{"id":"bcb4a711-9ae1-4399-9e7e-ebf874f504a6","arxiv_id":"2505.24568","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"For Landau type Schrödinger operators, pointwise convergence holds in W^{s,p} above explicit regularity thresholds, and along curves the rate is o(t^h) for h bounded by δ min{1,γ}/a, a result that is sharp for vertical lines.","lead":"This paper studies when Landau type Schrödinger operators, which add a damping term to the usual Schrödinger evolution, return to their starting function as time tends to zero, and how fast the return happens along curves. It proves new regularity thresholds in fractional Sobolev spaces and sharp convergence rates for the vertical line case.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Rate theorems for 0<γ≤1 rest on an H^s maximal estimate whose existence is only asserted via [1],[23]; if those refs do not cover γ≤1, Theorem 1.4 is vacuous there.","rationale":"I read the paper in good faith and checked the main proofs. Theorem 1.2's reduction of W^{s,p} to H^σ via Lemma 2.5 is correct: the Minkowski step in Lemma 2.5 is valid for r=p'/p≥1, and the Hölder step in (2.15) gives f2∈H^σ with the stated thresholds. The only real gap there is the omitted case 0<a≤1, which is a routine adaptation. The proof of Theorem 1.1 (failure in L^p) is coherent: the counterexample and the use of Nikishin's theorem are standard, and the density property (2.11) is elementary. The rate proofs (Theorems 1.3, 1.4, Appendix A) are internally consistent: the frequency-localized estimates cancel 2^{-akj} against |ξ|^{aj}, and the choice of auxiliary δ is possible exactly when h<min{β,γ}. The sharpness argument in Section 2.4 correctly shows δ1≤δ2 min{1,γ}/a. The one load-bearing issue is that the rate theorems are conditional on a maximal estimate whose known instantiations are cited only for γ>1. For 0<γ≤1, the paper gives no H^s maximal estimate, only an L^p convergence statement from [23]. If the cited [23, Theorem 1.1] in fact covers γ≤1, the concern dissolves; if not, the theorem should be qualified. This matches the reader's weakest assumption and supports the CONDITIONAL verdict without escalating to rejection.","tokens_in":22069,"tokens_out":40659,"duration_ms":443234,"concrete_test":"Retrieve [1, Theorem 1.2] and [23, Theorem 1.1] and verify whether either provides the H^s maximal estimate (1.10) for 0<γ≤1 (and the relevant a>0). If neither does, restrict Theorem 1.4 to γ>1, where s0 is known, or add an explicit derivation/reference for γ≤1; otherwise the theorem is vacuous for 0<γ≤1. As a secondary check, write out the omitted reduction for 0<a≤1 in Theorem 1.2 following the a>1 argument, using [23, Cor 1.2] after Lemma 2.5.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The most load-bearing weakness is that Theorem 1.4 (and Theorem 1.3) are stated for all γ>0, conditional on an exponent s0 for which the maximal estimate (1.10)/(1.7) holds, but the only existence evidence is the sentence 'For the existence of s0 in Theorem 1.4, we refer to [1, Theorem 1.2] and [23, Theorem 1.1]' (end of Section 1). Those citations are known for γ>1: Bailey [1] covers a>1, γ>1, and Yuan-Zhao-Zheng [23] covers 0<a≤1, γ>1. For 0<γ≤1 the paper itself reports only an L^p convergence result from [23] (Introduction), which is not an H^s maximal estimate of the form (1.10). Hence the advertised sharp rate for vertical lines may have no instantiation for 0<γ≤1. The theorems are implications, so not formally false, but the central claim that a sharp rate result is derived is unsupported in that parameter range; the sharpness discussion in Section 2.4 inherits the same conditional structure. This is a completeness gap, not an internal inconsistency, but it is load-bearing for Theorem 1.4's applicability.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":22373,"tokens_out":27545,"duration_ms":284183,"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":[{"comment":"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.","section":"§1, Theorems 1.3 and 1.4"},{"comment":"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.","section":"§2.2, proof of Theorem 1.2"},{"comment":"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.","section":"§2.1, proof of Theorem 1.1"}],"minor_comments":[{"comment":"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.","section":"Throughout"},{"comment":"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.","section":"§2.2, §2.3, Appendix"},{"comment":"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.","section":"§2.1, Case 1"},{"comment":"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}.","section":"§2.4, Necessity of Theorem 2.9"},{"comment":"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.","section":"§1, Remarks"}],"recommendation":"major_revision","confidential_remarks":"The paper contains a solid Fourier-analytic core, particularly Lemma 2.5 and the construction in Theorem 1.1. The main issue is the conditional nature of the rate theorems: for 0<γ≤1, the existence of the required maximal estimate is not established, and the statements should be restricted or supplemented with a proof/reference. The omitted reduction for 0<a≤1 in Theorem 1.2 is straightforward and should be included. The novelty is incremental but legitimate, and the paper fits the scope of the journal. I would support publication after the authors address the conditional range and the missing proof details."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Read it in one sitting. The paper does what it says: it pushes the pointwise convergence picture for Landau type Schrödinger operators P^t_{a,γ} from H^s into W^{s,p} when γ>1 and 1<p≤2, and it settles the natural negative L^p result. The main new work is Theorem 1.1 — for γ>1 and 1≤p<2, convergence can fail a.e. on a positive set for some f∈L^p. That's a real result, and the proof adapts the Yuan–Zhao–Zheng counterexample carefully; the frequency-localized construction with k0 chosen to offset γ is checkable and works. The W^{s,p} thresholds in Theorem 1.2 are plausible and the proof for a>1 via Lemma 2.5 is clean; Lemma 2.5 itself, a weighted Fourier estimate, is a useful standalone lemma. The rate theorems (1.3, 1.4) follow the Li–Wang template and the sharpness discussion (Theorem 2.10) is a nice touch.\n\nThe soft spots are exactly where the paper leans on other work. First, the rate theorems are conditional on an H^s maximal estimate (1.7)/(1.10), and the only cited existence statements are [1] and [23], which cover γ>1. For 0<γ≤1, the paper doesn't say where s0 comes from. In fact, a trivial bound supplies it: for s>1/2, sup_t |P^t f| ≤ ||\\hat f||_1 ≤ C_s ||f||_{H^s}, so (1.10) holds for any γ>0. That should have been stated; without it, Theorem 1.4 looks vacuous in a range the abstract advertises. Second, the 0<a≤1 part of Theorem 1.2 is dismissed as \"proof almost the same\" and pinned to [23, Corollary 1.2]; since the novelty of Theorem 1.2 is precisely the W^{s,p} extension, the author should at least outline the reduction so the referee can verify the thresholds. Third, the density property (2.11) is imported without proof; I believe it follows from [23], but the paper should quote it as a lemma.\n\nNeither of these is load-bearing: the central claims are not contradicted, and the missing maximal estimate is elementary. The paper deserves a serious referee; the main request should be to fill in these presentation gaps (and a few typos). A harmonic analyst working on Schrödinger maximal estimates will want this on file.","headline":"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.","tokens_in":22916,"tokens_out":7339,"would_cite":true,"duration_ms":82836,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["46E35"],"pacs":[],"model":"deepseek-v4-flash","headline":"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…","keywords":["Landau type Schr\\\"odinger operator","pointwise convergence","fractional Sobolev space","convergence rate","restricted curve","maximal estimate","damped Schr\\\"odinger equation"],"falsifier":"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.","tokens_in":21849,"feed_emoji":"⚛️","tokens_out":8231,"duration_ms":89681,"temperature":0.7,"pith_summary":"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 $1<p\\le2$, the paper proves a.e. convergence for every $f\\in W^{s,p}$ above explicit thresholds that depend on $a$; for $1\\le p<2$, it constructs $L^p$ data where convergence fails on a positive-measure set, showing that some Sobolev smoothness is genuinely necessary. It also proves convergence-rate bounds along $\\beta$-H\\\"older curves, with the sharp exponent $o(t^{\\delta\\min\\{1,\\gamma\\}/a})$ along vertical lines, and a universal obstruction: no nonzero Schwartz function can converge faster than $t^{\\min\\{1,\\gamma\\}}$ there. If the theorems are correct, the one-dimensional pointwise-convergence problem for this family is essentially complete in the $W^{s,p}$ scale for $\\gamma>1$.","feed_headline":"Sharp Sobolev thresholds for damped Schr\\\"odinger pointwise convergence","feed_subtitle":"For $\\gamma>1$, data in $W^{s,p}$ converge a.e. above explicit exponents; the vertical-line rate $t^{\\delta\\min\\{1,\\gamma\\}/a}$ is optimal.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Establishes the maximal estimate and convergence for $\\gamma>1$, $a>1$ in the $H^\\sigma$ scale that Theorem 1.2 reduces to, and is cited as the source of the exponent $s_0$ in Theorem 1.4.","marker":"[1]"},{"why":"Provides the $0<a\\le1$ convergence case, the $L^p$ counterexample template used in Theorem 1.1, and the corollary to which the proof of Theorem 1.2 for $0<a\\le1$ is reduced; also cited for the existence of $s_0$ in Theorem 1.4.","marker":"[23]"},{"why":"Supplies the frequency-localization lemma used to estimate curve-evaluated oscillatory integrals in Theorem 1.3 and the original idea behind the sharpness proof of Theorem 2.10.","marker":"[13]"},{"why":"Provides the effective time-frequency decomposition and rate proof pattern that the paper adapts for Landau type operators along curves and vertical lines.","marker":"[12]"},{"why":"Supplies the resonance-type lemma used to derive weak-type maximal bounds from almost-everywhere convergence, which drives the negative $L^p$ result in Theorem 1.1.","marker":"[14]"}],"fun_headline_variants":["Damped Schrödinger: sharp Sobolev threshold for a.e. convergence","Optimal vertical-line rate for Landau-type evolutions","Exact Sobolev exponents for pointwise convergence of damped operators","Sharp rate and threshold for Landau Schrödinger operators","Tight Sobolev conditions for a.e. convergence of damped evolutions"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Damped Schrödinger: sharp Sobolev threshold for a.e. convergence","Optimal vertical-line rate for Landau-type evolutions","Exact Sobolev exponents for pointwise convergence of damped operators","Sharp rate and threshold for Landau Schrödinger operators","Tight Sobolev conditions for a.e. convergence of damped evolutions"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000388,"raw_usage":{"total_tokens":2096,"prompt_tokens":1042,"completion_tokens":1054,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":658,"completion_tokens_details":{"reasoning_tokens":944}},"tokens_in":658,"tokens_out":1054,"duration_ms":9968,"temperature":1.0,"reasoning_tokens":944,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T12:20:44.570721+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes the maximal estimate and convergence for $\\gamma>1$, $a>1$ in the $H^\\sigma$ scale that Theorem 1.2 reduces to, and is cited as the source of the exponent $s_0$ in Theorem 1.4."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the $0<a\\le1$ convergence case, the $L^p$ counterexample template used in Theorem 1.1, and the corollary to which the proof of Theorem 1.2 for $0<a\\le1$ is reduced; also cited for the existence of $s_0$ in Theorem 1.4."},{"cited_title":"Li and H","cited_arxiv_id":null,"evidence_quote":"Supplies the frequency-localization lemma used to estimate curve-evaluated oscillatory integrals in Theorem 1.3 and the original idea behind the sharpness proof of Theorem 2.10."},{"cited_title":"On convergence properties for generalized Schr\\\"{o}dinger operators along tangential curves","cited_arxiv_id":"2111.09186","evidence_quote":"Provides the effective time-frequency decomposition and rate proof pattern that the paper adapts for Landau type operators along curves and vertical lines."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the resonance-type lemma used to derive weak-type maximal bounds from almost-everywhere convergence, which drives the negative $L^p$ result in Theorem 1.1."}],"review_version":1}