REVIEW 2 major objections 3 minor 3 cited by
The $L^p$-boundedness of wave operators for higher order Schr\"odinger operator with zero singularities in low odd dimensions
T0 review · 2 major / 3 minor · reviewed 2026-08-15 · deepseek-v4-flash
Pith's one-line read Zero-energy resonances determine exactly the L^p range of wave operators for higher-order Schrödinger operators in low odd dimensions.
desk verdict A serious, detailed proof of sharp L^p wave-operator bounds for all zero-resonance types in the remaining odd low-dimensional higher-order Schrödinger cases, conditional on a resolvent expansion imported from the authors' own preprint [4]. 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 carrying object is the operator $M_\pm(\lambda)=U+vR_0^\pm(\lambda^{2m})v$ on $L^2$, where $v=\sqrt{|V|}$ and $U=\operatorname{sgn}V$; the symmetric second resolvent identity writes the perturbed resolvent as $R^\pm(\lambda^{2m})V=R_0^\pm(\lambda^{2m})v(M_\pm(\lambda))^{-1}v$. The proof pivots on the asymptotic expansion of $(M_\pm(\lambda))^{-1}$ as $\lambda\to0^+$, a sum over resonance projections $Q_j$ with powers $\lambda^{2m-n-i-j}$, together with the cancellation relations $Q_j(x^\alpha v)=0$ for $|\alpha|\le\max\{0,\lfloor j+1/2\rfloor\}-1$. These cancellations turn the low-energy wave-operator kernel into oscillatory integrals with phases $|x|\pm|y|$; the sharp $p$-threshold appears when the remaining singular kernel is a weighted truncated Hilbert transform. Sharpness comes from a complementary lower bound for $|\langle e^{itH}P_{ac}(H)\psi,\psi\rangle|$, a Kato–Jensen-type time-decay estimate whose decay exponent matches the free dispersive rate exactly at $p=p_k$.
What would settle it
Test the predicted threshold in the case $n=3$, $m=2$, where a second-kind resonance is claimed to give boundedness exactly for $1<p<3$ and unboundedness for all $p>3$: construct an explicit potential $V$ satisfying Assumption 1.3 with a known second-kind zero resonance and check whether $\|W_+(H;(-\Delta)^2)\|_{L^p\to L^p}$ stays finite for some $p>3$; a single such example would refute Theorem 1.7.
Extended reading notes
Core claim
On the paper's own terms, the central discovery is that in odd dimensions $1\le n\le 4m-1$ the $L^p$ mapping properties of the wave operators $W_\pm(H;(-\Delta)^m)$ are controlled entirely by the kind of zero-energy singularity of $H$, through a single critical index $k_c$. With regular zero counted as $k=0$ and a zero eigenvalue as $k=m_n+1$, Theorem 1.5 gives full-range boundedness $1<p<\infty$ for $k\le k_c$, the finite range $1<p<\frac{n}{n-2m+k+k_c-1}$ for $k_c<k\le m_n$, and $1<p<\frac{2n}{n-1}$ in the eigenvalue case. Theorem 1.7 shows these ranges are sharp: for $p$ beyond the threshold the operators are unbounded, so the low-energy singularity itself is the obstruction. The paper also derives the corresponding $L^p$-$L^{p'}$ decay rates for the perturbed Schrödinger group and verifies that the results reproduce the known $m=1,2$ cases.
Load-bearing premise
The paper's low-energy conclusions stand on the asymptotic expansion of $(M_\pm(\lambda))^{-1}$ near zero energy that is quoted from the earlier preprint [4] and only summarized in Appendix A; if that expansion is incomplete in any term, the stated $p$-thresholds would not follow.
Editorial extensions
If this is right
- The dispersive estimate $\|e^{itH}P_{ac}(H)\|_{L^p\to L^{p'}}\lesssim |t|^{-(n/m)(1/p-1/2)}$ holds exactly for the endpoint ranges $p'$ listed in Corollary 1.10, so any nonlinear application of the perturbed group inherits the same resonance-dependent restriction.
- In dimensions $2m+1\le n\le4m-1$, $k_c=0$, so the mere presence of any zero resonance or eigenvalue shrinks the range from the full $1<p<\infty$ (or $1\le p\le\infty$ in the regular case) to a finite interval.
- The endpoint $p=p_k$ is not settled by the paper; endpoint boundedness or weak-type results at $p=p_k$ would be needed for applications requiring the optimal decay rate at the threshold.
- The sharpness proof links $L^p$-unboundedness to optimality of Kato–Jensen time decay, so these two phenomena are equivalent up to the paper's assumptions: improving one improves the other.
- The results unify the known $m=1,2$ cases and place the fourth-order line $n=1,3,5,7$ and its resonance classifications into a single formula.
Reading between the lines
- The paper leaves $p=p_k$ open; the classical Hilbert transform analogy suggests a weak-type $(p_k,p_k)$ bound may survive where strong boundedness fails, so endpoint dispersive applications could still be possible in a restricted sense.
- The parity of $n$ enters the free-resolvent coefficients, so even-dimensional thresholds likely involve logarithmic corrections or shifted critical indices; the paper's advertised sequel for even dimensions should reveal whether the formula $p_k$ persists in that setting.
- One testable consequence is that the leading angular term $\sum_{|\alpha|=|\beta|=\vartheta(j)} y^\alpha Q_j(vz^\beta)(x)/|y|^{(n-1)/2+\vartheta(j)}$ controls the threshold: potentials for which this term vanishes could have a better effective $p$-range than the generic formula, suggesting the resonance type alone may not be the full story for every potential.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The manuscript proves sharp L^p-boundedness ranges for the wave operators W_±(H;(-Δ)^m) for H=(-Δ)^m+V on R^n with odd 1≤n≤4m-1, under explicit decay and regularity assumptions on V and under a classification of zero-energy resonances. The main results are Theorems 1.5 and 1.7: for regular zero or resonance types k≤k_c the wave operators are bounded on L^p for all 1<p<∞; for k_c<k≤m_n the range is 1<p<n/(n-2m+k+k_c-1); and for zero being an eigenvalue the range is 1<p<2n/(n-1). Unboundedness is proved for p beyond these thresholds. The proof splits the stationary representation into low- and high-energy parts. The low-energy part is controlled by combining the asymptotic expansion of (M_±(λ))^{-1} near λ=0 (Theorem 2.5) with detailed kernel estimates for Q_j v R_±^0 (Lemmas 2.7 and 2.10) and oscillatory-integral arguments; the high-energy part is handled via existing results of Erdogan-Green and resolvent estimates. Unboundedness is reduced to optimal time-decay estimates for e^{itH}P_ac(H).
Significance. If the low-energy expansion is fully justified, the paper settles a natural open problem: the L^p-boundedness of wave operators for higher-order Schrödinger operators in all low odd dimensions with all zero-energy resonance types, and it gives explicit, parameter-free thresholds that agree with the known m=1,2 cases. The proof has a clear architecture, contains many detailed estimates, and the main claims are falsifiable and sharp in the stated range. The paper also correctly gives credit to the dependency of the resolvent expansion on the authors' preprint [4]; this dependency is the main source of risk rather than an internal inconsistency.
major comments (2)
- [Section 2.2 and Appendix A] The central low-energy machine is Theorem 2.5, the expansion (2.36) of (M_±(λ))^{-1} with remainder bounds (2.37)-(2.38). The proof in Appendix A is explicitly only an outline: the decisive facts that D_00 is strictly positive, D_11 is strictly negative, d is invertible and rigidly negative definite on ⊕_{j∈J_k''} Q_j L^2, and that Q v G_{4m-n} v Q is invertible in the eigenvalue case are all quoted from Lemma 2.6 of [4]. Every low-energy kernel estimate in Section 3 (Lemmas 3.3, 3.8, 3.11 and Propositions 3.5, 3.7) and the extraction of the principal term in Section 5, equations (5.3)-(5.6), starts from (2.36). Thus the main theorems are exactly as secure as Theorem 2.5. The manuscript should either include a complete proof of Theorem 2.5, including Lemma 2.6 of [4], or state the main results as conditional on that preprint and explain why this external foundation is acceptable for the journal.
- [Section 5, Remark 5.2] The if-and-only-if characterization of p-wave resonances is asserted without proof: a non-zero distributional solution φ of Hφ=0 in ∩_{s<-1/2} L^2_s \setminus L^2 exists if and only if ψ=Uvφ belongs to Q_{(2m-n+1)/2}L^2. This equivalence is used in Case II to guarantee Q_{(2m-n+1)/2}≠0 and hence that the principal term in (5.6) does not vanish. Since Theorem 1.7(ii) is stated precisely for the eigenvalue case with such a distributional solution, this equivalence is load-bearing and needs either a proof or a precise reference to a result establishing it.
minor comments (3)
- [Section 3.2.2, equation (3.42)] The displayed radial estimate contains a repairable typo: after applying the L^p-boundedness of the truncated Hilbert transform, the final integral should have the weight r^{n-1}, not r^{(n-1)/2}. With the correction the estimate is uniform in s,h and the conclusion of Proposition 3.13 is unaffected.
- [Section 1.4] The sentence 'W_+f=W_-f' is not literally correct; the proof treats W_- and then W_+ must be handled by the analogous stationary formula or by complex conjugation of the kernels. This does not affect the argument, but the wording should be corrected.
- [Appendix A] The appendix says 'we outline the processes' and refers to [4] for several key algebraic facts; in addition to the major concern above, the notation M^±_{i,j} and the statement of the Neumann-series remainder Γ^±(λ) would benefit from a slightly fuller explanation of why the displayed remainder bounds are uniform over the finite index set J_k.
Circularity Check
No circularity: the L^p ranges are derived from resolvent-expansion kernel estimates; the load-bearing expansion of (M±(λ))^{-1} is imported from the authors' preprint [4], but that is a dependency, not a reduction to the target result.
full rationale
The derivation chain is genuine. The low-energy wave operator is written via the symmetric second resolvent identity (1.17), then the asymptotic expansion (2.36) of (M±(λ))^{-1} from Theorem 2.5 is inserted to obtain the kernel representation (3.1)-(3.5). The p-thresholds are obtained by optimizing the genuinely estimated kernel bounds in Propositions 3.5 and 3.7, not by assuming the thresholds. The unboundedness proof in Section 5 also proceeds from the resolvent expansion in (5.3)-(5.6) and an oscillatory integral lemma, isolating a nonzero principal term; no fitted parameter or pre-assigned output is used. The only concerning dependence is Theorem 2.5 itself. The paper explicitly states this: Remark 2.6 says 'by repeating the proof for Theorem 2.7 of [4], we establish a slightly modified version, which we present as Theorem 2.5', and Appendix A says 'we outline the processes how to obtain the asymptotic expansions ... which actually was proved by Cheng et al [4]'. The invertibility facts used there are also quoted, e.g. 'D00 is strictly positive (see Lemma 2.6 in [4])' and 'D11 is strictly negative (see Lemma 2.6 in [4])'. This is a load-bearing self-citation with one overlapping author (Han Cheng), and the paper is not fully self-contained. However, under the stated rubric this is a dependency rather than circularity: Theorem 2.5 is a parameter-free expansion theorem whose hypotheses (Assumption 1.3) do not contain the target L^p-boundedness result, and it can be checked independently. The paper's L^p conclusions are not equivalent by construction to the expansion; they follow from nontrivial kernel estimates. Hence no circular step is exhibited.
Assumptions & free parameters
assumptions (3)
- domain assumption Assumption 1.3: absence of positive embedded eigenvalues, decay |V|≲⟨x⟩^{-β} with β from (1.7), and the extra weighted L^2 bound at n=4m-1.
- standard math Theorem 2.5: asymptotic expansion of (M±(λ))^{-1} for each resonance type k, with the stated operators M±_{i,j} and remainder Γ±_{i,j}.
- standard math Zero-resonance classification for odd n with 1≤n≤4m-1: m_n types, critical type k_c, and the orthogonal projections Q_j satisfying cancellation (2.32).
Cite this review
Pith. "Pith review of The $L^p$-boundedness of wave operators for higher order Schr\"odinger operator with zero singularities in low odd dimensions." pith.science (2026). https://pith.science/paper/JZO2V7S7
@misc{pith2026250507009,
author = {Pith},
title = {Pith review of: The $L^p$-boundedness of wave operators for higher order Schr\"odinger operator with zero singularities in low odd dimensions},
year = {2026},
howpublished = {\url{https://pith.science/paper/JZO2V7S7}},
note = {Machine review of arXiv:2505.07009}
}
abstract
This paper investigates the $L^p$-bounds of wave operators for higher-order Schr\"odinger operators $H = (-\Delta)^m + V$ on $\mathbb{R}^n$, with $m \ge 2$ and real-valued decaying potentials $V$. Our main objective is to establish the sharp $L^p$-boundedness of the wave operators $W_\pm(H; (-\Delta)^m)$ in the presence of all types of zero-resonance singularities, for all odd dimensions $1 \le n \le 4m - 1$. Specifically, for odd $n$ with $1 \le n \le 4m - 1$, there exist $m_n$ types of zero resonances for $H$, along with a critical type $k_c$ (both depending on $n$ and $m$). If zero is a regular point of $H$ or a $\mathbf{k}$-th kind resonance with $1 \le \mathbf{k} \le k_c$, the wave operators $W_\pm(H; (-\Delta)^m)$ are bounded on $L^p(\mathbb{R}^n)$ for all $1 < p < \infty$. If zero is a $\mathbf{k}$-th kind resonance with $k_c < \mathbf{k} \le m_n$, we show that the range of $p$-boundedness for $W_\pm(H; (-\Delta)^m)$ narrows to $1 < p < p_{\mathbf{k}}$, where $$p_{\mathbf{k}} = \frac{n}{n - 2m + \mathbf{k} + k_c - 1}.$$ Additionally, if zero is an eigenvalue of $H$ (i.e., $\mathbf{k} = m_n + 1$), then $W_\pm(H; (-\Delta)^m)$ are bounded on $L^p(\mathbb{R}^n)$ for all $1 < p < \frac{2n}{n - 1}$. Furthermore, it is shown that the wave operators $W_\pm(H; (-\Delta)^m)$ are unbounded on $L^p(\mathbb{R}^n)$ for all $p_{\mathbf{k}} < p \le \infty$ if $k_c < \mathbf{k} \le m_n$, and for all $\frac{2n}{n - 1} < p \le \infty$ if zero is an eigenvalue of $H$ with a non-zero solution $\phi$ to $H\phi = 0$ in $\bigcap_{s < -\frac{1}{2}} L^{2}_{s}(\mathbb{R}^n) \setminus L^2(\mathbb{R}^n)$(referred to as a $p$-wave resonance). The key idea of the proof is to reduce the $L^p$-unboundedness to establishing the optimality of time-decay estimates for $e^{itH}P_{ac}(H)$ in weighted $L^2$ spaces.
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Reference graph
Works this paper leans on
-
[1]
Agmon,Spectral properties of Schr¨ odinger operators and scattering theory, Ann
S. Agmon,Spectral properties of Schr¨ odinger operators and scattering theory, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4)2(1975), no. 2, 151–218
work page 1975
-
[2]
Beceanu,Structure of wave operators for a scaling-critiacal class of potentials, Amer
M. Beceanu,Structure of wave operators for a scaling-critiacal class of potentials, Amer. J. Math. 136(2014), no. 2 255-308
work page 2014
-
[3]
M. Beceanu and W. Schlag,Structure formulas for wave operators, Amer. J. Math.142(2020), no. 3, 751-807. [4]H. Cheng, S. Huang, T. Huang and Q. Zheng,Pointwise estimates for the fundamental solutions of higher order Schr¨ odinger equations in odd dimensions I: low dimensional case, arXiv e-prints, (2024), arXiv:2401.04969v10. [5]H. Cheng, A. Soffer, Z. Wu...
arXiv 2020
-
[6]
P. D’Ancona and L. Fanelli,L p-boundedness of the wave operator for the one dimensional Schr¨ odinger operator, Commun. Math. Phys.268(2006), no. 2, 415–438
work page 2006
-
[7]
M.B. Erdo˘ gan, M. Goldberg and W.R. Green,On theL p boundedness of wave operators for two- dimensional Schr¨ odinger operators with threshold obstructions, J. Funct. Anal.274(2018), 2139–2161
work page 2018
-
[8]
M.B. Erdo˘ gan, M. Goldberg and W.R. Green,Counterexamples toL p boundedness of wave operators for classical and higher order Schr¨ odinger operators, J. Funct. Anal.285(2023), no. 5, 110008, 18pp
work page 2023
-
[9]
M. B. Erdo˘ gan, M. Goldberg, W. R. Green,Dispersive estimates for higher order Schr¨ odinger operators with scaling-critical potentials. Math. Ann. (2025). https://doi.org/10.1007/s00208-025-03146-1
-
[10]
M.B. Erdo˘ gan and W.R. Green,TheL p-continuity of wave operators for higher order Schr¨ odinger opera- tors, Adv. Math.404(2022), 108450, 41pp
work page 2022
Show all 53 references
-
[11]
Erdo˘ gan and W.R
M.B. Erdo˘ gan and W.R. Green,TheL p-continuity of wave operators for higher order Schr¨ odinger opera- tors, J. Differential Equations355(2023), 144–161
2023
-
[12]
Erdo˘ gan, W.R
M.B. Erdo˘ gan, W.R. Green and K. Lamaster,Lp-continuity of wave operators for higher order Schr¨ odinger operators with threshold eigenvalues in high dimensions, Discrete and Continuous Dynamical Systems,45 (2025), no. 9, 3258–3274
2025
-
[13]
Erdo˘ gan, W.R
M.B. Erdo˘ gan, W.R. Green and E. Toprak,On the fourth order Schr¨ odinger equation in three dimensions: dispersive estimates and zero energy resonances, J. Differential Equations271(2021), 152–185. 56
2021
-
[14]
Feng,Dispersive estimates for inhomogeneous fourth-order Schr¨ odinger operator in 3D with zero energy obstructions, Nonlinear Anal.207(2021), Paper No
H. Feng,Dispersive estimates for inhomogeneous fourth-order Schr¨ odinger operator in 3D with zero energy obstructions, Nonlinear Anal.207(2021), Paper No. 112269, 33pp
2021
-
[15]
H. Feng, A. Soffer, Z. Wu and X. Yao,Decay estimates for higher-order elliptic operators, Trans. Amer. Math. Soc.373(2020), no. 4, 2805–2859
2020
-
[16]
H. Feng, A. Soffer and X. Yao,Decay estimates and Strichartz estimates of fourth-order Schr¨ odinger operator, J. Funct. Anal.274(2018), no. 2, 605–658
2018
-
[17]
Finco and K
D. Finco and K. Yajima,TheL p boundedness of wave operators for Schr¨ odinger operators with threshold singularities II. the even dimensional case, J. Math. Sci. Univ. Tokyo13(2006), no. 3, 277–346
2006
-
[18]
Galtbayar and K
A. Galtbayar and K. Yajima,TheL p-continuity of wave operators for one dimensional Sch¨ odinger oper- ators, J. Math. Sci. Univ. Tokyo.7(2000), 221-240
2000
-
[19]
Galtbayar and K
A. Galtbayar and K. Yajima,TheL p-boundedness of wave operators for fourth order Sch¨ odinger operators onR 4, J. Spectral theory,14(2024), 271-354
2024
-
[20]
Goldberg and W.R
M. Goldberg and W.R. Green,TheL p boundedness of wave operators for Schr¨ odinger operators with threshold singularities, Adv. Math.303(2016), 360–389
2016
-
[21]
Goldberg and W.R
M. Goldberg and W.R. Green,On theL p boundedness of wave operators for four-dimensional Schr¨ odinger operators with a threshold eigenvalue, Ann. Henri Poincar´ e,18(2017), 1269–1288
2017
-
[22]
Goldberg and W.R
M. Goldberg and W.R. Green,L p boundedness of wave operators for fourth order Schr¨ odinger operators, Trans. Amer. Math. Soc.374(2021), 4075–4092
2021
-
[23]
Goldberg and W
M. Goldberg and W. Schlag,Dispersive estimates for Schr¨ odinger operators in dimensions one and three, Comm. Math. Phys.251(2004), no. 1, 157–178
2004
-
[24]
Goldberg and M
M. Goldberg and M. Visan,A counterexample to dispersive estimates for Schr¨ odinger operators in higher dimensions,Comm. Math. Phys.266(2006), 211-238
2006
-
[25]
I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series and products. Santiego New York: Academic Press, 6th edn. 2002
2002
-
[26]
Grafakos, Classical Fourier Analysis
L. Grafakos, Classical Fourier Analysis. Graduate Texts in Mathematics, second ed., vol. 249, New York: Springer, 2008
2008
-
[27]
Greenand E
W.R. Greenand E. Toprak,On the fourth order Schr¨ odinger equation in four dimensions: dispersive estimates and zero energy resonances, J. Differential Equations267(2019), no. 3, 1899–1954
2019
-
[28]
Jensen and T
A. Jensen and T. Kato,Spectral properties of Schr¨ odinger operators and timedecay of the wave functions, Duke Math. J.46(1979), no. 3, 583-611
1979
-
[29]
Jensen,Spectral properties of Schr¨ odinger operators and time-decay of the wave functions results in L2(Rm),m≥5, Duke Math
A. Jensen,Spectral properties of Schr¨ odinger operators and time-decay of the wave functions results in L2(Rm),m≥5, Duke Math. J.47(1980), no. 1, 57-80
1980
-
[30]
Jensen and G
A. Jensen and G. Nenciu,A unified approach to resolvent expansions at thresholds,Rev. Math. Phys.13 (2001), no. 6, 717-754
2001
-
[31]
Jensen and K
A. Jensen and K. Yajima,A remark onL p−boundedness of wave operators for two-dimensional Schr¨ odinger operators, Comm. Math. Phys.225(2002), no. 3, 633-637
2002
-
[32]
Journ´ e, A
J.-L. Journ´ e, A. Soffer, and C. D. Sogge,Decay estimates for Schr¨ odinger operators, Comm. Pure Appl. Math.44(1991), no. 5, 573–604
1991
-
[33]
S. T. Kuroda,Scattering theory for differetial operators. I. Operator theory, J. Math. Soc. Japan25(1973), no. 1, 75-104
1973
-
[34]
P. Li, A. Soffer and X. Yao,Decay estimates for fourth-order Schr¨ odinger operators in dimension two, J. Funct. Anal.284(2023), no. 6, 109816, 83 pp
2023
-
[35]
Mizutani, Z
H. Mizutani, Z. Wan and X. Yao,L p boundedness of wave operators for fourth-order Schr¨ odinger operators on the line, Adv. Math.451(2024),109806, 68 pp
2024
-
[36]
Mizutani, Z
H. Mizutani, Z. Wan and X. Yao,L p boundedness of wave operators for fourth-order Schr¨ odinger operators with zero resonance onR 3, J. Funct. Anal.289(2025), no. 8, 111013, 68 pp
2025
-
[37]
Mizutani, Z
H. Mizutani, Z. Wan and X. Yao,Counterexamples and weak(1,1)estimates of wave operators for fourth- order Schr¨ odinger operators in dimension three, J. Spectral theory,14(2024), no. 4, 1409-1450
2024
-
[38]
Reed and B
M. Reed and B. Simon,Methods of Modern Mathematical Physics III: Scatteing Theory, Academic Press, New York, NY, 1972
1972
-
[39]
Rodnianski and W
I. Rodnianski and W. Schlag,Time decay for solutions of Schr¨ odinger equations with rough and time- dependent potentials, Invent. Math.155(2004), no. 3, 451–513. 57
2004
-
[40]
Schlag,Dispersive estimates for Schr¨ odinger operators in dimension two, Comm
W. Schlag,Dispersive estimates for Schr¨ odinger operators in dimension two, Comm. Math. Phys.257 (2005), no. 1, 87–117
2005
-
[41]
Schlag,Dispersive estimates for Schr¨ odinger operators: a survey, Mathematical aspects of nonlinear dispersive equations, Ann
W. Schlag,Dispersive estimates for Schr¨ odinger operators: a survey, Mathematical aspects of nonlinear dispersive equations, Ann. of Math. Stud., vol. 163, Princeton Univ. Press, Princeton, NJ, 2007, pp. 255– 285
2007
-
[42]
Soffer, Z
A. Soffer, Z. Wu and X. Yao,Decay estimates for bi-Schr¨ odinger operators in dimension one, Ann. Henri Poincar´ e.23(2022), no. 8, 2683–2744
2022
-
[43]
E. M. Stein and G. Weiss,Introdution to Fourier Analysis in Euclidean Spaces, Princeton University Press, 1970
1970
-
[44]
E. M. Stein and T. S. Murphy,Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton University Press, 1993
1993
-
[45]
R. A. Weder,TheW k,p-continuity of the Schr¨ odinger wave operator on the line, Comm. Math. Phys.208 (1999), no. 2, 507–520
1999
-
[46]
R. A. Weder,L p−Lp′ estimates for the Schr¨ odinger equation on the line and inverse scattering for the non linear Schr¨ odinger equation with a potential, J. Funct. Analysis170(2000), 37–68
2000
-
[47]
Yajima,TheW k,p-continuity of wave operators for Schr¨ odinger operators, J
K. Yajima,TheW k,p-continuity of wave operators for Schr¨ odinger operators, J. Math. Soc. Japan47 (1995), no. 3, 551–581
1995
-
[48]
Yajima,TheW k,p-continuity of wave operators for Schr¨ odinger operators.III.Even-dimensional cases m≥4, J
K. Yajima,TheW k,p-continuity of wave operators for Schr¨ odinger operators.III.Even-dimensional cases m≥4, J. Math. Sci. Univ. Tokyo.2(1995), no. 2, 311–346. 551–581
1995
-
[49]
Yajima,L p−boundedness of wave operators for two-dimensional Schr¨ odinger operators, Comm
K. Yajima,L p−boundedness of wave operators for two-dimensional Schr¨ odinger operators, Comm. Math. Phys.208(1999), no. 3, 125-152
1999
-
[50]
Yajima,TheL p boundedness of wave operators for Schr¨ odinger operators with threshold singularities I
K. Yajima,TheL p boundedness of wave operators for Schr¨ odinger operators with threshold singularities I. the odd dimensional case, J. Math. Soc. Japan.13(2006), 43–93
2006
-
[51]
Yajima,Remark on theL p-boundedness of wave operators for Schr¨ odinger operators with threshold singularities, Doc
K. Yajima,Remark on theL p-boundedness of wave operators for Schr¨ odinger operators with threshold singularities, Doc. Math.21(2016), 391–443
2016
-
[52]
Yajima,L 1 andL ∞-boundedness of wave operators for three dimensional Schr¨ odinger operators with threshold singularities, Tokyo J
K. Yajima,L 1 andL ∞-boundedness of wave operators for three dimensional Schr¨ odinger operators with threshold singularities, Tokyo J. Math.41(2018), no. 2, 385-406
2018
-
[53]
Yajima,TheL p boundedness of wave operators for two dimensional Schr¨ odinger operators with threshold singularities, J
K. Yajima,TheL p boundedness of wave operators for two dimensional Schr¨ odinger operators with threshold singularities, J. Math. Soc. Japan.74(2022), no. 4, 1169–1217
2022
-
[54]
K. Yajima,TheL p-boundedness of wave operators for four dimensional Schr¨ odinger operators with thresh- old resonances, The physics and mathematics of Elliott Lieb—the 90th anniversary. Vol. II, 517–563. EMS Press, Berlin, 2022
2022
-
[55]
Zheng, X
Q. Zheng, X. Yao and D. Fan,Convex hypersurfaces andL p estimates for Schr¨ odinger equations, J. Funct. Anal.208(2004), no. 1, 122-139. Han Cheng, Institute of Applied Physics and Computational Mathematics, Beijing, 100088, Peo- ple’s Republic of China Email address:chmathh@1...
2004
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