangle}\\hat u_0(\\xi)$ for a unique distribution $\\hat u_0$ on $\\hat G$, and this correspondence is a bijection between the intersection of the kernels of $\\bar\\partial_{z_i}$ and the class $e^{-2\\pi\\langle\\cdot,\\cdot\\rangle}\\cdot C_c^\\infty(\\hat G)'$. Within the mixed-Fourier-norm space $X(G\\times Y)=F^{-1}(\\Xi(\\hat G,\\mathcal Y(Y)))$, the holomorphic subspace $AX(G\\times Y)$ therefore consists exactly of those $u$ whose $\\hat u_0$ satisfies $|\\hat u_0|\\rho\\in\\Xi(\\hat G)$, where $\\rho(\\xi)=\\|e^{-2\\pi\\langle\\cdot,\\xi\\rangle}\\|_{\\mathcal Y(Y)}$; Proposition 6 makes $u\\mapsto\\hat u_0$ an isometry, and under a Bochner-measurability assumption an isometric isomorphism onto $\\Xi(\\hat G,\\rho)$. The same factorization yields the support property $\\operatorname{supp}\\hat u_0\\subset\\hat G_+$ and the boundedness property that allows extension from $G\\times Y$ back to the original domain by the classical holomorphic extension theorem.","pith_inferences":["A testable extension is a bundle-valued version of Proposition 6: the paper's global product assumption $G\\times Y$ is exactly where a nontrivial free Abelian action would break the factorization, and the authors do not show how the spaces glue across charts.","Because the construction defines spaces directly on the Fourier side instead of through square integrability, it offers a route to Bergman-type spaces when the Parseval identity is unavailable; the cost is that $\\Xi(\\hat G,\\mathcal Y(Y))$ is only shown closed under convergence in measure in general.","The Bochner-measurability hypothesis in Proposition 6(2) is the natural place to test the boundary between isometry and full isomorphism; checking it for Orlicz or Morrey-type $\\mathcal Y(Y)$ spaces would likely produce examples where surjectivity fails.","The explicit weights $\\rho$ computed for $L^p$ spaces make the support and boundedness conclusions quantitatively checkable, potentially yielding new endpoint cases in weighted Bergman space theory."],"forward_implications":["For a holomorphic function in a mixed-Fourier-norm space, the whole membership question collapses to one scalar condition: $|\\hat u_0|\\rho$ must lie in $\\Xi(\\hat G)$.","Whenever the map $\\xi\\mapsto e^{-2\\pi\\langle\\cdot,\\xi\\rangle}/\\rho(\\xi)$ is Bochner-measurable into $\\mathcal Y(Y)$, the Fourier transform is an isometric isomorphism onto $\\Xi(\\hat G,\\rho)$, and completeness of the latter makes $AX(G\\times Y)$ complete.","In the elliptic disc model with $\\Xi(\\hat G)=\\ell^q$ and $\\mathcal Y(Y)=X((0,1))$, the support property gives the description $A_{q;X}(\\mathbb D)=\\{f\\in\\operatorname{Hol}(\\mathbb D):\\{\\hat f_\\xi\\}_{\\xi\\ge 0}\\subset X((0,1)),\\ \\|f\\|<\\infty\\}$, with the classical extension theorem supplying values at the puncture.","The parabolic and hyperbolic half-plane models fit the same construction, with explicit weights $\\rho$ for $\\mathcal Y(Y)=L^p$: a power law that is $+\\infty$ on $\\xi\\le 0$ in the parabolic case, and a Gamma-function weight in the hyperbolic strip."],"supporting_citations":[{"why":"Supplies the Schwartz-Bruhat space and Paley-Wiener theorem for locally compact abelian groups, giving the Fourier isomorphism used for the half-Fourier transform.","marker":"[15]"},{"why":"Gives the wavefront-set regularity theorem used in Lemma 7 to pass from weak Cauchy-Riemann equations to holomorphy.","marker":"[6]"},{"why":"Introduces mixed-norm spaces of analytic functions in the disc; the elliptic example of the paper extends this theory.","marker":"[8]"},{"why":"Defines weighted holomorphic mixed-norm spaces in the disc via Fourier coefficients and supplies the Paley-Wiener results used as a model.","marker":"[11]"},{"why":"Characterizes weighted mixed-norm spaces of analytic functions by Fourier-coefficient conditions, supporting the Paley-Wiener support property.","marker":"[12]"},{"why":"Presents the commutative Toeplitz-algebra theory on the Bergman space that motivates the three disc geometries.","marker":"[19]"},{"why":"Supplies the completeness of convergence-in-measure spaces used in Proposition 3.","marker":"[1]"}],"fun_headline_variants":["Half-Fourier transform reduces holomorphy to one weighted profile","CR equations become a single weight check via half-Fourier","Mixed-Fourier holomorphy: one profile decides all","Holomorphic functions in mixed-Fourier spaces: one weight suffices","Half-Fourier view: holomorphy collapses to one weighted condition"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument assumes the domain is globally a product $G\\times Y$ with $Y\\subset\\mathbb R^n$ and that the target domain $\\Omega$ admits a bi-holomorphism $\\Phi:G\\times Y\\to\\Omega$ onto an open dense subset; if the free Abelian group action has no global slice, the factorization and Paley-Wiener conclusions are not established.","fun_headline_variants_meta":{"raw":{"variants":["Half-Fourier transform reduces holomorphy to one weighted profile","CR equations become a single weight check via half-Fourier","Mixed-Fourier holomorphy: one profile decides all","Holomorphic functions in mixed-Fourier spaces: one weight suffices","Half-Fourier view: holomorphy collapses to one weighted condition"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00115,"raw_usage":{"total_tokens":4794,"prompt_tokens":996,"completion_tokens":3798,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":612,"completion_tokens_details":{"reasoning_tokens":3707}},"tokens_in":612,"tokens_out":3798,"duration_ms":27121,"temperature":1.0,"reasoning_tokens":3707,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T14:21:51.237018+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"In the parabolic half-plane model with $\\mathcal Y(Y)=L^p(\\mathbb R_+,\\nu_\\lambda)$ and $\\Xi(\\hat G)=L^q(\\mathbb R)$, take $\\hat u_0(\\xi)=e^{-\\pi(\\xi-\\xi_0)^2}$ for some $\\xi_0>0$ and compute both sides of the claimed isometry $\\|F^{-1}(e^{-2\\pi\\langle\\xi,y\\rangle}\\hat u_0(\\xi))\\|_{L_{q;X}(\\Pi)}=\\bigl(\\int|\\hat u_0(\\xi)|^q\\rho(\\xi)^q\\,d\\xi\\bigr)^{1/q}$ using the paper's explicit formula for $\\rho$; any mismatch for some $p,q,\\lambda$ would refute Proposition 6.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the Schwartz-Bruhat space and Paley-Wiener theorem for locally compact abelian groups, giving the Fourier isomorphism used for the half-Fourier transform."},{"cited_title":"H¨ ormander","cited_arxiv_id":null,"evidence_quote":"Gives the wavefront-set regularity theorem used in Lemma 7 to pass from weak Cauchy-Riemann equations to holomorphy."},{"cited_title":"Karapetyants, S","cited_arxiv_id":null,"evidence_quote":"Introduces mixed-norm spaces of analytic functions in the disc; the elliptic example of the paper extends this theory."},{"cited_title":"Karapetyants, I","cited_arxiv_id":null,"evidence_quote":"Characterizes weighted mixed-norm spaces of analytic functions by Fourier-coefficient conditions, supporting the Paley-Wiener support property."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Presents the commutative Toeplitz-algebra theory on the Bergman space that motivates the three disc geometries."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the completeness of convergence-in-measure spaces used in Proposition 3."}],"review_version":1}