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REVIEW 3 major objections 7 minor 25 references

New Method of Smooth Extension of Local Maps on Linear Topological Spaces. Applications and Examples

T0 review · 3 major / 7 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read This paper proves that a C^q germ on a Banach space with a C^q blid map always has a global C^q representative, and derives differentiable linearization on spaces like C^q[0,1] that have no smooth bump functions.

desk verdict Useful blid-map framework, but this preprint is an announcement: the main theorems are deferred to prior papers, and the proof of the ZLZ linearization application has a genuine missing factor that leaves Theorem 4.3 unsupported as written. read the letter →

arxiv 1908.07713 v1 pith:CPQYDPOB submitted 2019-08-21 math.DS

Genrich Belitskii , Victoria Rayskin This is my paper · ORCID
classification math.DS MSC 46T2037C05
keywords blidmapssmoothextensionoflocalbumpfunctionsBanachspaceslineartopologicalBorellemmadifferentiablelinearizationcohomologicalequations
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

The paper proposes a replacement for smooth bump functions in infinite-dimensional local analysis. It defines a 'blid map'—a globally defined map that equals the identity in a neighborhood of zero and has bounded image—and proves that whenever a Banach space carries a C^q blid map, every C^q germ at zero into any Banach space has a global C^q representative. The same composition idea extends to linear topological spaces under a 'blid-differentiable' property. If the constructions are sound, this covers spaces such as $C^q[0,1]$ that have no smooth bump functions, and it supplies a Borel-type reconstruction of a map from its Taylor jet, global solutions of cohomological equations, and differentiable linearization at hyperbolic fixed points.

What carries the argument

The central object is the blid map (bounded local identity): a $C^q$ map $H:X\to X$ with $H(x)=x$ near $0$ and $\sup_x\|H(x)\|<\infty$. The basic mechanism is the one-line composition $F=f\circ H$, which turns a germ into a global map because $H$ pushes the whole space into the germ's domain while behaving as the identity near zero. For non-smooth spaces like $C^q[0,1]$, the paper constructs blid maps by an iterated-integral Taylor formula, e.g. $H(x)(t)=\sum_{j=0}^{q-1}\frac{t^j}{j!}h(x^{(j)}(0))x^{(j)}(0)+\int_0^t dt_1\cdots\int_0^{t_{q-1}} h(x^{(q)}(s))x^{(q)}(s)\,ds$, where $h$ is a real bump function; this operator is what gives $C^q[0,1]$ a smooth blid map despite having no smooth bump function. The same operator family also supports the Borel Lemma and the linearization argument.

What would settle it

Compute explicitly the Fréchet derivative of the operator $H$ in Example 2.7 on $C^q[0,1]$. If for some $x$ the derivative fails to exist or is unbounded as a linear operator even on bounded subsets, the claimed $C^\infty$ blid map is not delivered and the extension theorem's example collapses. A simpler check: test the claimed bound $\sup_x\|H(x)\|_k < a e^k$ in Lemma 3.7 numerically on a sequence of functions with growing derivatives.

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Extended reading notes

Core claim

The central claim is that for a Banach space $X$, the existence of a $C^q$ blid map $H$ is sufficient to globalize every $C^q$ germ at $0$. If $f$ is defined on a neighborhood $U$ of $0$ and $H(X)$ lies in $U$, the map $F(x)=f(H(x))$ is a global representative; boundedness of $H$'s derivatives transfers boundedness of $f$'s derivatives. The same argument, with 'differentiable local identity' in place of bounded local identity, proves Proposition 3.5 for linear topological spaces. In Section 4, the paper uses blid maps to prove a Borel Lemma (every sequence of continuous homogeneous polynomials is the Taylor jet of a smooth map) and to replace the smooth-bump-function hypothesis in the differentiable linearization theorem of [ZLZ] by a differentiable blid map with bounded derivative; the corollary gives differentiable linearization on $C^q[0,1]$.

Load-bearing premise

Every example that makes the theorems non-vacuous rests on the asserted smoothness and boundedness of the explicit blid constructions, especially the iterated-integral operator on $C^q[0,1]$ and its cousins in Lemmas 3.7, 3.9, and 3.11; if any of these is not differentiable to the claimed order with bounded derivatives, Theorems 2.3 and 4.3 lose their supporting examples.

Editorial extensions

If this is right

  • Every $C^q$ germ at $0$ on a Banach space with a $C^q$ blid map extends to a global $C^q$ map, with bounded derivatives when the local representative and blid map have bounded derivatives.
  • The Borel Lemma holds on any Banach space with a $C^\infty$ blid map whose derivatives of all orders are bounded: any prescribed sequence of homogeneous polynomials occurs as the Taylor jet of some $C^\infty$ map.
  • For a hyperbolic linear automorphism $A$, a formally solvable cohomological equation $g(Ax)-g(x)=f(x)$ has a global $C^\infty$ solution on spaces with a $C^\infty$ blid map with bounded derivatives.
  • Differentiable linearization at a hyperbolic fixed point holds on $C^q[0,1]$: the conjugating homeomorphism is differentiable at the fixed point, with the stated remainder estimate, even though no smooth bump function exists there.

Reading between the lines

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

  • Beyond the paper, the iterated-integral construction on $C^q[0,1]$ is likely adaptable to $C^q(M)$ for a compact manifold $M$, using local coordinates and a partition of unity, which would broaden Theorem 2.3's examples beyond intervals.
  • If smooth blid maps were built on $\ell^p$ with non-even $p$, the same theorems would immediately give global extension and differentiable linearization on those spaces; the paper leaves this as Question 5.1, so this is an extrapolation, not a claim.
  • The proof of Theorem 4.3 suggests a testable threshold: a blid map with only first-order bounded derivative suffices for the linearization estimate, so spaces that admit $C^1$ blid maps but not smoother ones may still linearize differentiably.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 7 minor

Summary. The paper introduces 'blid maps' (global bounded local identity maps) as a replacement for smooth bump functions in infinite-dimensional spaces. It claims (Theorem 2.3) that any C^q germ on a Banach space possessing a C^q blid map has a global C^q representative, with a bounded-derivative version when the blid has bounded derivatives. Section 3 extends the idea to linear topological spaces using bounded and compact differentiability, and gives examples on C^q(R), C^∞[0,1], and C^∞(R). Section 4 states a Borel lemma and a cohomological-equation theorem, both deferred to a companion paper, and gives a proof of a differentiable-linearization theorem (Theorem 4.3) intended to replace the smooth-bump assumption in the Zhang-Lu-Zhang theorem by the existence of a blid map. The paper concludes with additional examples and open problems.

Significance. The blid-map concept is simple and potentially useful: if the extension theorem and the examples are correct, it provides a localization tool for spaces such as C^q[0,1] that lack smooth bump functions, and it would improve the Zhang-Lu-Zhang linearization theorem by removing the smooth-bump hypothesis. The composition mechanism F=f∘H is transparent, and the concrete formulas in Examples 2.5-2.7 and Lemmas 3.7-3.11 are plausible. However, the central extension theorem and the Borel lemma are stated without proof and deferred to the authors' own prior work, and the proof of Theorem 4.3 has a serious structural gap. The paper is therefore not yet acceptable in its current form, though the underlying ideas appear worth pursuing.

major comments (3)
  1. [Section 4.3, Theorem 4.3] The proof of Theorem 4.3 does not establish the hypotheses of the Zhang-Lu-Zhang theorem. Setting f=DF-Λ, the proof defines \tilde f(x)=f(δH(x/δ)) and then verifies estimates for \tilde f. But condition (7.6) in [ZLZ] is a condition on the difference D\tilde F-Λ for some global map \tilde F, and no such \tilde F is constructed. The natural candidate \tilde F(x)=Λx+φ(δH(x/δ)) with φ=F-Λ gives D\tilde F(x)-Λ = f(δH(x/δ))DH(x/δ), not f(δH(x/δ)); the factor DH(x/δ) is absent from \tilde f and from the estimates. In addition, the displayed estimates bound ||D\tilde f(x)||, whereas (7.6) as reproduced in (4) bounds ||DF(x)-Λ|| = ||f(x)||; the function f need not be differentiable (the theorem assumes only that DF is α-Hölder), so D\tilde f may not exist. Thus the proof verifies a different object from the one needed, and Theorem 4.3 is not supported by the text as written. The argument can perhaps be repaired by working directly with φ(δH(x/δ)) and estimating f(δH(x/δ))DH(x/δ), but that repair is not present.
  2. [Theorem 2.3; Theorems 4.1 and 4.2] The manuscript's central extension mechanism is not proved here. Theorem 2.3 is stated with a reference to [BR1] and no proof, yet it is the basis for the claimed applications; Borel Lemma 4.1 is explicitly said to be proved in [BR1], and Theorem 4.2 is deferred to [BR1] as well. The one-line composition F=f∘H would prove the first statement of Theorem 2.3 if the definition of blid map were precisely fixed, but the boundedness assertion and the existence of the required blid maps for the examples are not demonstrated in this paper. Since these results are load-bearing for the paper's advertised contributions, the manuscript should either include their proofs or state exactly which hypotheses from [BR1] are being used and why the reader can rely on them.
  3. [Section 3, Lemmas 3.7-3.11 and Proposition 3.6] The blid-property examples in Section 3 are asserted rather than proved. In Lemmas 3.7, 3.9, and 3.11, formulas for H (or H_k) are displayed and it is stated without verification that each is a differentiable local identity with the stated boundedness; these lemmas are the only support for Corollaries 3.8, 3.10, and 3.12. The proof of Proposition 3.6 is also a sketch: the displayed inequality (1) is garbled, and the step from monotonicity of the norms to d(H_c(x),0)<c omits the necessary estimates of the first k+1 terms and of the tail. These gaps are probably fixable, but as written the extension claims for the metric spaces rest on unproved assertions.
minor comments (7)
  1. [Lemma 3.11] In the first sum, the index is p but the factorial is written as j!, so the expression is inconsistent; it should be p!.
  2. [References] The reference list ends with six unrelated-looking entries [1]-[6] on molecular biology and grid computing; these appear to be template artifacts and should be removed.
  3. [Section 1] The introduction refers to 'Section ??' for the open problems; this should be Section 5.
  4. [Keywords] The keywords contain 'map extinctions' instead of 'map extensions'.
  5. [Example 5.3] The text has 'C[1,0]' where C[0,1] is meant.
  6. [Proof of Proposition 3.6] The line 'k >1 − lnc/ ln 2(1)' does not display equation (1); the equation number should be attached to the displayed inequality.
  7. [Section 3.1] The phrase 'the space X=C^q(R) of all smooth functions' is imprecise for finite q; it should say 'q times continuously differentiable functions'.

Circularity Check

2 steps flagged · score 6.0 of 10

Theorem 4.3's proof verifies a substituted defect f(δH(x/δ)) without constructing any map whose derivative defect equals it, while the paper's key blid examples and Borel Lemma are deferred to the authors' own [BR1].

  1. other [Section 4.3, Proof of Theorem 4.3]
    "Let DF − Λ = f . Define ˜f (x) := f (δH (x/δ)). We will show that if f satisfies (7.6), then so does ˜f ."

    The ZLZ condition (7.6) is a condition on the derivative defect D F̃ − Λ of the modified map, not on an arbitrary composition. The proof substitutes f into the old defect, yet never constructs F̃ with D F̃ − Λ = f(δH(x/δ)). For the natural modification F̃(x)=Λx+φ(δH(x/δ)), φ=F−Λ, the derivative defect is f(δH(x/δ)) DH(x/δ), not f(δH(x/δ)); the factor DH is absent from the paper's definition, and f need not even be differentiable when DF is only α-Hölder. The displayed estimates also bound ||D tilde f||, whereas (7.6) requires a bound on ||tilde f||. Thus the proof verifies a different object from the one required by (7.6), so the linearization conclusion is not derived from the blid map in the written argument.

  2. self citation load bearing [Section 2, Examples 2.4–2.7; Section 4.1, Borel Lemma; Section 4.2, Theorem 4.2]
    "However, there are examples of Banach spaces that have blid-maps, but do not have bump functions of the same smoothness. We will illustrate this idea with the following examples (for details and proofs see [BR1]). ... In this section we state the Borel Lemma proved in [BR1]."

    The advertised examples (e.g., the C^q[0,1] blid map in Example 2.7), the Borel Lemma 4.1, and the solvability theorem 4.2 are load-bearing results of the paper, but their proofs are not contained here; each is referred to the authors' own prior publication [BR1]. No machine-checked, code-reproduced, or externally verified version is supplied, so the paper's extension and linearization claims for non-smooth spaces rest on an unverified self-citation chain rather than on a derivation in this text.

full rationale

The core extension mechanism, F=f∘H, is a one-line composition and is not circular by itself. However, the paper's advertised contributions to non-smooth spaces depend on unproved blid-map constructions and on a Borel Lemma whose proofs are deferred to the authors' own [BR1]; these are load-bearing self-citations. More seriously, the proof of Theorem 4.3 reduces the ZLZ condition (7.6) for a new map to the old defect f by defining tilde f = f(δH(x/δ)), but it never produces a map tilde F whose derivative defect is tilde f. Because ZLZ applies to derivative defects, the estimates in the proof verify the wrong object, and the factor DH is missing from any natural modification. This is a derivation gap in the central claim, not merely a stylistic self-citation. Score 6 reflects partial circularity: the blid-based linearization result is not established by the written argument, and the supporting existence results are imported from the authors' own prior work.

Assumptions & free parameters 0 free parameters · 4 assumptions · 1 invented entities

No empirical free parameters are fitted. The dependent assumptions are standard functional analysis, the external ZLZ theorem, and the authors' earlier Borel lemma. The one newly employed object is the blid map, which is given by explicit constructions in the example spaces.

assumptions (4)
  • standard math Frechet differentiability, the chain rule, and the standard local theory of differentiable maps on Banach spaces apply.
    Used throughout Section 2 and Section 4 without proof; this is standard functional analysis.
  • domain assumption The theorem of Zhang, Lu and Zhang (ZLZ Theorem 7.1) is valid, and condition (7.6) is sufficient for its conclusion.
    Section 4.3 uses ZLZ Theorem 7.1 as a black box; Theorem 4.3 is derived by showing condition (7.6) transfers to a modified map.
  • domain assumption Borel Lemma 4.1 for Banach spaces, stated as proved in [BR1], is correct and applicable.
    Section 4.1 and the applications in Sections 4.2 and 4.3 depend on this lemma, whose proof is not included in this preprint.
  • standard math The norm families defining C^q(R), C^infty[0,1], and C^infty(R) are complete, and the metric constructed from them induces the topology used.
    Section 3 relies on this to discuss neighborhoods, boundedness, and differentiability on these function spaces.
invented entities (1)
  • blid maps (bounded local identity maps) independent evidence
    purpose: Replacement for smooth bump functions in localization and extension of locally defined maps on Banach and linear topological spaces.
    The paper provides explicit constructions of blid maps on C[0,1], C^q[0,1], C^infty[0,1], and C^infty(R), which are concrete and checkable, even though existence on arbitrary Banach spaces is left open in Question 5.2.

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Pith. "Pith review of New Method of Smooth Extension of Local Maps on Linear Topological Spaces. Applications and Examples." pith.science (2026). https://pith.science/paper/CPQYDPOB

@misc{pith2026190807713,
  author       = {Pith},
  title        = {Pith review of: New Method of Smooth Extension of Local Maps on Linear Topological Spaces. Applications and Examples},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CPQYDPOB}},
  note         = {Machine review of arXiv:1908.07713}
}
abstract

The question of extension of locally defined maps to the entire space arises in many problems of analysis (e.g., local linearization of functional equations). A known classical method of extension of smooth local maps on Banach spaces uses smooth bump functions. However, such functions are absent in the majority of infinite-dimensional spaces. We suggest a new approach to localization of Banach spaces with the help of locally identical maps, which we call blid maps. In addition to smooth spaces, blid maps also allow to extend local maps on non-smooth spaces (e.g., $C^q [0, 1]$, $q=0, 1, 2,...$). For the spaces possessing blid maps, we show how to reconstruct a map from its derivatives at a point (see the Borel Lemma). We also demonstrate how blid maps assist in finding global solutions of cohomological equations having linear transformation of the argument. We present application of blid maps to local differentiable linearization of maps on Banach spaces. We discuss differentiable localization for metric spaces (e.g., $C^{\infty}(\R)$), prove an extension result for locally defined maps and present examples of such extensions for the specific metric spaces. In conclusion, we formulate open problems.

Discussion (0). Continue with ORCID to comment.

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