{"id":"6b6f4a6c-2d64-4b6c-809d-130d0bd044a6","arxiv_id":"1908.06184","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"Mixed ultraholomorphic extension theorems hold for sector openings controlled by new mixed growth indices γ(M,N) and γ(σ,ω), even when all unmixed indices vanish.","lead":"This paper proves extension theorems for ultraholomorphic function classes in a mixed setting, where the growth of a function and the growth of its derivatives are controlled by two different weights. It introduces new mixed growth indices that determine how wide the sectors can be, and it covers cases where the usual single-weight indices vanish.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 5.12 depends on the unproved equality γ(M,N)=γ(ω_M,ω_N) in Lemma 3.7; if the cited results do not cover it, the main weight-sequence theorem is unsupported.","rationale":"I agree with the reader's weakest assumption. The most fragile point is not the weight-function construction but the bridge between the two settings. The proof of Proposition 3.19 can be repaired by choosing r1 between r and μ(ω), so it is not the main issue. The equality in Lemma 3.7 is essential and unverified in this text; since Theorem 5.12 is a headline result, the paper should either prove or precisely cite a proof covering the real-parameter case. The weight-function main theorem 5.7 has a detailed proof and I see no reason to reject it on internal grounds. I therefore maintain the CONDITIONAL verdict: the paper may well be correct, but the transfer from weight functions to weight sequences rests on an unshown identification of growth indices.","tokens_in":32663,"tokens_out":13272,"duration_ms":125642,"concrete_test":"Obtain [10] and [12] and verify that the statements used in Lemma 3.7 hold for every real r>0 and under exactly the hypotheses M,N∈LC, μ_p≤ν_p, M (mg). Then independently rederive the implication (M,N)_γ^r ⇒ (ω_M,ω_N)_γ^r from the definitions of ω_M and the matrix Ω in §2.4, and check the reverse inequality in the (mg) case. As a computational sanity check, compute both γ(M,N) and γ(ω_M,ω_N) for the explicit sequences in Lemma A.1 (or A.2) with parameters satisfying (A.1), using the formulas for μ_k and the associated functions; if the values differ, the reduction in Theorem 5.12 collapses.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's main weight-sequence theorem (Theorem 5.12) is not proved directly; it is deduced from the weight-function theorem 5.7 by setting σ=ω_M, ω=ω_N and invoking Lemma 3.7 for the equality γ(M,N)=γ(ω_M,ω_N) when M has (mg). Lemma 3.7 is stated with a citation only: it refers to [10, Cor. 4.6(iii)] and [12, Lemmas 5.8–5.9] and gives no derivation. The two indices are defined by different mechanisms — a discrete supremum of quotient sums for γ(M,N), an integral kernel estimate for γ(ω_M,ω_N) — so the equality is not formal; it needs the full force of the associated-weight-function machinery. The rest of the paper (Remark 3.8, Theorem 5.12, Theorem A.3) repeatedly leans on this link. If the cited lemmas are proved only for integer ramification r∈N_{>0}, or require extra hypotheses such as (γ1)/(ω1), or if the equality fails for some admissible pair, then Theorem 5.12 and the claimed 'mixed weight sequence' corollary are unproved. This is the single most load-bearing unverified step.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops mixed ultraholomorphic extension theorems for Roumieu-type classes defined by two weight functions or two weight sequences. For normalized weight functions σ, ω satisfying (ω3) and γ(σ,ω)>0, Theorem 5.7 constructs, for every 0<γ<γ(σ,ω), a continuous linear right inverse for the asymptotic Borel map from the sequence space Λ^{Σhat} into the ultraholomorphic space A^{Ωhat}(Sγ), yielding the inclusion B(A{Ωhat}(Sγ)) ⊇ Λ{Σhat}. For weight sequences M,N with μp≤Cνp, M having (mg), and γ(M,N)>0, Theorem 5.12 obtains the analogous inclusion B(A{Nhat}(Sγ)) ⊇ Λ{Mhat}. The paper introduces mixed growth indices γ(M,N) and γ(σ,ω), an order of quasianalyticity μ(ω), and uses descendant/heir constructions to treat sectors beyond the unmixed growth index. Appendix A constructs pairs with γ(M,N)>0 while all unmixed indices γ(M), γ(N), γ(ωM), γ(ωN) vanish, showing that the results go beyond the authors' earlier unmixed theorems.","tokens_in":32955,"tokens_out":6838,"duration_ms":71149,"significance":"If the main theorems are correct, they provide a genuine extension of Thilliez's weight-sequence results and of the authors' weight-function results to a mixed setting, with a natural mixed index controlling the sector opening. The Appendix A examples are valuable: they show that the mixed statement has content even when every unmixed growth index is zero, so the paper is not merely a routine transposition of known arguments. The constructions are concrete and the main proofs in Sections 4-6 are largely detailed, in particular the flat-function estimates and the Laplace-type moment method leading to Theorem 5.7. The paper is honest about the places where it relies on earlier work, but two such places are load-bearing enough that the present manuscript is not yet fully self-contained.","major_comments":[{"comment":"Lemma 3.7 asserts, for M,N∈LC with μp≤νp, that γ(M,N)≤γ(ωM,ωN), and that equality holds when M has (mg). This statement is quoted from [12, Lemmas 5.8-5.9] and [10, Corollary 4.6(iii)] without reproducing the argument, but it is load-bearing: Theorem 5.12 is deduced exclusively by applying Theorem 5.7 to σ=ωM and ω=ωN and invoking the equality γ(M,N)=γ(ωM,ωN). The two indices are defined by different mechanisms, a discrete supremum over quotient sums for γ(M,N) and an integral estimate for γ(ωM,ωN), so the equality is not a formality. Please provide a proof, or state the precise cited result with all hypotheses, showing the equality for every r>0 under the assumptions used in Theorem 5.12; without this, the weight-sequence half of the paper's main claim is unsupported.","section":"Section 3.1, Lemma 3.7; used in the proof of Theorem 5.12"},{"comment":"The proof of Proposition 3.19 contains an invalid pointwise bound. From μ(ω)>r, i.e. ∫_1∞ ω(u)/u^{1+1/r}du<∞, the paper claims that there exist t0≥1, ε>0 and C≥1 with ω(t)≤Ct^{1/r-ε} for all t≥t0. This implication is false: for example, ω(t)=t^{1/r}/log^2(e+t) (suitably regularized to be continuous, nondecreasing, and vanish at 0) satisfies the integrability condition but grows faster than t^{1/r-ε} for every ε>0. Since this pointwise bound is the only argument supplied for the converse direction, the proof as written is incorrect. The proposition may still be true by a different argument, and it is used to identify μ(ω) as the upper bound for the values r satisfying (3.10), so the proof should be repaired or replaced.","section":"Section 3.10, Proposition 3.19"}],"minor_comments":[{"comment":"In the sentence 'γ(ω) = sup {r > 0 : M satisfies (ωγr)}', the symbol M should be ω; as written it introduces an undefined sequence in the weight-function context.","section":"Section 2.2, definition of γ(ω)"},{"comment":"The displayed inclusion 'B(A{ˆΩ}(Sγ) ⊇ Λ{ˆΣ}' is missing a closing parenthesis; it should read B(A{ˆΩ}(Sγ)) ⊇ Λ{ˆΣ}.","section":"Theorem 6.4, statement"},{"comment":"The phrase 'if even n∈LC holds true' is unclear because the lowercase sequence n was defined in Section 2.1 only informally; it should be explicitly identified as (Np/p!)p∈N before being used in the formula μ(ωN)=μ(N)=μ(n)+1.","section":"Section 3.10, Lemma 3.20"},{"comment":"In the second part of Theorem A.3, the paper states γ(ωM,ωN)≥γ(M,N) for sequences without (mg); it would be clearer to state whether equality is expected or open in this non-(mg) case, since the subsequent discussion relies on this distinction.","section":"Appendix A, Theorem A.3"}],"recommendation":"major_revision","confidential_remarks":"The editor may wish to verify that [12, Lemmas 5.8-5.9], which is an arXiv preprint at the time of writing, indeed proves the equality in Lemma 3.7 with the full generality and for all r>0 as used in Theorem 5.12. The manuscript is part of a series by the same authors, and the present contribution is incremental relative to that series, but the mixed indices and the Appendix A examples give it independent value if the cited equality is confirmed."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nYou should know this paper if you care about Borel maps in ultraholomorphic classes. The authors unify the weight-sequence and weight-function approaches to mixed extension theorems and introduce mixed growth indices γ(M,N) and γ(σ,ω) that genuinely sharpen what was known. The main theorems (5.7, 5.12, 6.2, 6.4) construct right inverses for the Borel map in sectors of opening up to π times the mixed index, and the Appendix A examples show this can happen even when all unmixed indices vanish. That is a real advance, not repackaging.\n\nThe proofs follow the authors' earlier machinery and are mostly careful. The flat-function construction in Section 4 adapts to the mixed setting cleanly. The idea that the order of quasianalyticity μ(N), μ(ω) bounds the mixed index, and the descendant/heir constructions give the optimal loss of regularity, are conceptually solid.\n\nThe soft spots are real but manageable. The load-bearing equality γ(M,N)=γ(ω_M,ω_N) for M with (mg) is stated in Lemma 3.7 with a citation to prior work, not proved. Since Theorem 5.12 deduces the weight-sequence result from the weight-function theorem via this equality, a referee should verify the cited [12, Lemmas 5.8–5.9] actually cover arbitrary r>0 and the exact hypotheses. I don't see evidence that they don't, but it is the first place I would push. The proof of Proposition 3.19 also has a sloppy bound: the split of the integral is not written correctly for small t. The result is true and easily fixed, but as written it is wrong.\n\nCitations are heavy on the authors' own work, but the cited results are prior published theorems with proofs elsewhere; I see no circularity.\n\nWho should read this: anyone working on ultradifferentiable or ultraholomorphic extension problems. It deserves a serious referee. Send it out, with a request to check Lemma 3.7 carefully and fix Proposition 3.19.","headline":"A genuine unification of mixed ultraholomorphic extension theorems with new growth indices, worth a careful refereeing despite a couple of citation-dependent steps.","tokens_in":33524,"tokens_out":3443,"would_cite":true,"duration_ms":31226,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["26E10","30D60","46A13","46E10"],"pacs":[],"model":"deepseek-v4-flash","headline":"For pairs of weight functions or sequences, the Borel map has continuous linear right inverses in sectors governed by a new mixed growth index, even when all unmixed indices vanish.","keywords":["ultraholomorphic classes","weight sequences","weight functions","mixed growth indices","order of quasianalyticity","Borel map","sectorial extensions","Roumieu type"],"falsifier":"Compute the two indices directly for a pair constructed as in Lemma A.1, one with moderate growth and one with $\\mu_p\\le C\\nu_p$; if $\\gamma(M,N)\\neq\\gamma(\\omega_M,\\omega_N)$, Lemma 3.7 fails and Theorem 5.12 loses its stated derivation. Alternatively, look for a continuous linear right inverse of the Borel map on a sector of opening slightly larger than $\\pi\\gamma(M,N)$: the theorems predict none exists, so any successful extension there would refute their sharpness.","tokens_in":32449,"feed_emoji":"📐","tokens_out":13439,"duration_ms":115627,"temperature":0.7,"pith_summary":"This paper establishes ultraholomorphic extension theorems in a mixed setting, where the size of the derivatives at the origin is controlled by one weight while the growth inside the sector is controlled by another. The central result, Theorem 5.7, states that for two normalized weight functions $\\sigma$ and $\\omega$ with mixed growth index $\\gamma(\\sigma,\\omega)>0$, every sector of opening $\\pi\\gamma$ with $0<\\gamma<\\gamma(\\sigma,\\omega)$ carries a continuous linear right inverse for the asymptotic Borel map between the associated weight-matrix classes; Theorem 5.12 transfers the same conclusion to weight sequences $M,N$ with $M$ of moderate growth. The interest is that the mixed index can be positive even when the unmixed indices $\\gamma(\\sigma)$, $\\gamma(\\omega)$, $\\gamma(M)$, $\\gamma(N)$ all vanish, so the results cover situations inaccessible to the single-weight theorems. A byproduct is a concrete role for the order of quasianalyticity $\\mu(\\omega)$ and $\\mu(N)$: it bounds the mixed index and, when only the weight defining the function space is fixed, controls the largest sector opening attainable with a suitably chosen smaller weight.","feed_headline":"Borel maps gain right inverses in two-weight classes","feed_subtitle":"A new mixed growth index, positive even when single-weight indices vanish, sets the sector opening.","key_machinery":"The load-bearing object is the mixed growth index: for weight functions $\\gamma(\\sigma,\\omega)=\\sup\\{r>0:\\exists C>0\\ \\forall t\\ge0\\ \\int_1^\\infty \\omega(tu)u^{-1-1/r}\\,du\\le C\\sigma(t)+C\\}$, and for weight sequences $\\gamma(M,N)=\\sup\\{r>0:\\sup_{p}\\frac{(\\mu_p)^{1/r}}{p}\\sum_{k\\ge p}\\nu_k^{-1/r}<\\infty\\}$. It measures how strongly the smaller weight $\\sigma$ or $M$ dominates the tail of the larger weight $\\omega$ or $N$, and it is exactly the quantity that bounds the sector opening $\\pi\\gamma$ for which extension is possible. The construction combines optimal flat functions $G_a$, produced by a Poisson-integral outer-function formula once the $r=1$ mixed condition holds, with their moment functions $m_a(\\lambda)=\\int_0^\\infty t^{\\lambda-1}e_a(t)\\,dt$; the two-sided moment estimates of Proposition 5.5 let the formal Borel transform of a given sequence converge and produce the extension operator. The associated weight matrices $\\Sigma=\\{S^x\\}$, $\\Omega=\\{W^x\\}$ and their factorial-twisted versions $\\hat\\Sigma$, $\\hat\\Omega$ translate the function-level theorem into classes of sequences, while a descendant construction supplies optimal smaller weights in the fixed-right-weight variants.","core_discovery":"The paper's central discovery is that the asymptotic Borel map $B$, sending an ultraholomorphic function to its sequence of derivatives at the origin, has continuous linear right inverses in a mixed weight-function and weight-sequence framework, with the admissible sector opening governed by a new mixed growth index. In the weight-function case (Theorem 5.7), for normalized weight functions $\\sigma,\\omega$ with $\\gamma(\\sigma,\\omega)>0$ and satisfying $(\\omega 3)$, for every $0<\\gamma<\\gamma(\\sigma,\\omega)$ there exists $k_0>0$ such that for every $x,h>0$ one can construct a continuous linear map $E:\\Lambda_{\\hat S^x,h}\\to A_{\\hat W^{8x},k_0h}(S_\\gamma)$ with $B\\circ E=\\mathrm{id}$, hence $B(A_{\\hat\\Omega}(S_\\gamma))\\supseteq\\Lambda_{\\hat\\Sigma}$. The weight-sequence version (Theorem 5.12) yields $B(A_{\\hat N}(S_\\gamma))\\supseteq\\Lambda_{\\hat M}$ for $M,N$ with $\\mu_p\\le C\\nu_p$, $M$ satisfying (mg), and $\\gamma(M,N)>0$. The proof builds sectorially flat functions whose modulus is trapped between a lower bound dictated by the input weight and an upper bound dictated by the output weight; their moments feed a Laplace-type formula that converts an arbitrary sequence into the desired holomorphic function. The paper also shows by explicit construction that the mixed index may be arbitrarily large while every unmixed index vanishes, so these theorems are genuinely new and not consequences of the single-weight results.","pith_inferences":["Inference: The equality $\\gamma(M,N)=\\gamma(\\omega_M,\\omega_N)$ for $M$ with moderate growth is the bridge that carries Theorem 5.12 across from Theorem 5.7; since Lemma 3.7 is only cited, a direct proof or explicit counterexample for this equality would settle exactly how far the sequence version stands on its own.","Inference: One can read the mixed-index mechanism as a quantitative trade-off: by asking the Taylor coefficients at the origin to obey a smaller weight, the sector opening gains access to the full interval up to the order of quasianalyticity; this suggests that surjectivity failures for large openings can be repaired by exactly the amount of regularity loss encoded in the index gap.","Inference: The same flat-function and moment construction should transfer to Beurling-type ultraholomorphic classes and to ramified variants, since the underlying mixed condition already has a ramified formulation in the ultradifferentiable literature; adapting the outer-function estimates would be the main step.","Inference: Condition (3.6) gives a testable criterion for when the optimal smaller weight itself has moderate growth; checking whether $N^{1/r}$ satisfies it for specific non-moderate-growth sequences like the one in Example 3.16 would reveal whether the loss of regularity can be kept within the moderate-growth class."],"forward_implications":["For any pair of normalized weight functions $\\sigma,\\omega$ with $\\gamma(\\sigma,\\omega)>0$ and $(\\omega 3)$, Theorem 5.7 gives a continuous linear map $E:\\Lambda_{\\hat S^x,h}\\to A_{\\hat W^{8x},k_0h}(S_\\gamma)$ with $B\\circ E=\\mathrm{id}$, so $B(A_{\\hat\\Omega}(S_\\gamma))\\supseteq\\Lambda_{\\hat\\Sigma}$ for every $0<\\gamma<\\gamma(\\sigma,\\omega)$.","For weight sequences $M,N\\in LC$ with $\\mu_p\\le C\\nu_p$, $M$ of moderate growth and $\\gamma(M,N)>0$, the same construction yields $B(A_{\\hat N}(S_\\gamma))\\supseteq\\Lambda_{\\hat M}$ for every $0<\\gamma<\\gamma(M,N)$, with the constant $k_1$ independent of $h$.","The mixed indices interpolate: $\\gamma(N)\\le\\gamma(M,N)\\le\\mu(N)$ and $\\gamma(\\omega)\\le\\gamma(\\sigma,\\omega)\\le\\mu(\\omega)$, so the extension range fills the entire gap between the usual growth index and the order of quasianalyticity whenever these differ.","When only the weight defining the function space is fixed, extension for sectors of opening up to $\\pi r$ with $r<\\mu(N)$ (resp. $r<\\mu(\\omega)$) is still possible by choosing an optimal smaller weight; this is Theorems 6.2 and 6.4.","The paper constructs explicit pairs of sequences with $\\gamma(M,N)$ arbitrarily large while $\\gamma(M)=\\gamma(N)=\\gamma(\\omega_M)=\\gamma(\\omega_N)=0$, so the mixed framework yields conclusions that the unmixed theorems cannot reach."],"supporting_citations":[{"why":"Supplies the flat-function and moment-estimate machinery, including the theorem that Theorem 5.7 directly generalizes.","marker":"[14]"},{"why":"Introduces the mixed growth conditions and the equality between sequence and weight-function indices cited in Lemma 3.7.","marker":"[12]"},{"why":"Provides the characterizations of growth indices used to identify the vanishing unmixed indices in the Appendix examples.","marker":"[10]"},{"why":"Contains the single-weight-sequence ultraholomorphic extension result that the mixed sequence theorem extends.","marker":"[32]"},{"why":"Gives the descendant sequence construction used to choose optimal smaller weights in the fixed-weight variants.","marker":"[24]"},{"why":"Introduces the order of quasianalyticity that bounds the mixed indices and sets the sector openings in Theorems 6.2 and 6.4.","marker":"[26]"},{"why":"Contains the mixed Borel-map surjectivity condition for ultradifferentiable classes that motivates the mixed index.","marker":"[31]"},{"why":"Supplies the Laplace-type transform and moment-function method used to write the right inverse explicitly.","marker":"[18]"},{"why":"Contains the mixed Whitney extension results whose growth condition appears as the $r=1$ case of the new indices.","marker":"[7]"}],"fun_headline_variants":["Mixed weight Borel maps get right inverses even when single indices fail","New mixed growth index opens sectors even when single-weight indices vanish","Ultraholomorphic Borel map gains right inverse via mixed weight indices","Borel map right inverse exists for mixed weights even if single-weight fails"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The weight-sequence theorem inherits its validity from the equality between the mixed index of two sequences and the mixed index of their associated weight functions whenever the smaller sequence has moderate growth; this equality is cited from earlier work and not proved in the paper.","fun_headline_variants_meta":{"raw":{"variants":["Mixed weight Borel maps get right inverses even when single indices fail","New mixed growth index opens sectors even when single-weight indices vanish","Ultraholomorphic Borel map gains right inverse via mixed weight indices","Borel map right inverse exists for mixed weights even if single-weight fails"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000956,"raw_usage":{"total_tokens":4105,"prompt_tokens":1004,"completion_tokens":3101,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":620,"completion_tokens_details":{"reasoning_tokens":3024}},"tokens_in":620,"tokens_out":3101,"duration_ms":20968,"temperature":1.0,"reasoning_tokens":3024,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T12:53:38.876189+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the two indices directly for a pair constructed as in Lemma A.1, one with moderate growth and one with $\\mu_p\\le C\\nu_p$; if $\\gamma(M,N)\\neq\\gamma(\\omega_M,\\omega_N)$, Lemma 3.7 fails and Theorem 5.12 loses its stated derivation. Alternatively, look for a continuous linear right inverse of the Borel map on a sector of opening slightly larger than $\\pi\\gamma(M,N)$: the theorems predict none exists, so any successful extension there would refute their sharpness.","supporting_citations":[{"cited_title":"Jiménez-Garrido, J","cited_arxiv_id":null,"evidence_quote":"Supplies the flat-function and moment-estimate machinery, including the theorem that Theorem 5.7 directly generalizes."},{"cited_title":"The surjectivity of the Borel mapping in the mixed setting for ultradifferentiable ramification spaces","cited_arxiv_id":"1904.04947","evidence_quote":"Introduces the mixed growth conditions and the equality between sequence and weight-function indices cited in Lemma 3.7."},{"cited_title":"Thilliez","cited_arxiv_id":null,"evidence_quote":"Contains the single-weight-sequence ultraholomorphic extension result that the mixed sequence theorem extends."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces the order of quasianalyticity that bounds the mixed indices and sets the sector openings in Theorems 6.2 and 6.4."},{"cited_title":"Schmets and M","cited_arxiv_id":null,"evidence_quote":"Contains the mixed Borel-map surjectivity condition for ultradifferentiable classes that motivates the mixed index."},{"cited_title":"Lastra, S","cited_arxiv_id":null,"evidence_quote":"Supplies the Laplace-type transform and moment-function method used to write the right inverse explicitly."},{"cited_title":"Chaumat and A.-M","cited_arxiv_id":null,"evidence_quote":"Contains the mixed Whitney extension results whose growth condition appears as the $r=1$ case of the new indices."}],"review_version":1}