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Ultraholomorphic extension theorems in the mixed setting

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Pith's one-line read 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.

desk verdict A genuine unification of mixed ultraholomorphic extension theorems with new growth indices, worth a careful refereeing despite a couple of citation-dependent steps. read the letter →

arxiv 1908.06184 v1 pith:LEUXCPRS submitted 2019-08-16 math.CV math.FA

Javier Jiménez-Garrido , Javier Sanz , Gerhard Schindl This is my paper · ORCID
classification math.CVmath.FA MSC 26E1030D6046A1346E10
keywords ultraholomorphicclassesweightsequencesfunctionsmixedgrowthindicesorderofquasianalyticityBorelmapsectorialextensionsRoumieutype
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • 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.
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Editorial analysis

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Referee Report

2 major / 4 minor

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.

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 (2)
  1. [Section 3.1, Lemma 3.7; used in the proof of Theorem 5.12] 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.
  2. [Section 3.10, Proposition 3.19] 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.
minor comments (4)
  1. [Section 2.2, definition of γ(ω)] 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.
  2. [Theorem 6.4, statement] The displayed inclusion 'B(A{ˆΩ}(Sγ) ⊇ Λ{ˆΣ}' is missing a closing parenthesis; it should read B(A{ˆΩ}(Sγ)) ⊇ Λ{ˆΣ}.
  3. [Section 3.10, Lemma 3.20] 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.
  4. [Appendix A, Theorem A.3] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the mixed extension theorems are proved by explicit construction, and the cited equality in Lemma 3.7 is a prior-theorem dependency, not a self-referential reduction.

full rationale

The paper's main weight-function result (Theorem 5.7) is proved by an explicit construction using the flat functions Ga from Theorem 4.6, which are built from Lemma 4.4 following [14] and [32]; no step in that construction assumes the Borel-surjectivity conclusion. The mixed indices γ(M,N) and γ(σ,ω) are defined by independent growth conditions, and the extension theorems are stated for any γ below the index, so the result is not the definition renamed. The sequence case (Theorem 5.12) is obtained by applying Theorem 5.7 to associated weight functions and invoking Lemma 3.7 for the equality γ(M,N)=γ(ω_M,ω_N). That equality is the paper's most load-bearing cited input, but it is cited to the authors' [10, Cor. 4.6(iii)] and [12, Lemmas 5.8–5.9], where proofs are claimed; citing a prior theorem, even one's own, is not the same as assuming the target conclusion. If the equality were false or the cited proof required extra hypotheses, Theorem 5.12 would be unsupported, which is a correctness risk rather than circularity. The reductions to the unmixed cases (M=N, σ=ω) and the Appendix A examples are consistency checks, not inputs to the main proofs. Accordingly, no step in the derivation chain is equivalent to its own input by construction, and no fitted quantity is renamed as a prediction.

Assumptions & free parameters 1 free parameters · 5 assumptions · 5 invented entities

The central claim rests on standard assumptions about weight sequences and functions, plus a collection of prior results from the authors and others. The main new objects are the mixed growth indices and the associated orders of quasianalyticity. The paper does not introduce empirical free parameters; the listed parameters appear only in the construction of counterexamples.

free parameters (1)
  • γ and γ' in Appendix A examples
    Arbitrary positive reals chosen to satisfy inequality (A.1) and to produce sequences M,N with desired properties. These are construction parameters, not fitted to data.
assumptions (5)
  • domain assumption Weight sequences are assumed to belong to LC (normalized, log-convex, (M_k)^{1/k}→∞).
    Throughout Section 2; standard assumption in Denjoy-Carleman theory.
  • domain assumption Weight functions are continuous, nondecreasing, with ω(0)=0 and ω(t)→∞.
    Section 2.2.
  • standard math Prior extension theorems [14, Thm 7.4], [32, Thm 3.2.1], [11, Thm 6.12] are valid.
    The paper generalizes these results and uses their proofs as templates.
  • standard math The descendant construction [24, Lemma 4.2] yields a maximal sequence L_{N,r}.
    Used in Remark 3.14 and Theorem 6.2.
  • standard math The equality γ(M,N)=γ(ω_M,ω_N) when M has (mg), stated in Lemma 3.7.
    Quoted from [10, Cor. 4.6(iii)] and [12, Lemmas 5.8-5.9]; no proof in this paper.
invented entities (5)
  • Mixed growth index γ(M,N)
    purpose: Controls the maximum sector opening for mixed weight-sequence extension theorems.
    New definition in Section 3.1; used in Theorem 5.12.
  • Mixed growth index γ(σ,ω)
    purpose: Controls the maximum sector opening for mixed weight-function extension theorems.
    New definition in Section 3.1; used in Theorem 5.7.
  • Order of quasianalyticity μ(ω)
    purpose: Analog of μ(N) for weight functions; marks the upper limit of the mixed index and the onset of flat-function control.
    New definition in Section 3.10.
  • Descendant sequence L_{N,r}
    purpose: Optimal (maximal) weight sequence with respect to (L,N)γr, used in Theorem 6.2.
    From [24] via Remark 3.14.
  • Heir weight κ_ω^r
    purpose: Minimal weight satisfying (κ,ω)γr, used in Theorem 6.4.
    Defined in (3.12).

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Pith. "Pith review of Ultraholomorphic extension theorems in the mixed setting." pith.science (2026). https://pith.science/paper/LEUXCPRS

@misc{pith2026190806184,
  author       = {Pith},
  title        = {Pith review of: Ultraholomorphic extension theorems in the mixed setting},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LEUXCPRS}},
  note         = {Machine review of arXiv:1908.06184}
}
read the original abstract

The aim of this work is to generalize the ultraholomorphic extension theorems from V. Thilliez in the weight sequence setting and from the authors in the weight function setting (of Roumieu type) to a mixed framework. Such mixed results have already been known for ultradifferentiable classes and it seems natural that they have ultraholomorphic counterparts. In order to have control on the opening of the sectors in the Riemann surface of the logarithm for which the extension theorems are valid we are introducing new mixed growth indices which are generalizing the known ones for weight sequences and functions. As it turns out, for the validity of mixed extension results the so-called order of quasianalyticity (introduced by the second author for weight sequences) is becoming important.

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