{"id":"88dadac1-94ad-4578-84d1-c8ecd5905fc3","arxiv_id":"2502.04859","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"New explicit Beurling-Malliavin multipliers for Holder weights give the improved upper bound beta+ <= 3^{3/4}/4 L^2 ≈ 0.56987677 L^2 for fast boundary controls of the 1D Schrodinger equation.","lead":"This paper proves explicit Beurling-Malliavin multiplier estimates for weights whose logarithm is Holder continuous, and uses them to bound the cost of fast boundary controls for the 1D Schrodinger equation. The new upper bound 0.570L^2 improves the previous best 1.5L^2 and narrows the gap to the conjectured optimal 0.25L^2.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The Poisson-transform formulas in §2.4 are false as printed (missing square roots): P_tΩ(0)=√(πt), not √π t, so Theorem 2.4's proof cannot be checked as written.","rationale":"Read in good faith: the paper's plan is sensible and the control-theoretic application is a significant potential improvement. The Poisson smoothing recipe, the use of Theorem 2.2, and the moment-method reduction are all coherent. However, the burden of the quantitative result sits on the closed-form Section 2.4 identities, and those identities as printed are wrong. This is not merely a missing derivation; a one-line evaluation at x=0 disproves the displayed formula. Because the final constant 3^{3/4}/4 arises from the corrected identities, there is strong circumstantial evidence that the intended proof is repairable, and no external/consensus issue is at stake. But a paper whose key displayed computation is false cannot be accepted as-is. The reader's weakest-assumption flag is exactly the right spot, and our check sharpens it from 'unproved' to 'false as stated'. The verdict CONDITIONAL remains appropriate: the result should be accepted only after the formulas are corrected and the m=±1/4 issue is fixed.","tokens_in":17454,"tokens_out":19450,"duration_ms":182761,"concrete_test":"Evaluate (2.26) at x=0 for a specific value, e.g. t=4, by direct quadrature of the Poisson integral: P_4Ω(0)=(4/π)√(2π)∫_0^∞√y/(16+y²)dy = 2√π, whereas the printed formula gives 4√π. Alternatively, substitute y=tu in the defining integral and use ∫_0^∞√u/(1+u²)du=π/√2 to obtain P_tΩ(0)=√(πt). If the computation confirms √(πt), then (2.26) requires correction and the subsequent identities need re-derivation before Theorem 2.4 can be accepted.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In Section 2.4, for Ω(x)=√(2π)√x 1_{x>0}, equation (2.26) asserts P_tΩ(x)=√π(√(x²+t²)+x). At x=0 the defining formula (1.7) gives P_tΩ(0)=(t/π)√(2π)∫_0^∞√y/(t²+y²)dy=(t/π)√(2π)·π/(√2√t)=√(π t), whereas the displayed RHS is √π t. Hence (2.26) is not an identity. The same missing square root propagates: the displayed formula for Qtilde_tΩ just below (2.27) has the wrong x-dependence (its t-derivative at x=0 is independent of t, while a scale change in the defining integral gives a t^{-1/2} derivative), and the derivative formula (2.28) together with the claimed L∞ value √π 3^{3/4}/(4√t) is inconsistent with the printed (2.26) by a factor √t. A consistent set of formulas is P_tΩ=√π√(r+x), Qtilde_tΩ=-√π+√π√(r-x), and (H(P_tΩ))'=-√π√(r-x)/(2r), with r=√(x²+t²), which reproduces the claimed sup and the final constant. But these corrected formulas are not what the manuscript displays, and the unproved \"explicit computation\" cannot be checked as written. Since Theorem 2.4's lower bound feeds directly into (3.19)–(3.21) and hence into the control cost estimate of Theorem 3.1, the proof as submitted does not establish the central claim at the stated level of rigor. A secondary, independent typo: m=±1/4 in (3.20) lies outside the intervals where (3.19) supplies a lower bound; m should be ±3/4 for the division by Fψ(m) to be justified.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":17887,"tokens_out":9841,"duration_ms":98584,"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":[{"comment":"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.","section":"§2.4, Eq. (2.26)–(2.30)"},{"comment":"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.","section":"§2.4, Theorem 2.4"},{"comment":"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.","section":"§3.3, Eq. (3.20)"}],"minor_comments":[{"comment":"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.","section":"§1.1, Theorem 1.3"},{"comment":"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.","section":"§2.4, Eq. (2.25)"},{"comment":"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.","section":"§3.3, Eq. (3.28)"},{"comment":"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.","section":"§2.2, Eq. (2.20)"},{"comment":"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.","section":"§3.2, Theorem 3.1"}],"recommendation":"major_revision","confidential_remarks":"The paper is potentially publishable after the Section 2.4 formulas are corrected and proved, and after the statement of Theorem 2.4 is aligned with its proof. The m=±1/4 typo in (3.20) should also be corrected. I recommend requesting a major revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: the reader's conditional verdict is fair—maybe a bit generous, because the fix to the §2.4 formulas is not actually in the manuscript. The paper has a concrete false formula that undermines the proof as written, but the strategy is good enough to deserve a serious referee.\n\nThe genuinely new thing is the explicit BM1 multiplier for the one-sided exponential weight. Earlier explicit constructions gave even multipliers, which is wasteful here. Using Poisson smoothing to turn an α-Hölder log-weight into a well-prepared one (Theorem 2.3) is clean, and the control application gets a real improvement: β+ ≤ 0.5699 L² vs the previous 1.5 L². That is a big step toward the conjectured L²/4.\n\nNow the soft spots. Equation (2.26) is not an identity. Evaluating at x=0 by the defining integral gives √(πt); the displayed expression gives √π t. The correct formula appears to be P_tΩ(x)=√π√(r+x) with r=√(x²+t²). With that correction, the subsequent Q̃_t and (H(P_tΩ))′ expressions become consistent and do give the claimed sup and final constant—but that is not what the manuscript says. Since Theorem 2.4's lower bound feeds directly into (3.19)–(3.21), the proof as printed cannot be checked. I agree with the stress-test note: this looks like a typo, not a broken argument, but it has to be fixed.\n\nMinor issues: m=±1/4 should be ±3/4 to lie in the (−1,−1/2) or (1/2,1) intervals where the lower bound is valid; and Theorem 2.4 is stated for every T>0 but the proof only covers small T. The application only needs small T, so the overclaim is harmless, but the statement should be patched.\n\nFor whom: anyone working on quantitative Beurling–Malliavin or on cost of fast controls for 1D Schrödinger. The moment method part is standard, and the author is properly careful about where the lower bound is needed. Self-citations are modest and not problematic.\n\nRecommendation: send to a good referee in harmonic analysis or control. The errors look repairable, and the result is significant enough to justify the referee time. I would not desk-reject.","headline":"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.","tokens_in":18453,"tokens_out":4769,"would_cite":false,"duration_ms":45747,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["42A45","35Q41","42A70","44A15","93B05"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["Beurling–Malliavin multiplier theorem","Hölder continuous weights","modified Hilbert transform","Poisson transform","fast boundary controls","Schrödinger equation","moment method","cost of controllability"],"falsifier":"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.","tokens_in":17226,"feed_emoji":"⚛️","tokens_out":17705,"duration_ms":158190,"temperature":0.7,"pith_summary":"The paper sets out to turn the Beurling–Malliavin multiplier theorem into an explicit, quantitative statement for weights whose logarithm is Hölder continuous, and to use that statement to improve a concrete constant in control theory. The concrete payoff is a new upper bound $\\beta_+ \\le (3^{3/4}/4)L^2 \\approx 0.5699L^2$ for the exponential cost of fast boundary controls of the 1D Schrödinger equation on a segment of length $L$, replacing the previous best $3L^2/2$ and shrinking the gap to the conjectured lower bound $L^2/4$ by about three quarters. The argument replaces a general weight by its Poisson-smoothed version, whose modified Hilbert transform has controllable derivative, then applies a quantitative multiplier theorem for such “well-prepared” weights. For the specific weight arising in the Schrödinger control problem, the Poisson transform is explicit, which is what produces the sharp numerical constant. The interest is that explicit versions of the multiplier theorem are rare, and here the constants feed directly into a long-standing small-time controllability question.","feed_headline":"Fast Schrödinger controls now cost at most 0.57 L²","feed_subtitle":"Previous best was 1.5 L²; explicit Beurling–Malliavin multipliers shrink the gap to the conjectured L²/4.","key_machinery":"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 $\\varepsilon$.","core_discovery":"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 $\\beta_+\\le3^{3/4}L^2/4$.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Founds the multiplier theorem this paper makes quantitative and supplies the non-relaxable logarithmic-integral condition.","marker":"[1]"},{"why":"Provides the quantitative well-prepared-weight estimate and outer-function criterion that Theorem 2.2 builds on.","marker":"[14]"},{"why":"The stated source of Theorem 2.1, the $L^2$-based multiplier bound for well-prepared weights.","marker":"[16]"},{"why":"Introduces the modified Hilbert transform and its inversion formula, used throughout the smoothing argument.","marker":"[17]"},{"why":"Gives the two-step decomposition of the multiplier theorem and the outer-function sufficient condition that produces the support of $\\psi$.","marker":"[23]"},{"why":"Holds the previous fast-control upper bound that Theorem 3.1 improves, and supplies the $|b_k|$ lower bound needed by the moment method.","marker":"[29]"},{"why":"Establishes the lower bound $L^2/4$ that the new upper bound approaches.","marker":"[26]"},{"why":"Brings the moment-method reduction linking biorthogonal multipliers to boundary control cost.","marker":"[6]"}],"fun_headline_variants":["Explicit multipliers cut Schrödinger control cost to 0.57 L²","Schrödinger control cost down to 0.57 L² via explicit multipliers","Explicit Beurling–Malliavin gives Schrödinger control at 0.57 L²","Hölder logs yield explicit multipliers, Schrödinger cost 0.57 L²","Explicit multipliers: Schrödinger control cost 0.57 L²"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Explicit multipliers cut Schrödinger control cost to 0.57 L²","Schrödinger control cost down to 0.57 L² via explicit multipliers","Explicit Beurling–Malliavin gives Schrödinger control at 0.57 L²","Hölder logs yield explicit multipliers, Schrödinger cost 0.57 L²","Explicit multipliers: Schrödinger control cost 0.57 L²"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001276,"raw_usage":{"total_tokens":5323,"prompt_tokens":1157,"completion_tokens":4166,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":773,"completion_tokens_details":{"reasoning_tokens":4057}},"tokens_in":773,"tokens_out":4166,"duration_ms":29447,"temperature":1.0,"reasoning_tokens":4057,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-08T21:12:19.526625+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"On Fourier transforms of measur es with compact support","cited_arxiv_id":null,"evidence_quote":"Founds the multiplier theorem this paper makes quantitative and supplies the non-relaxable logarithmic-integral condition."},{"cited_title":"Fractal uncertainty principle with e xplicit exponent","cited_arxiv_id":null,"evidence_quote":"Provides the quantitative well-prepared-weight estimate and outer-function criterion that Theorem 2.2 builds on."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"The stated source of Theorem 2.1, the $L^2$-based multiplier bound for well-prepared weights."},{"cited_title":"A note on Hilbert transforms","cited_arxiv_id":null,"evidence_quote":"Introduces the modified Hilbert transform and its inversion formula, used throughout the smoothing argument."},{"cited_title":"The Beurlin g-Malliavin multiplier theorem: the seventh proof","cited_arxiv_id":null,"evidence_quote":"Gives the two-step decomposition of the multiplier theorem and the outer-function sufficient condition that produces the support of $\\psi$."},{"cited_title":"New blow-up rates for fast controls of Schr¨ odinger and heat equations","cited_arxiv_id":null,"evidence_quote":"Holds the previous fast-control upper bound that Theorem 3.1 improves, and supplies the $|b_k|$ lower bound needed by the moment method."},{"cited_title":"How violent are fast controls for Schr¨ odinger and pla te vibrations? Arch","cited_arxiv_id":null,"evidence_quote":"Establishes the lower bound $L^2/4$ that the new upper bound approaches."},{"cited_title":"Fattorini and David L","cited_arxiv_id":null,"evidence_quote":"Brings the moment-method reduction linking biorthogonal multipliers to boundary control cost."}],"review_version":1}