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On the rate of convergence for Landau type Schr\"odinger Operators

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Pith's one-line read For $\gamma>1$, the paper proves that Landau type Schr\"odinger evolutions converge pointwise to their initial data in $W^{s,p}$ above explicit smoothness thresholds, and that the convergence rate along vertical lines is sharp at…

desk verdict A solid, workmanlike extension of Landau-type Schrödinger convergence results to W^{s,p} with a new negative L^p result; the rate theorems are conditional but a trivial maximal estimate covers the missing γ≤1 case, so the gap is presentation, not substance. read the letter →

arxiv 2505.24568 v1 pith:SCVZK5JE submitted 2025-05-30 math.AP

Yucheng Pan , Wenchang Sun This is my paper
classification math.AP MSC 46E35
keywords LandautypeSchr\"odingeroperatorpointwiseconvergencefractionalSobolevspaceraterestrictedcurvemaximalestimatedampedequation
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 studies whether the damped fractional Schr\"odinger evolution $P^t_{a,\gamma}f(x)=\frac{1}{2\pi}\int \hat f(\xi)e^{ix\xi}e^{it|\xi|^a}e^{-t^\gamma|\xi|^a}d\xi$ recovers its initial data pointwise as $t\to0^+$. The target is the regularity question in fractional Sobolev spaces $W^{s,p}(\mathbb R)$: how much smoothness $s$ is needed, and how fast can the recovery be in the rate sense. For $\gamma>1$ and $11$.

What carries the argument

The argument is carried by the maximal operator $P^*_{a,\gamma}f(x)=\sup_{0<t<1}|P^t_{a,\gamma}f(x)|$, whose boundedness converts into almost-everywhere convergence. The damping factor $e^{-t^\gamma|\xi|^a}$ is exploited through the identity $|e^{it|\xi|^a}e^{-t^\gamma|\xi|^a}-1|\lesssim t^{\min\{1,\gamma\}}|\xi|^a$, which seeds the rate estimates. To move from $W^{s,p}$ to the $H^\sigma$ results for $a>1$, the paper proves a weighted Hausdorff-Young type lemma, $\|(1+|\cdot|^2)^{s/2}\hat u\|_{L^{p'}}\lesssim\|u\|_{W^{s,p}}$, which transfers fractional Sobolev smoothness into weighted $L^{p'}$ decay of the Fourier transform. For the curve-dependent rates, a frequency-localization lemma from the literature is used to compare an oscillatory integral evaluated at $\Gamma(x,t)$ with a sum of integrals evaluated at nearby points $x+l/2^k$. The negative result uses a resonance-type lemma that promotes almost-everywhere convergence of a family of linear operators into weak-type bounds for the maximal operator; those weak-type bounds are then contradicted by a family of frequency-localized test functions.

What would settle it

Take $f_R$ with $\hat{f_R}=\chi_{[R,R+1]}$, as in the paper's necessity argument, and compute the liminf as $t\to0^+$ of $t^{-\delta\min\{1,\gamma\}/a}|P^t_{a,\gamma}f_R(x)-f_R(x)|$ on a positive-measure set. The paper's sharpness argument predicts this liminf is positive and forces $\delta_1\le\delta_2\min\{1,\gamma\}/a$; finding any $\delta\in[0,a)$ where the liminf is zero on a positive-measure set would falsify the claimed sharp rate. A second check is to exhibit, or rule out, an exponent $s_0$ satisfying (1.10) for some $0<\gamma\le1$, since Theorem 1.4 is empty without such an instance.

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Extended reading notes

Core claim

The central claim is a precise Sobolev threshold for almost-everywhere convergence of damped fractional Schr\"odinger evolutions. For $\gamma>1$ and $1<p\le2$, the paper proves that $\lim_{t\to0}P^t_{a,\gamma}f(x)=f(x)$ for almost every $x\in\mathbb R$ whenever $f\in W^{s,p}(\mathbb R)$ with $s>\frac{2-p}{2p}+\frac{a(1-1/\gamma)}4$ for $0<a<1$, $s>\frac1p-\frac{1}{2\gamma}$ for $a=1$, and $s>\frac{2-p}{2p}+\min\{\frac{a(1-1/\gamma)}4,\frac14\}$ for $a>1$. The paper also proves a negative counterpart: for $1\le p<2$ there exists $f_0\in L^p(\mathbb R)$ and a positive-measure set $E$ such that the limit fails for every $x\in E$, so the $W^{s,p}$ smoothness is not merely an artifact of the method. On the rate side, assuming the relevant maximal estimate holds, the paper shows that along a curve $\Gamma(x,t)$ of H\"older order $\beta$, the difference $P^t_{a,\gamma}f(\Gamma(x,t))-f(x)$ is $o(t^h)$ for the claimed ranges of $h$, and that along vertical lines the sharp rate is $o(t^{\delta\min\{1,\gamma\}/a})$ for $0\le\delta<a$. The sharpness discussion shows that larger rate exponents are impossible, including the universal statement that a nonzero Schwartz function cannot converge faster than $t^{\min\{1,\gamma\}}$ along vertical lines.

Load-bearing premise

The rate theorems assume, rather than prove, that some Sobolev exponent $s_0$ exists for which the maximal estimate (1.7)/(1.10) holds; for $\gamma>1$ this is cited from earlier work, but for $0<\gamma\le1$ the paper gives no supporting reference, so the rate results may have no known instance there.

Editorial extensions

If this is right

  • For $\gamma>1$, the a.e. convergence question in the $W^{s,p}$ scale is closed in one dimension: the displayed thresholds are sufficient, and the $L^p$ counterexample shows that dropping Sobolev smoothness entirely breaks convergence.
  • If the vertical-line rate $o(t^{\delta\min\{1,\gamma\}/a})$ is sharp, then no general method can observe faster reconstruction than $t^{\min\{1,\gamma\}}$ for highly regular data, because the universal barrier forces any nonzero Schwartz function with rate better than $t^{\min\{1,\gamma\}}$ to vanish.
  • The same proofs are claimed to extend to dimensions $n\ge2$, with the frequency interval replaced by a cube, so the rate threshold and the sharpness construction are not one-dimensional accidents.
  • For $0<\gamma\le1$, the rate theorems remain conditional on the existence of an exponent $s_0$ for which the maximal estimate holds; they would become unconditional only when such an $s_0$ is established.
  • The compatibility of the thresholds at $p=2$ with the previously known $H^s$ results means the new $W^{s,p}$ statement is a genuine interpolation of the $L^2$-based theory rather than a separate phenomenon.

Reading between the lines

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

  • The threshold formulas suggest that for $a>1$ the $W^{s,p}$ problem reduces to the $H^\sigma$ problem up to the Sobolev embedding gap $(2-p)/(2p)$; if that reduction is sharp, the known sharpness of the $H^\sigma$ exponent would transfer, making the Theorem 1.2 thresholds optimal up to the endpoint, although the paper does not explicitly claim this.
  • The case $0<a\le1$ in Theorem 1.2 is dispatched by a reduction to a cited corollary with the proof omitted; a careful reader would want that interpolation step written out, since the $p<2$ endpoint behavior is not automatic.
  • The sharpness construction uses indicator functions in frequency; a natural extension is to test whether the same rate barrier survives for smooth, well-localised Fourier symbols, where cancellation effects might produce faster decay for special data.
  • The curve version invites numerical checking on explicit examples such as $\Gamma(x,t)=x-t^\beta$: the barrier theorem predicts that the universal rate cannot exceed $t^{\min\{\beta,\gamma\}}$, which could be observed directly at high frequencies.
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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 / 5 minor

Summary. The paper studies the pointwise convergence of Landau type Schrödinger operators P^t_{a,γ} on the real line, defined by P^t_{a,γ}f(x) = (1/2π)∫ \hat{f}(ξ)e^{ixξ}e^{it|ξ|^a}e^{-t^γ|ξ|^a}dξ. The main unconditional results are Theorem 1.1, a negative result showing that for γ>1 and 1≤p<2 there is an L^p function for which the convergence fails on a positive measure set, and Theorem 1.2, a positive W^{s,p} convergence result for γ>1 and 1<p≤2 with thresholds that extend the earlier H^s results of Bailey and of Yuan–Zhao–Zheng. The paper also states two conditional convergence-rate theorems: Theorem 1.3 gives rates along Hölder curves and Theorem 1.4 gives the rate o(t^{δ min{1,γ}/a}) along vertical lines for f∈H^{s+δ}, both under a maximal estimate hypothesis (1.7)/(1.10). Section 2.4 contains a sharpness analysis: a frequency-localized counterexample shows the exponent in (2.42) cannot be improved, and Theorem 2.10 shows that for nonzero Schwartz functions the rate along vertical lines (or along x−t^β) cannot exceed t^{min{1,γ}} (or t^{min{β,γ}}). The appendix provides the proof of Theorem 1.4.

Significance. If the main theorems are correct, the paper delivers a natural generalization of the known H^s convergence results to the fractional Sobolev scale W^{s,p}, and it provides a systematic treatment of convergence rates for Landau type Schrödinger operators, including sharp exponents in the frequency-localized inequality and a universal lower bound for Schwartz functions. The proof of Lemma 2.5 is a self-contained alternative derivation of a useful Fourier estimate, and the counterexample in Theorem 1.1 is carefully constructed. The main weakness is that the rate theorems are conditional on maximal estimates whose existence is documented only for γ>1; for 0<γ≤1 the advertised sharp rate may have no instantiation. The significance is therefore real but somewhat narrower than the abstract suggests until that gap is closed.

major comments (3)
  1. [§1, Theorems 1.3 and 1.4] The rate theorems are stated for all γ>0, but the existence of the exponent s0 in the maximal estimates (1.7) and (1.10) is only supported for γ>1. The sentence at the end of Section 1 refers to [1, Theorem 1.2] and [23, Theorem 1.1], yet according to the paper's own introduction those results cover γ>1 (for a>1 and 0<a≤1 respectively). For 0<γ≤1 the introduction reports only an L^p convergence result from [23], which does not imply an H^s maximal estimate of the form (1.10). Consequently, for 0<γ≤1, Theorems 1.3 and 1.4 are vacuous unless an additional reference or proof is supplied. The authors should either restrict the statements to γ>1 or provide a proof/quote establishing (1.7)/(1.10) for 0<γ≤1.
  2. [§2.2, proof of Theorem 1.2] The case 0<a≤1 in Theorem 1.2 is dispatched by saying that it follows from [23, Corollary 1.2] with the proof omitted. Since Theorem 1.2 is a main result and the W^{s,p} thresholds in that range are part of the claim, the authors should either state the cited corollary precisely or include the short reduction (split f into low and high frequencies as in the a>1 case, apply Lemma 2.5 to control the high-frequency part in H^σ, and use the known H^s convergence from [23]). Without this, the reader cannot verify that the thresholds in Theorem 1.2 match those of the cited result.
  3. [§2.1, proof of Theorem 1.1] Property (2.11) is used to obtain the final contradiction, but it is only asserted to follow 'from the proof of [23, Theorem 1.3]' with no argument. This is a simple density fact for any measurable set E_ε with |E_ε|>3/4, but it should be stated as a small lemma and proved. As written, the proof of Theorem 1.1 depends on an unproved external assertion.
minor comments (5)
  1. [Throughout] There are several typographical errors: 'vaild' in the introduction, 'spilt' in Section 2.1, and 'fuction' in the statement of Theorem 1.3. These should be corrected.
  2. [§2.2, §2.3, Appendix] The displayed equations contain numerous typesetting issues, especially in the proof of Lemma 2.5 where the denominator '|x|^{sp′+p′/p}' appears garbled. The intended computation is clear, but the paper should be typeset carefully so the exponents are legible.
  3. [§2.1, Case 1] In the proof of Theorem 1.1, the definition of t is garbled: 'fixt=xθ a(k0+1)-1−k0/a' should read t = x θ^{a(k0+1)-1-k0}/a. The subsequent estimate in (2.6) depends on this definition and should be written out cleanly.
  4. [§2.4, Necessity of Theorem 2.9] The counterexample in the necessity discussion uses functions f_R whose H^{s+δ} norm grows like R^{s+δ}. Thus the argument proves optimality of the exponent in the frequency-localized inequality (2.42) on the family {f_R}, while Theorem 2.10 gives a separate universal lower bound for Schwartz functions. The paper should clarify the relationship between these two notions of sharpness and the actual convergence rate for a fixed f∈H^{s+δ}.
  5. [§1, Remarks] The statement 'the proofs of Theorem 1.3 and Theorem 1.4 are both applicable to higher dimensions' is given without any indication of the modifications needed. Either provide a brief explanation or mark this as a remark for the scalar case only.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main convergence extension (Theorem 1.2) reduces high frequencies to external H^s results via an independently proved embedding lemma (Lemma 2.5), and the rate theorems are explicitly conditional on stated maximal estimates rather than disguised identities.

full rationale

The paper's derivation chain is not circular. Theorem 1.2 is proved by decomposing f into a low-frequency part f1, handled by dominated convergence, and a high-frequency part f2, shown to lie in H^sigma by Lemma 2.5; convergence of f2 is then imported from the external benchmark results [1] and [23]. Lemma 2.5 is proved in the text via Hausdorff-Young and Littlewood-Paley-type estimates, so the W^{s,p} to H^sigma reduction is not an assumption equivalent to the conclusion. Theorem 1.3 and Theorem 1.4 are explicitly conditional statements: they assume the existence of s0 for which the maximal estimate (1.7) or (1.10) holds, and the appendix derives (1.11) from (1.10) by a dyadic frequency decomposition and Taylor estimates. The rate conclusion is not used to define the maximal-estimate hypothesis, so there is no equation reducing to itself. The sharpness discussion in Section 2.4 uses the independent test function f_R with frequency interval [R,R+1] and does not rename the theorem's conclusion as an input. The only self-citation, [17], appears in the introduction as contextual mention of a discrete analogue and is not used in any proof. A genuine completeness limitation, rather than circularity, is that the existence sentence for s0 in Theorem 1.4 refers to [1, Theorem 1.2] and [23, Theorem 1.1], which according to the paper's own introduction treat gamma>1; for 0<gamma<=1 the stated maximal-estimate hypothesis is therefore not instantiated by the cited results, leaving Theorem 1.4 conditional in that parameter range. This affects applicability but does not make the derivation circular.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

The central results add no free fitted parameters of their own, but the proof machinery pulls in several prior theorems and one unproved structural property from [23]. The main external input is the maximal estimate hypothesis s0, which the paper conditionally assumes.

free parameters (1)
  • k0 (choice of integer in Theorem 1.1 counterexample) = 2⌈γ/(min{a,1}(γ-1))⌉
    Hand-chosen to make the exponents a(k0+1)(γ-1)-2γ and related terms positive, so the phase and damping estimates in Section 2.1 close. It is an auxiliary proof parameter, not a fitted constant of the theorems.
assumptions (5)
  • standard math Standard Fourier analysis tools: Hausdorff-Young, Minkowski integral inequality, Littlewood-Paley theorem, dominated convergence, Nikishin's theorem
    Used throughout Section 2; Lemma 2.5 and Theorem 2.1 rest on these.
  • domain assumption Known convergence theorems of Bailey [1] and Yuan-Zhao-Zheng [23] for H^s (p=2)
    Theorem 1.2 reduces the a>1 case to [1, Theorem 1.3] and the 0<a≤1 case to [23, Corollary 1.2]; Theorems 1.3/1.4 cite [1, Theorem 1.2] and [23, Theorem 1.1] for the existence of s0.
  • domain assumption Existence of s0 making the maximal estimate (1.7)/(1.10) true
    This is stated as a hypothesis in Theorems 1.3 and 1.4. The paper does not prove it for 0<γ≤1, only citing prior work for γ>1.
  • standard math Lemma 2.7 (Lagrangian-type estimate for oscillatory integrals) taken from [13, Lemma 5.1]
    Used in Section 2.3 to control the short-time part of the convergence rate; it is quoted as a prior lemma without proof.
  • ad hoc to paper Property (2.11): for the Nikishin set E_ε with |E_ε|>3/4, for small λ there exists x0 with |E_ε∩[x0,x0+λ]|≥λ/2
    The paper says 'we can see from the proof of [23, Theorem 1.3]' but does not reproduce the argument. It is load-bearing for the contradiction in Theorem 1.1.

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Pith. "Pith review of On the rate of convergence for Landau type Schr\"odinger Operators." pith.science (2026). https://pith.science/paper/SCVZK5JE

@misc{pith2026250524568,
  author       = {Pith},
  title        = {Pith review of: On the rate of convergence for Landau type Schr\"odinger Operators},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SCVZK5JE}},
  note         = {Machine review of arXiv:2505.24568}
}
abstract

We study the pointwise convergence of Landau type Schr\"odinger operators within the fractional Sobolev space $W^{s,p}(\mathbb R)$. Our results extend those established by Bailey (Rev. Mat. Iberoam., 29 (2): 531-546, 2013) and Yuan, Zhao and Zheng (Nonlinear Anal., 208: Paper No. 112312, 28, 2021). Furthermore, we also analyze the convergence rate of Landau type Schr\"odinger operators along curves and derive a sharp result for the case of convergence along vertical lines.

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Works this paper leans on

24 extracted references · 22 canonical work pages

  1. [1]

    A. D. Bailey. Boundedness of maximal operators of Schr¨ odinger type with complex time.Rev. Mat. Iberoam., 29(2):531–546, 2013

    work page 2013

  2. [23]

    J. Yuan, T. Zhao, and J. Zheng. On the dimension of divergence sets of Schr¨ odinger equation with complex time.Nonlinear Anal., 208:Paper No. 112312, 28, 2021

    work page 2021

  3. [2]

    Bourgain

    J. Bourgain. A note on the Schr¨ odinger maximal function.J. Anal. Math., 130:393– 396, 2016

    work page 2016

  4. [3]

    Z. Cao, D. Fan, and M. Wang. The rate of convergence on Schr¨ odinger operator. Illinois J. Math., 62(1-4):365–380, 2018. 22 YUCHENG PAN AND WENCHANG SUN

    work page 2018

  5. [4]

    Carleson

    L. Carleson. Some analytic problems related to statistical mechanics. InEuclidean harmonic analysis (Proc. Sem., Univ. Maryland, College Park, Md., 1979), volume 779 ofLecture Notes in Math., pages 5–45. Springer, Berlin, 1980

    work page 1979

  6. [5]

    Cazenave, J

    T. Cazenave, J. a.-P. Dias, and M. Figueira. Finite-time blowup for a complex Ginzburg-Landau equation with linear driving.J. Evol. Equ., 14(2):403–415, 2014

    work page 2014

  7. [6]

    B. E. J. Dahlberg and C. E. Kenig. A note on the almost everywhere behavior of solutions to the Schr¨ odinger equation. InHarmonic analysis (Minneapolis, Minn., 1981), volume 908 ofLecture Notes in Math., pages 205–209. Springer, Berlin-New York, 1982

    work page 1981

  8. [7]

    Di Nezza, G

    E. Di Nezza, G. Palatucci, and E. Valdinoci. Hitchhiker’s guide to the fractional Sobolev spaces.Bull. Sci. Math., 136(5):521–573, 2012

    2012

Show all 24 references
  1. [8]

    X. Du, L. Guth, and X. Li. A sharp Schr¨ odinger maximal estimate inR 2.Ann. of Math. (2), 186(2):607–640, 2017

    2017

  2. [9]

    Du and R

    X. Du and R. Zhang. SharpL 2 estimates of the Schr¨ odinger maximal function in higher dimensions.Ann. of Math. (2), 189(3):837–861, 2019

    2019

  3. [10]

    Duan and P

    J. Duan and P. Holmes. On the Cauchy problem of a generalized Ginzburg-Landau equation.Nonlinear Anal., 22(8):1033–1040, 1994

    1994

  4. [11]

    T. Li, Y. Niu, and Y. Xue. The convergence rate for Schr¨ odinger operators with complex time.J. Korean Math. Soc., 61(6):1145–1170, 2024

    2024

  5. [12]

    Li and H

    W. Li and H. Wang. On convergence properties for generalized Schr¨ odinger operators along tangential curves.arXiv e-prints, page arXiv:2111.09186, Nov. 2021

    2021 arXiv

  6. [13]

    Li and H

    W. Li and H. Wang. A study on a class of generalized schr¨ odinger operators.Journal of Functional Analysis, 281(9):109203, 2021

    2021

  7. [14]

    E. M. Nikiˇ sin. A resonance theorem and series in eigenfunctions of the laplacian. Mathematics of the USSR-Izvestiya, 6(4):788, 1972

    1972

  8. [15]

    Niu and Y

    Y. Niu and Y. Xue. Estimates for Schr¨ odinger maximal operators along curve with complex time.J. Korean Math. Soc., 57(1):89–111, 2020

    2020

  9. [16]

    Niu and Y

    Y. Niu and Y. Xue. Maximal Schr¨ odinger operators with complex time.Ann. Funct. Anal., 11(3):662–679, 2020

    2020

  10. [17]

    Pan and W

    Y. Pan and W. Sun. Pointwise convergence of Landau type Schr¨ odinger operators. Bull. Iranian Math. Soc., 51(1):Paper No. 3, 13, 2025

    2025

  11. [18]

    Schr¨ odinger

    E. Schr¨ odinger. An undulatory theory of the mechanics of atoms and molecules. Physical review, 28(6):1049, 1926

    1926

  12. [19]

    Sj¨ olin

    P. Sj¨ olin. Regularity of solutions to the Schr¨ odinger equation.Duke Math. J., 55(3):699–715, 1987

    1987

  13. [20]

    Sj¨ olin

    P. Sj¨ olin. Maximal operators of Schr¨ odinger type with a complex parameter.Math. Scand., 105(1):121–133, 2009

    2009

  14. [21]

    E. M. Stein.Singular integrals and differentiability properties of functions, volume No. 30 ofPrinceton Mathematical Series. Princeton University Press, Princeton, NJ, 1970

    1970

  15. [22]

    B. G. Walther. Higher integrability for maximal oscillatory Fourier integrals.Ann. Acad. Sci. Fenn. Math., 26(1):189–204, 2001

    2001

  16. [24]

    J. Yuan, T. Zhao, and J. Zheng. Pointwise convergence along non-tangential direction for the Schr¨ odinger equation with complex time.Rev. Mat. Complut., 34(2):389–407, 2021. ON THE RATE OF CONVERGENCE FOR LANDAU TYPE SCHR ¨ODINGER OPERATORS 23 School of Mathematical Sciences ...

    2021

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