REVIEW 4 minor 24 references
Functions with ultradifferentiable powers
T0 review · 0 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash
Pith's one-line read If two coprime powers of a function belong to a moderate Denjoy–Carleman class, then the function itself does.
desk verdict A solid, careful extension of Joris's theorem to Denjoy-Carleman classes under moderate growth, with a clean approximation characterization; the main result is new and the proof is essentially complete. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The key machinery is a characterization of $\mathcal{C}_M$ regularity by holomorphic approximation (Proposition 3.3.2): $f$ belongs to $\mathcal{C}_M$ on $[-1,1]$ exactly when, for each small $\varepsilon>0$, there is a function $f_\varepsilon$ holomorphic in the ellipse $\Omega_\varepsilon$ (the image of a strip under the sine map), uniformly bounded by a constant $K$, approximating $f$ on $[-1,1]$ with error $c_1h_M(c_2\varepsilon)$, where $h_M(t)=\inf_{j\ge0}t^jM_j$. To prove the theorem, one starts from holomorphic approximants $g_\varepsilon$ and $h_\varepsilon$ of $f^m$ and $f^{m+1}$, forms the quotient-like approximant $u_\varepsilon=\chi_\varepsilon g_\varepsilon h_\varepsilon/\max(|g_\varepsilon|,r_\varepsilon)^2$ with $r_\varepsilon=\delta_\varepsilon^{1/(m+1)}$, and controls its $\bar\partial$-derivative through an integral estimate on $|g'_\varepsilon|^2$ in the region where $|g_\varepsilon|<r_\varepsilon$. The moderate-growth condition enters as the inequality $h_M(t)\le h_M(\kappa_st)^s$, which converts the smallness $\delta_\varepsilon=c_4h_M(c_5\varepsilon)$ into approximation of $f$ with decay $h_M(c_2\varepsilon)$.
What would settle it
To test the role of the hypothesis, compute $h_M(t)=\inf_{j\ge0}t^jM_j$ for $M^\lambda_j=\exp(\lambda j^2/4)$ and check inequality (8) with $s=2$: it fails. With $g_\lambda(x)=\exp(-(\ln x)^2/\lambda)$ for $x>0$ and $g_\lambda(x)=0$ for $x\le0$, the germ $f=g_\lambda^{1/p}$ satisfies $f^p\in\mathcal{C}_{M^\lambda}(\mathbb{R},0)$ and $f^q\in\mathcal{C}_{M^\lambda}(\mathbb{R},0)$ for $p<q$, yet $f\notin\mathcal{C}_{M^\lambda}(\mathbb{R},0)$, as shown in Remark 2.2.3. A counterexample with a sequence that does satisfy (8) would falsify the theorem.
Extended reading notes
Core claim
The central result, Theorem 2.2.1, states that for a weight sequence $M=(M_j)_{j\ge0}$ with $M_0=1$, increasing, logarithmically convex, and satisfying the moderate-growth condition $M_{j+k}\le A^{j+k}M_jM_k$, the following holds for a complex-valued germ $f$ at the origin in $\mathbb{R}$: if $f^p$ and $f^q$ belong to the Denjoy–Carleman class $\mathcal{C}_M(\mathbb{R},0)$ for coprime positive integers $p$ and $q$, then $f$ belongs to $\mathcal{C}_M(\mathbb{R},0)$. The statement is local, so it applies to functions on any open interval. The paper also shows the hypothesis on two coprime powers is not an artifact: with a single power the conclusion fails even for strongly regular sequences, and without moderate growth it fails for the sequence $M^\lambda_j=\exp(\lambda j^2/4)$.
Load-bearing premise
The load-bearing premise is the moderate-growth condition (4) on the weight sequence $M$; every decay estimate in the proof passes through the equivalent inequality (8), and the paper shows the conclusion is false without it.
Editorial extensions
If this is right
- On any open subset of $\mathbb{R}$, the local conclusion gives: if $f^p$ and $f^q$ are in $\mathcal{C}_M$ with $\gcd(p,q)=1$, then $f$ is in $\mathcal{C}_M$.
- The proof shows a stronger one-pair reduction: it is enough that $f^m$ and $f^{m+1}$ belong to $\mathcal{C}_M$ for the single integer $m$ such that every integer $j\ge m$ is a nonnegative combination of $p$ and $q$.
- In the non-quasianalytic case, the same conclusion holds for germs in $\mathbb{R}^n$ (Corollary 2.2.5), by reducing the problem to one variable along suitable curves.
- The theorem applies to the standard factorial-type strongly regular weight sequences $M_j=(j!)^\alpha$, so for those sequences coprime powers in the class force the function into the same class.
- The moderate-growth and non-quasianalytic assumptions cannot simply be dropped: a single power fails even for strongly regular sequences, and the exponential-sequence example shows the two-power theorem fails without moderate growth.
Reading between the lines
- Editorial inference: the approximation characterization of Proposition 3.3.2 is likely a reusable tool for local questions; any regularity statement that can be phrased as uniform holomorphic approximation in the ellipses $\Omega_\varepsilon$ with $h_M$-error could be attacked the same way, for instance stability of $\mathcal{C}_M$ under other nonlinear maps.
- Editorial inference: tracking the constants through Lemmas 4.2.2 and 4.2.4 should yield a quantitative form of the theorem; the $\mathcal{C}_M$ constants of $f^p$ and $f^q$ determine the $\mathcal{C}_M$ constant of $f$ through the moderate-growth constant $\kappa_s$ and the integer $m$, and a reader needing such bounds could extract them from the proof.
- Editorial inference: the quasianalytic case is not reached by the curve-reduction argument; a natural next question is whether the quasianalytic analogue of the power theorem follows from the algebraic power-series route instead, since the paper notes that the quasianalytic case needs only derivation stability, not moderate growth.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper establishes a Denjoy-Carleman analogue of Joris's theorem. For a weight sequence M satisfying the moderate-growth condition (4), if a germ f at the origin in R has two powers f^p and f^q, with gcd(p,q)=1, belonging to the Denjoy-Carleman class C_M, then f itself belongs to C_M. The proof follows the 2018 MathOverflow proof by "fedja" of the C^∞ case, and is built on a new approximation-theoretic characterization of C_M regularity on an interval (Proposition 3.3.2), together with estimates for solutions of the Cauchy-Riemann equation (Lemma 3.1.1), a subharmonicity lemma (Lemma 3.2.3), the Hadamard three-lines theorem (Lemma 3.2.4), and the moderate-growth inequality (8). The paper also contains a counterexample showing that the moderate-growth assumption is necessary for the two-power theorem (Remark 2.2.3), a single-power counterexample for strongly regular sequences (Proposition 2.1.1), and corollaries for functions of several variables under a non-quasianalyticity assumption (Corollary 2.2.5).
Significance. The main theorem is a natural and nontrivial extension of a classical result of Joris, and it answers a question raised in [24] about the role of moderate growth in the Denjoy-Carleman setting. The proof is self-contained in its main line, with all lemmas proved in detail; Proposition 3.3.2, an approximation-theoretic characterization of C_M regularity, is likely to be of independent interest. The counterexamples are well chosen and demonstrate that the hypotheses are not idle. The paper is clearly written and the central argument is internally coherent; the issues I found are typographical and local, not load-bearing. Once the notational slips described below are corrected, the paper will be a solid contribution to the literature on ultradifferentiable functions.
minor comments (4)
- [Section 4.2, display after (35) and Lemma 4.2.3] The displayed definition of u_epsilon appears to be missing a conjugate on g_epsilon: to have u_epsilon = h_epsilon/g_epsilon on the set {|g_epsilon| > r_epsilon}, as used in Lemma 4.2.2, the numerator should be \bar{g}_epsilon h_epsilon rather than g_epsilon h_epsilon. Accordingly, in formula (41) of Lemma 4.2.3 the factor g'_epsilon should be \bar{g'}_epsilon. The subsequent estimates use only |g'_epsilon|, so the argument is unaffected, but the displayed formulas should be corrected for consistency.
- [Section 3.2, proof of Lemma 3.2.4] The stated value of the constant a3, namely a3 = max(a1^{1/2}, L^{1/2}), is not consistent with the immediately preceding estimate |g(w)| ≤ a1 (K h_M(a2 epsilon))^{1/2} with K = max(1, L/a1). The correct value is a3 = max(a1, (a1 L)^{1/2}). This is a typographical error in an explicit constant and does not affect the existence of the required a3 and a4.
- [Section 4.3, end of proof] The final step of the proof is somewhat compressed: after bounding the error by c12 delta_{2epsilon}^{1/s}, the paper uses the moderate-growth property to conclude |f - f_epsilon| ≤ c'_1 h_M(c'_2 epsilon). A one-sentence explanation of how the polynomial and logarithmic factors from Lemma 4.2.4 are absorbed by the fast decay of delta_epsilon (using h_M(t) = O(t^N) for every N) would improve readability.
- [Section 2.1, paragraph before (12)] There is a minor typo: "it is not difficult the check" should read "it is not difficult to check".
Circularity Check
No significant circularity: Theorem 2.2.1 is proved from scratch using external analytic tools.
full rationale
The main derivation chain of Theorem 2.2.1 is self-contained. The proof first reduces f^p, f^q in C_M(R,0) to f^m, f^{m+1} in C_M([-1,1]) via the algebra property of C_M (Section 4.1, Eq. (32)). It then invokes Proposition 3.3.2, whose forward direction is supplied by Dynkin's pseudanalytic extension theorem [5] and whose converse is proved directly by Cauchy estimates, the moderate-growth inequality (9), and convergence in the Banach space C_{M,\sigma}([-b,b]). The construction of holomorphic approximants in Section 4.2 uses only the assumptions on f^m and f^{m+1}, the estimates (38), and the external boundary estimates of Lemmas 3.1.1-3.2.4; no step assumes the conclusion f in C_M. Moderate growth (4) enters exactly where the proof must convert powers of h_M into h_M(C epsilon): in Lemma 3.2.4 via (8), in the factor (9) used at (31), and in the final comparison delta_{2epsilon}^{1/s} <= c h_M(c epsilon) in Section 4.3. These are consequences of the stated hypothesis, not of the target statement. The author's own earlier results appear only in the ancillary counterexample Proposition 2.1.1 (using [22, Lemma 3.6] and [21], plus [14] for reduction to one variable), not in the proof of the main theorem; consequently they are not load-bearing for the central claim. The cited equivalence (8) to moderate growth is attributed to an external source [16, Proposition 3.6]. Thus there is no self-definitional step, no fitted input relabeled as a prediction, and no self-citation chain forcing the conclusion. The paper's own Remark 2.2.3 demonstrates that the moderate-growth hypothesis is genuinely needed by exhibiting M^lambda classes where the conclusion fails, which further shows the proof is not vacuous. Minor notational and constant issues noted by a careful reader (a possible conjugate in u_epsilon and the exact value of a3) do not affect the existence of constants and do not constitute circularity.
Assumptions & free parameters
assumptions (4)
- standard math Dynkin's pseudoanalytic extension theorem provides a C^1 extension g of f with |bar-delta g(z)| <= c1 h_M(c2 dist(z,[-1,1])) for f in C_M([-1,1]).
- standard math Komatsu's equivalence: moderate growth (4) implies the inequality h_M(t) <= (h_M(kappa_s t))^s for any s >= 1.
- standard math For coprime positive integers p and q, the additive semigroup generated by p and q contains all sufficiently large integers.
- standard math Hadamard's three-lines theorem for bounded holomorphic functions on a strip.
Cite this review
Pith. "Pith review of Functions with ultradifferentiable powers." pith.science (2026). https://pith.science/paper/6HMPQCVW
@misc{pith2026190900177,
author = {Pith},
title = {Pith review of: Functions with ultradifferentiable powers},
year = {2026},
howpublished = {\url{https://pith.science/paper/6HMPQCVW}},
note = {Machine review of arXiv:1909.00177}
}
abstract
We study the regularity of smooth functions $f$ defined on an open set of $\mathbb{R}^n$ and such that, for certain integers $p\geq 2$, the powers $f^p :x\mapsto (f(x))^p$ belong to a Denjoy-Carleman class $\mathcal{C}_M$ associated with a suitable weight sequence $M$. Our main result is a statement analogous to a classic theorem of H. Joris on $\mathcal{C}^\infty$ functions: if a function $f:\mathbb{R}\to\mathbb{R}$ is such that both functions $f^p$ and $f^q$ with $\gcd(p,q)=1$ are of class $\mathcal{C}_M$ on $\mathbb{R}$, and if the weight sequence $M$ satisfies the so-called moderate growth assumption, then $f$ itself is of class $\mathcal{C}_M$. Various ancillary results, corollaries and examples are presented.
Reference graph
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Reviewed August 14, 2026 · model on record in the stance chip above.
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