REVIEW 3 major objections 5 minor 1 cited by
Effective multipliers for weights whose log are H\"older continuous. Application to the cost of fast boundary controls for the 1D Schr{\"o}dinger equation
T0 review · 3 major / 5 minor · reviewed 2026-08-08 · deepseek-v4-flash
Pith's one-line read Quantitative Beurling–Malliavin multipliers for Hölder weights give a new fast-boundary-control bound $\beta_+ \le 3^{3/4}L^2/4 \approx 0.570L^2$ for the 1D Schrödinger equation.
desk verdict Significant improvement in the control-cost bound, but Section 2.4's closed-form Poisson formulas are wrong as printed—worth refereeing but not publishable without corrections. 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 load-bearing mechanism is the pair $(P_t,\widetilde Q_t)$ — the Poisson transform and the modified conjugate Poisson transform — together with the modified Hilbert transform $H$, an extension of the Hilbert transform adapted to $L^1(\mathbb{R},\langle x\rangle^{-2}dx)$. They are linked by the identity $H(P_t\Omega)=\widetilde Q_t\Omega+C_t(\Omega)$. A weight is called well-prepared when $\|(H\Omega)'\|_{L^\infty}$ is at most a constant multiple of the target exponential type; Theorem 2.2 then supplies $\psi$ with the required support and pointwise bounds. The new step is to show that $P_t\Omega$ is well-prepared whenever $\log\omega=-\Omega$ is Hölder continuous, with explicit control on the deviation $|P_t\Omega-\Omega|$. The outer-function criterion (Lemma 1.11) converts these bounds into a genuine Fourier transform supported in $[0,\sigma]$. In the Schrödinger example the exact derivative formula shows the supremum of $H(P_t\Omega)'$ is attained at $x=-t/\sqrt3$ and equals $\sqrt{\pi}3^{3/4}/(4\sqrt{t})$, which fixes the smoothing scale $t$ in terms of the control time $T$ and the slack parameter $\varepsilon$.
What would settle it
Check the closed forms numerically: for $\Omega(x)=\sqrt{2\pi x}\mathbf{1}_{x\ge0}$, evaluate $P_t\Omega(x)$ for several $t>0$ and compare with $\sqrt{\pi}(\sqrt{x^2+t^2}+x)$, and locate the maximum of $P_t\Omega-\Omega$; if the formula fails, or if the maximum exceeds $\sqrt{\pi t}$, the claimed constant $3^{3/4}/4$ does not follow.
Extended reading notes
Core claim
The central claim is that, for $\omega=e^{-\Omega}$ with $\Omega$ having a Hölder-continuous logarithm of exponent $\alpha\in(0,1)$, the Beurling–Malliavin multiplier can be produced with explicit two-sided pointwise bounds. For any $0<\sigma'<\sigma<1/10$ there is a nonzero $\psi\in L^2(\mathbb{R})$ with $\operatorname{supp}\psi\subset[0,\sigma]$, $|\mathcal{F}\psi|$ bounded above by a constant times $\omega$, and $|\mathcal{F}\psi(x)|\ge C(\sigma-\sigma')^6\omega(x)$ on one of the intervals $(-1,-1/2)$ or $(1/2,1)$, with all constants explicit in the Hölder constant $K_0$, $\alpha$, $\sigma$, and $\sigma'$. The proof smooths $\Omega$ to $P_t\Omega$ and shows that choosing $t=(K_0/(\pi\sigma'\cos(\pi\alpha/2)))^{1/(1-\alpha)}$ forces $\|(H(P_t\Omega))'\|_{L^\infty}\le\pi\sigma'$, so that the smoothed weight is well-prepared. For the one-sided weight $\Omega(x)=\sqrt{2\pi x}\mathbf{1}_{x\ge0}$, the exact formula $P_t\Omega(x)=\sqrt{\pi}(\sqrt{x^2+t^2}+x)$ and the derivative formula $H(P_t\Omega)'(x)=-\sqrt{\pi}(\sqrt{t^2+x^2}-x)/(2\sqrt{t^2+x^2})$ give the supremum $\sqrt{\pi}3^{3/4}/(4\sqrt{t})$ at $x=-t/\sqrt3$, which fixes $t=3\sqrt3/(16\pi(T(1-\varepsilon))^2)$. Feeding this into the moment method for the Schrödinger control problem yields $\beta_+\le3^{3/4}L^2/4$.
Load-bearing premise
The load-bearing premise is the unproved closed-form computation in Section 2.4, especially the formula $P_t\Omega(x)=\sqrt{\pi}(\sqrt{x^2+t^2}+x)$ and the claim that the maximum of $P_t\Omega-\Omega$ is attained at $x=0$; the constant $3^{3/4}/4$ in Theorem 3.1 rests on those identities.
Editorial extensions
If this is right
- For the 1D Schrödinger equation on a segment of length $L$, the fast-control cost exponent now satisfies $\beta_+\le0.5699L^2$, replacing the previous $3L^2/2$ and leaving a gap of about $0.32L^2$ to the conjectured optimum $L^2/4$.
- The boundary controls obtained are constructive: they are built from biorthogonal functions $\psi_l$ that are products of the canonical factor $P_l$ and a translated multiplier $\psi$, so the exponential rate comes with an explicit rational-in-$T$ prefactor.
- For every weight with $\alpha$-Hölder logarithm, the multiplier theorem now carries explicit two-sided pointwise bounds with constants depending only on $K_0$, $\alpha$, $\sigma$, and $\sigma'$, making quantitative comparison with any future construction possible.
- The same Poisson-smoothing strategy is pointed out in the paper as applicable to fractional Schrödinger and heat control problems, provided the moment-method product $P_l$ can be estimated in those settings.
Reading between the lines
- A direct numerical check could test the sharpness of the constant: for the one-sided square-root weight, compute the supremum of $P_t\Omega-\Omega$; a value strictly below the asserted $\sqrt{\pi t}$ would allow a smaller exponent in Theorem 3.1, while a value above it would invalidate the claimed bound.
- The translation step that selects $m=\pm1/4$ exploits the one-sided decay of the weight; a symmetric weight would pay the exponential on both sides, so part of the improvement likely comes from the asymmetry of the constructed multipliers.
- If a lower bound for $|\mathcal{F}\psi|$ could be obtained on the whole real line rather than one interval, the same machinery might reduce the heat-equation cost constant $0.6966L^2$; the paper identifies the whole-line lower bound as the blocking difficulty.
- The exact constant $3^{3/4}/4$ is an artefact of the explicit Poisson formulas for the specific weight; other weights with explicit Poisson and conjugate Poisson transforms may yield different sharp constants through the same scheme.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper proves a quantitative Beurling–Malliavin multiplier theorem for weights whose logarithm is Hölder continuous, by smoothing the weight using the Poisson transform and then applying a known "well-prepared" multiplier construction of Jin–Zhang. It then specializes to the one-sided weight e^{-sqrt(2π x_+)} and, using explicit Poisson/conjugate-Poisson computations, constructs a multiplier with a lower bound that leads, via the moment method, to the upper bound β_+ ≤ 3^{3/4}L^2/4 < 0.57 L^2 for the small-time boundary control cost of the 1D Schrödinger equation. The main structural argument is self-contained modulo standard harmonic analysis and the independent well-prepared theorem, and I found no circular reasoning. However, the explicit formulas in Section 2.4 are not correct as printed, and the proof of Theorem 2.4 is stated with greater generality than what is actually proved, so the main application is not fully established in the submitted form.
Significance. If the Section 2.4 computations are corrected and fully derived, the paper is significant: it gives one of the rarer quantitative, explicit versions of the Beurling–Malliavin multiplier theorem and provides a concrete improvement of the known upper bound for fast control costs, narrowing the gap to the conjectured L^2/4. The Poisson-smoothing strategy is elegant and plausibly transferable to other one-sided weights and PDE control problems. The moment-method application is carefully executed, with explicit L^2 estimates and a clean use of Hilbert's inequality. The main obstacle is the unverified and, as printed, incorrect explicit formulas in Section 2.4; the theorem statements also need to be aligned with the proofs.
major comments (3)
- [§2.4, Eq. (2.26)–(2.30)] The displayed formula P_tΩ(x)=√π(√(x²+t²)+x) is not an identity. At x=0, the defining formula (1.7) gives P_tΩ(0)=√(πt), whereas the displayed right-hand side equals √π t. The same scaling error propagates into the formulas for Qtilde_tΩ and (2.28); for instance, differentiating the printed (2.26) would give √π x/√(x²+t²), not the expression in (2.28). Consequently, the asserted sup norm ||(H(P_tΩ))'||_∞=√π 3^{3/4}/(4√t) and the bound P_tΩ(x)-Ω(x)≤√(πt) used in (2.29)–(2.30) are not established by the manuscript. Since (2.29)–(2.30) feed directly into the lower bound of Theorem 2.4 and then into (3.19)–(3.21) and Theorem 3.1, this is a load-bearing gap. The author should either correct and prove the intended formulas (which may well yield the claimed constant) or restrict the claims to what is actually verified.
- [§2.4, Theorem 2.4] Theorem 2.4 is stated for every T>0 and every ε∈(0,1), but the proof applies Theorem 2.2, which requires 0<σ'<σ<1/10, and the proof explicitly says "for any T>0 small enough and any ε>0 close enough to 0". Thus the stated generality is not proved. The Schrödinger application only needs small T, but the theorem as stated should be restricted, or the proof extended to cover the full claimed range.
- [§3.3, Eq. (3.20)] The point m=±1/4 used in the definition of g_l does not lie in either interval (-1,-1/2) or (1/2,1) on which the lower bound (3.19) is asserted, so the division by Fψ(m) in (3.20) is not justified by (3.19). The intended choice appears to be m=±3/4. This is a local correction, but it is essential for constructing the biorthogonal family, so it should be fixed.
minor comments (5)
- [§1.1, Theorem 1.3] Theorem 1.3 should explicitly assume 0<α<1, since the displayed constants contain (1-α)^{-1}; the current assumption "there exists α>0" is incomplete.
- [§2.4, Eq. (2.25)] The notation Ω(x)=√(2πx) 1_{(0,+∞)}(x) is misleading for x<0; writing Ω(x)=√(2π x_+) or using an indicator with the understanding that the square root is taken on the positive part would be clearer.
- [§3.3, Eq. (3.28)] In (3.28), the norm ||y0||^2_{H^{-1}(0,T)} should be ||y0||^2_{H^{-1}(0,1)}; the spatial interval is (0,1), not the time interval.
- [§2.2, Eq. (2.20)] Equation (2.20) contains an extra factor 1/π in the displayed integrand; since the following bound (2.21) is weaker, the final estimate is unaffected, but the displayed formula should be corrected for readability.
- [§3.2, Theorem 3.1] The constant in Theorem 3.1 should be typeset unambiguously, e.g. as √[4]{27}/4 or 3^{3/4}/4; the printed "4√27/4" is easy to misread.
Circularity Check
No significant circularity: the central claim is proved from independent external results and analytic estimates, not from the paper's own conclusions; self-citations are contextual only.
full rationale
The derivation chain is not circular. Theorem 1.3 is obtained by applying Theorem 2.2, whose statement and proof are explicitly imported from the independent external work of Jin and Zhang [14] and only slightly refined in the manuscript; no conclusion of the present paper is used in the hypotheses of that theorem. The transformation of a Hölder weight into a well-prepared weight in Theorem 2.3 is a direct estimate on the Poisson transform and the modified Hilbert transform using only the defining formulas (1.7), (1.14), (1.15), (1.22), and the Hölder hypothesis (1.3); it does not presuppose the multiplier whose existence is being proved. Section 2.4 is an application to a specific weight; its closed-form identities are analytic computations from the definitions, not fitted parameters, and the lower bound feeds into the moment-method construction of the controls through (3.19)-(3.25). The control cost bound in Theorem 3.1 is then a standard moment-method estimate using the canonical product P_l from (3.10)-(3.12); no target constant is inserted by hand. The author's self-citations [20], [21], and [22] concern lower bounds for the same equation or fractional variants and are used only as context or future-work discussion, not as the load-bearing premise for Theorem 3.1; the relevant prior comparison bounds are the external results of Tenenbaum-Tucsnak [29] and Miller [26]. Thus no step reduces, by construction or by self-citation, to its own input. One non-circularity caveat should be noted for completeness: the key closed-form formula (2.26) in Section 2.4 is introduced by the phrase 'An explicit computation shows' and is not derived, so the proof as printed contains an omitted computation; whether that computation is correct is a correctness and rigor issue, not a circularity issue, and it does not affect the circularity score.
Assumptions & free parameters
free parameters (3)
- Poisson smoothing scale t =
(K0/(πσ' cos(πα/2)))^{1/(1-α)}
- Auxiliary constant A in Theorem 2.2 =
6/(π(σ-σ'))
- Parameter ε in Theorem 2.4 and 3.1 =
arbitrary in (0,1), optimized to 0
assumptions (4)
- domain assumption Jin-Zhang well-prepared multiplier theorem (displayed as Theorem 2.1, cited as [16, Theorem 3.2]): for weights with L(ω)>-∞ and ||H(Ω)'|| ≤ π/(2σ), there is ψ with supp Fψ⊂[0,σ], |Fψ|≤ω, and the L2 lower bound (2.4).
- domain assumption Outer function criterion (Lemma 1.11 from Mashreghi, Nazarov and Havin's seventh proof of BM1): if ω^2 e^{2πiσx} is an outer function, there exists ψ∈L2 with supp Fψ⊂[0,σ] and |Fψ|=ω.
- domain assumption Moment method setup for the Dirichlet Laplacian on (0,1): eigenvalues π^2k^2, control coefficients |b_k|≥C, and the equivalence between null controllability and a biorthogonal family.
- standard math Standard Hilbert/Poisson transform identities: F(P_r)=e^{-2πr|ξ|}, F(Q_r)=-i sign(ξ)e^{-2πr|ξ|}, inversion H(H(f))=-f + constant, and the representation (1.16)-(1.22).
Cite this review
Pith. "Pith review of Effective multipliers for weights whose log are H\"older continuous. Application to the cost of fast boundary controls for the 1D Schr{\"o}dinger equation." pith.science (2026). https://pith.science/paper/M7GHND3K
@misc{pith2026250204859,
author = {Pith},
title = {Pith review of: Effective multipliers for weights whose log are H\"older continuous. Application to the cost of fast boundary controls for the 1D Schr\"odinger equation},
year = {2026},
howpublished = {\url{https://pith.science/paper/M7GHND3K}},
note = {Machine review of arXiv:2502.04859}
}
read the original abstract
We give a simple proof of the Beurling-Malliavin multiplier theorem (BM1) in the particular case of weights that verify the usual finite logarithmic integral condition and such that their log are H{\"o}lder continuous with exponent less than 1. Our proof has the advantage to give an explicit version of BM1, in the sense that one can give precise estimates from below and above for the multiplier, in terms of the exponential type we want to reach, and the constants appearing in the H{\"o}lder condition of our weights. The same ideas can be applied to a particular weight, that will lead to an improvement on the estimation of the cost of fast boundary controls for the 1D Schr{\"o}dinger equation on a segment. Our proof is mainly based on the use of a modified Hilbert transform together with its link with the harmonic extension in the complex upper half plane and some modified conjugate harmonic extension in the upper half plane.
Forward citations
Cited by 1 Pith paper
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Optimal cost of fast boundary controls for the one-dimensional heat equation
The exact small-time null-control cost for the 1D heat equation is exp((kappa_* L^2 + o(1))/T) with kappa_* = Gamma(1/4)^4 / (8 pi^3) approximately 0.6966.
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