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REVIEW 3 major objections 4 minor 41 references

Applications of renormalisation to orthonormal Strichartz estimates and the NLS system on the circle

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Pith's one-line read This paper claims that subtracting the spatial mean from the density — a renormalised density — strictly improves orthonormal Strichartz estimates on the circle, yielding an L² estimate up to the Hilbert–Schmidt class and an L³ estimate in

desk verdict New renormalised-density Strichartz estimates on the circle that look right; the advertised NLS application has a self-containment gap, but the main claims hold up better than the reader's report suggests. read the letter →

arxiv 2604.00252 v2 pith:XHJRF2KV submitted 2026-03-31 math.AP math-phmath.MP

Sonae Hadama , Andrew Rout This is my paper · ORCID
classification math.APmath-phmath.MP MSC 42B3735Q5535B45
keywords OrthonormalStrichartzestimatesRenormaliseddensityCubicNLSsystemSchattenclassesGlobalwell-posednessIll-posednessTorusDispersive
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

The paper introduces a renormalised density on the circle: subtract the spatial mean, equivalently a multiple of the trace, from the usual density. Because constants commute with everything, the renormalised density generates the same NLS system for trace-class states, but it satisfies better orthonormal Strichartz estimates than the standard density. The paper proves an L² estimate for all Hilbert–Schmidt data and an L³ estimate for Schatten exponents between 2 and 3, with the remaining lower range left as a conjecture. As an application, it claims the cubic renormalised NLS system on the circle is globally well-posed for Schatten exponents up to 2 and ill-posed above 2. On tori of dimension at least two, the same renormalisation is shown to yield almost no improvement over the non-renormalised estimates.

What carries the argument

The key object is the renormalised density ρA(x)=ρA(x)−(1/2π)TrA, defined for trace-class A and extended by density to S². Its value is that subtracting the constant removes the zero-frequency Fourier mode; duality then only needs test functions V with vanishing space-time mean, and Parseval's identity turns the L² estimate into a bound on an integral operator with kernel e^{−it(m²−n²)} whose zero-frequency term is absent. For the L³ estimate, the same mean-zero reduction converts the problem into counting solutions of a Diophantine equation ks−(k−α)s′=β, whose divisor-counting bound yields the N^{1+ε} estimate. For well-posedness, the propagator U_V(t,s) for a real L² potential and its Stri

What would settle it

Take γ_{0,N}=N^{−1/2}Σ_{n=1}^N |φ_n><φ_n| with φ_n=2^{−1/2}(e_n+e_{−n}); the paper's own computation gives ∥ρ(U(t)γ_{0,N}U*(t))∥_{L²}∼1 while ∥γ_{0,N}∥_{S^α}→0 for α>2, proving sharpness of the L² estimate. To test the missing well-posedness range, compute the contraction constant of the map V↦ρ(U_{±V}(t)γ0U*_{±V}(t)) on S^{3/2} for small T: a Lipschitz constant >1 for all small T would show the S²-based proof cannot reach α<2.

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Extended reading notes

Core claim

The central discovery is that the renormalised density ρ(U(t)γ0U*(t)) = ρ(U(t)γ0U*(t)) − (1/2π)Trγ0 obeys an L²_{t,x} bound on the two-torus for every Hilbert–Schmidt operator γ0, sharp in the sense that the bound fails for α>2, and a frequency-truncated L³ bound with exponent σ>2/3−1/α for α∈[2,3]. This improves the non-renormalised L³ condition, which required roughly 1/α>1−σ. Interpreting the Hilbert–Schmidt case via the unique bounded extension from trace-class data, the paper uses these estimates to build the solution map for the renormalised cubic NLS system by a fixed point in the density, yielding global well-posedness in S^α for α≤2 and discontinuity of the data-to-density map for α

Load-bearing premise

The load-bearing premise is that the contraction argument for the renormalised system, which is written only for γ0∈S² and relies on a cited unpublished preprint for a key Strichartz estimate, extends to all Schatten classes S^α with 1≤α<2; if that extension is not possible, the stated critical exponent 2 is unsupported.

Editorial extensions

If this is right

  • If Theorem 1.19 is correct, the cubic renormalised NLS system on the circle has a sharp dichotomy: global well-posedness in S^α for 1≤α≤2 and ill-posedness for α>2, with the solution map discontinuous above the threshold.
  • The L² theorem gives a genuine extension of the density to Hilbert–Schmidt operators, so the ill-defined diagonal of the kernel is handled by a bounded operator construction.
  • The L³ threshold σ>2/3−1/α for α∈[2,3], together with the paper's counterexamples below the threshold, fixes the growth estimate for the frequency-truncated renormalised density on the circle.
  • On T^d with d≥2, the necessary condition 1/α≥1−σ/(d−1) means renormalisation does not substantially improve the orthonormal Strichartz range in high dimension.
  • If Conjecture 1.16 is true, the L³ estimate would also hold for α∈[3/2,2) with σ>2/3−1/α, completing the admissible range down to α=3/2.

Reading between the lines

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

  • A decisive open test is whether the fixed-point contraction in the paper's Proposition 4.2 can be run in S^α for 1≤α<2; if it cannot, Theorem 1.19's well-posedness range may shrink to the proved S² statement even though the density estimates themselves hold in that range.
  • Because the proof of the key density estimates for the potential propagator relies on a cited unpublished preprint by one of the authors, a reader who wants to rely on the application should check whether those ingredients can be replaced by published arguments; this is a reproducibility question, not a mathematical one.
  • The mean-zero duality trick is not specific to the circle; a natural test is whether a similar renormalisation recovers a full range of orthonormal Strichartz estimates on other compact manifolds with a spectral gap.
  • For the quintic NLS system, the paper notes that renormalisation produces an extra cubic term; a plausible extension is that the well-posedness threshold shifts and is governed by the cubic correction, which could be tested by the same fixed-point scheme.
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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 / 4 minor

Summary. The paper introduces a renormalised density for the NLS system on the circle and proves orthonormal Strichartz estimates for it. The main new estimates are Theorem 1.12, an L^2_{t,x} bound for the renormalised density of U(t)γ0U(t)* with sharp Schatten threshold α=2, and Theorem 1.14, an L^3_{t,x} estimate for the frequency-truncated renormalised density with sufficient conditions for α∈[2,3] and a matching necessary condition for α∈[3/2,3]. The authors apply these estimates to the cubic NLS system: Proposition 4.1 gives well-posedness in S^1 for the unrenormalised system and ill-posedness for α>1, while Proposition 4.2 gives S^2 well-posedness for the renormalised system and ill-posedness for α>2, presented in the introduction as the critical-exponent result Theorem 1.19. A final section gives an alternative proof of a torus result of Nakamura and a necessary condition showing that in d≥2 the renormalised density yields only a small improvement over the non-renormalised one.

Significance. If the central estimates are correct, the paper makes a genuine contribution: the renormalisation of the density is a natural and useful device, and the sharp L^2 threshold α=2 on T is a clear improvement over the non-renormalised α=1 result. The L^3 estimate in Theorem 1.14 is technically interesting, and the counting argument behind Lemma 3.3 is elegant. The paper also gives a clean alternative proof of Nakamura's torus result and a transparent counterexample in higher dimensions. These parts are strong and, as far as I have checked, the proofs of Theorems 1.12 and 1.14 are self-contained and convincing. However, the advertised optimal well-posedness application is not fully established within the manuscript: two key lemmas in Section 4 depend on an unpublished preprint, and the precise statement proved in Proposition 4.2 is weaker in two respects than the theorem announced in the introduction.

major comments (3)
  1. [§4.1, Lemmas 4.8 and 4.10] Both lemmas are load-bearing for Proposition 4.2: Lemma 4.8 supplies the L^2 density bound for U_V, and Lemma 4.10 supplies the difference estimate needed for the contraction argument. The proof of Lemma 4.8 explicitly invokes [6, Cor. 4.9] and [6, Lemma 4.1] to replace a full time integral by a truncated one and to bound a double integral in S^2; Lemma 4.10 again invokes [6, Cor. 4.9]. These are results from an unpublished preprint, arXiv:2504.19552, and are not proved or stated in the present paper. Since the fixed-point argument for (RNLSS) has no independent estimate without these inputs, the well-posedness claim is conditional on results outside the manuscript. Please either include proofs of the two [6] results in an appendix or explicitly state Proposition 4.2 as conditional on [6].
  2. [§1.4, Theorem 1.19 vs §4.2, Proposition 4.2(i)] Theorem 1.19 advertises global well-posedness in S^α for every α∈[1,2], but Proposition 4.2(i), cited as the precise statement, only proves existence, uniqueness, and Lipschitz dependence in S^2. No argument is given that the data-to-solution map is continuous in the S^α topology for α<2, nor that the S^2 solution obtained for γ0∈S^α is the unique solution in C_t S^α. The inclusion S^α⊂S^2 and unitarity of U_V make a repair plausible, but the statement as written is not proved. The introduction should either restrict the well-posedness claim to S^2 or include the S^α continuity argument.
  3. [§4.2, Proposition 4.2(ii) vs Theorem 1.19 ill-posedness] The ill-posedness half of Theorem 1.19 is stated in terms of discontinuity of the solution map, but Proposition 4.2(ii) proves only that the data-to-density map S_2:E^α_R→L^2_{t,x} has no continuous extension. This is a weaker property: the sequence γ0,N has S^α norm tending to 0 while the associated density has L^2 norm bounded below, which obstructs continuity of γ0↦ργ but does not immediately obstruct continuity of γ0↦γ(t) in C_t S^α. If 'ill-posed' is intended in the usual flow-map sense, an additional argument is needed; otherwise the theorem should be reformulated to define ill-posedness via the density map, as Proposition 4.2 does.
minor comments (4)
  1. [§4.1, Eq. (4.6)] In the expansion of Lemma 4.8, the second term is printed as 'ϱ_1 + ϱ_1 + ϱ_2'; presumably the two ϱ_1 terms are ϱ_1 and its conjugate (or DγU* and UγD*). The notation should be cleaned up to avoid ambiguity.
  2. [§5.2, Proposition 5.5] The introduction says the improvement for d≥2 is 'minimal', but Proposition 5.5 only establishes a necessary condition, leaving a gap between 1/α ≥ 1−σ/(d−1) and the known sufficient condition 1/α > 1−σ/d. The wording 'minimal' overstates what is proved; 'the improvement is at most of order σ/(d(d−1))' would be more accurate.
  3. [§3.3, Conjecture 1.16] The remark that proving the α=3/2 case with σ>0 suffices for the conjecture is not justified in the text. The interpolation argument should be indicated, or the remark removed.
  4. [§1.3.1, Proposition 1.6] The necessity part of Proposition 1.6 is asserted without proof, with a reference to [31]. Since the proposition is stated as a result of this paper, it would be helpful to include the short counterexample or make the reference more precise.

Circularity Check

1 steps flagged · score 4.0 of 10

The core renormalised Strichartz estimates are derived self-containedly; the advertised well-posedness application is not, because Lemma 4.8 depends on two technical results from the authors' unpublished preprint [6].

  1. self citation load bearing [Section 4.1, proof of Lemma 4.8 (around (4.7)); also used in Lemma 4.10]
    "Hence it follows from [6, Corollary 4.9] that ... (4.7), where we note that the second integral is now taken over the interval [0,t]."

    Lemma 4.8 is the key input that lets the fixed-point map in Proposition 4.2 be bounded in S^2. The displayed S^2 bound for the truncated time integral is not proven here; it is imported from [6], an unpublished preprint co-authored by S. Hadama. The same preprint's Lemma 4.1 is then invoked a few lines later to conclude the bound for the double integral. Thus the well-posedness half of Theorem 1.19 reduces to an unverified self-citation for exactly the estimate that makes the S^2 contraction work.

full rationale

The main orthonormal Strichartz estimates (Theorems 1.12 and 1.14) are proven from scratch: Theorem 1.12 uses Lemma 3.2 and a duality argument; Theorem 1.14 uses the counting estimate (3.9) and interpolation, with an independent necessity example. These arguments do not import the target result, so the paper's central estimates are not circular. The only load-bearing self-citation is in Section 4: the proof of Lemma 4.8 (and Lemma 4.10) calls on [6, Cor. 4.9] and [6, Lemma 4.1] to pass from full time integrals to truncated ones in S^2 norms. Since [6] is an unreviewed preprint by one of the present authors and the needed statements are not reproduced or proved, the well-posedness part of Theorem 1.19/Proposition 4.2 is not self-contained. This is a foundation risk rather than a fitted-prediction circularity: no parameter is fitted and the central estimates do not reduce to their inputs. I also note Remark 3.4 is self-referential ('we know is optimal' before the optimality proof is given), but it is not used as a premise in the proof. The advertised range alpha in [1,2] does follow from the S^2 well-posedness once the latter is available, since S^alpha is contained in S^2 for alpha < 2; the bottleneck is upstream. Overall score 4: some load-bearing self-citation, while the central estimates have independent content.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No free parameters are fitted to data. The central claims rest on standard harmonic analysis, Schatten space theory, Bourgain's torus Strichartz estimates, and a set of technical lemmas from the cited literature, including an unpublished preprint by a co-author.

assumptions (6)
  • standard math Standard harmonic analysis tools: Fourier series, Parseval, interpolation, Christ–Kiselev lemma, X^{s,b} spaces
    Used throughout Sections 3–4 and Appendix A without proof.
  • standard math Bourgain's L4 and L6 Strichartz estimates on the torus (equations (1.8) and (1.9))
    Invoked in Propositions 1.5, 1.6 and Lemma 4.4; cited to [7], [41].
  • standard math Duality and trace identities for Schatten spaces, including Lemmas 2.4 and 2.5
    Central duality framework used to prove the renormalised density estimates; Lemma 2.4 cited to [15].
  • standard math Nakamura's orthonormal Strichartz estimates on the torus [31] are used as a benchmark, not as an input
    The paper contrasts its results with [31, Theorem 1.4] and gives an alternative proof in Section 5.1.
  • domain assumption Technical estimates from [6] (Borie–Hadama–Sabin preprint), specifically [6, Cor 4.9] and [6, Lemma 4.1]
    These estimates are used in the proof of Lemma 4.8 and Lemma 4.10 without being stated in the paper; [6] is an unpublished preprint by one co-author.
  • standard math Existence and unitarity of the propagator U_V(t,s) for real-valued V∈L^2_{t,x} (Appendix A)
    Proved in Appendix A using X^{s,b} spaces and mass conservation; required for all well-posedness arguments.

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Pith. "Pith review of Applications of renormalisation to orthonormal Strichartz estimates and the NLS system on the circle." pith.science (2026). https://pith.science/paper/XHJRF2KV

@misc{pith2026260400252,
  author       = {Pith},
  title        = {Pith review of: Applications of renormalisation to orthonormal Strichartz estimates and the NLS system on the circle},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XHJRF2KV}},
  note         = {Machine review of arXiv:2604.00252}
}
abstract

In this paper, we introduce a renormalisation procedure for the density associated with the system of nonlinear Schr\"odinger equations (NLSS) on a circle. We show that this renormalised density satisfies better orthonormal Strichartz estimates than the non-renormalised density, which was considered in Nakamura (2020). As an application, we determine the critical Schatten exponent below which the cubic renormalised NLSS on the circle is globally well-posed and above which it is ill-posed. Finally, we show that the improvement for orthonormal Strichartz estimates satisfied by the renormalised density on $\mathbb{T}^d$ for $d \ge 2$ is minimal.

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Works this paper leans on

41 extracted references · 2 linked inside Pith

  1. [6]

    Borie, S

    A. Borie, S. Hadama, and J. Sabin. Scattering for the positive density Hartree equation. Preprint, arXiv:2504.19552 [math.AP] (2025), 2025

    arXiv 2025

  2. [1]

    R. L. Acosta Babb and A. Rout. The well-posedness and convergence of higher-order Hartree equations in critical Sobolev spaces onT 3.Discrete Contin. Dyn. Syst., 48:376–403, 2026

    2026

  3. [2]

    N. Bez, Y. Hong, S. Lee, S. Nakamura, and Y. Sawano. On the Strichartz estimates for orthonormal systems of initial data with regularity.Adv. Math., 354:37, 2019. Id/No 106736

    2019

  4. [3]

    N. Bez, S. Kinoshita, and S. Shiraki. Boundary Strichartz estimates and pointwise convergence for orthonormal systems.Trans. Lond. Math. Soc., 11:43, 2024. Id/No e70002

    2024

  5. [4]

    N. Bez, S. Lee, and S. Nakamura. Maximal estimates for the Schr¨ odinger equation with orthonormal initial data.Sel. Math., New Ser., 26(4):24, 2020. Id/No 52

    2020

  6. [5]

    N. Bez, S. Lee, and S. Nakamura. Strichartz estimates for orthonormal families of initial data and weighted oscillatory integral estimates.Forum Math. Sigma, 9:52, 2021. Id/No e1

    2021

  7. [7]

    Bourgain

    J. Bourgain. Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. I: Schr¨ odinger equations.Geom. Funct. Anal., 3(2):107–156, 1993

    1993

  8. [8]

    Bourgain and C

    J. Bourgain and C. Demeter. Improved estimates for the discrete Fourier restriction to the higher dimensional sphere.Ill. J. Math., 57(1):213–227, 2013

    2013

Show all 41 references
  1. [9]

    Chen and A

    T. Chen and A. Bowles Urban. On the well-posedness and stability of cubic and quintic nonlinear Schr¨ odinger systems onT 3.Ann. Henri Poincar´ e, 25(2):1657–1692, 2024

    2024

  2. [10]

    T. Chen, Y. Hong, and N. Pavlovi´ c. Global well-posedness of the NLS system for infinitely many fermions. Arch. Ration. Mech. Anal., 224(1):91–123, 2017

    2017

  3. [11]

    T. Chen, Y. Hong, and N. Pavlovi´ c. On the scattering problem for infinitely many fermions in dimensionsd≥3 at positive temperature.Ann. Inst. Henri Poincar´ e, Anal. Non Lin´ eaire, 35(2):393–416, 2018

    2018

  4. [12]

    X. Dong. The Hartree equation with a constant magnetic field: well-posedness theory.Lett. Math. Phys., 111(4):43, 2021. Id/No 101

    2021

  5. [13]

    M. B. Erdoˇ gan and N. Tzirakis.Dispersive partial differential equations. Wellposedness and applications, volume 86 ofLond. Math. Soc. Stud. Texts. Cambridge: Cambridge University Press, 2016

    2016

  6. [14]

    G. Feng, S. S. Mondal, M. Song, and H. Wu. Orthonormal Strichartz inequalities and their applications on abstract measure spaces.Math. Z., 312(2):42, 2026. Id/No 55

    2026

  7. [15]

    R. L. Frank, M. Lewin, E. H. Lieb, and R. Seiringer. Strichartz inequality for orthonormal functions.J. Eur. Math. Soc. (JEMS), 16(7):1507–1526, 2014

    2014

  8. [16]

    R. L. Frank and J. Sabin. Restriction theorems for orthonormal functions, Strichartz inequalities, and uniform Sobolev estimates.Am. J. Math., 139(6):1649–1691, 2017

    2017

  9. [17]

    S. Hadama. Asymptotic stability of a wide class of stationary solutions for the Hartree and Schr¨ odinger equations for infinitely many particles.Ann. Henri Lebesgue, 8:181–218, 2025

    2025

  10. [18]

    Hadama and Y

    S. Hadama and Y. Hong. Global well-posedness of the nonlinear Hartree equation for infinitely many particles with singular interaction.J. Funct. Anal., 289(9):38, 2025. Id/No 111102

    2025

  11. [19]

    Hadama and T

    S. Hadama and T. Yamamoto. Probabilistic Strichartz estimates in Schatten classes and their applications to the Hartree equation.J. Math. Phys., 67(1):22, 2026. Id/No 011504

    2026

  12. [20]

    A. Hoshiya. Orthonormal Strichartz estimate for dispersive equations with potentials.J. Funct. Anal., 286(11):63, 2024. Id/No 110425

    2024

  13. [21]

    A. Hoshiya. Orthonormal Strichartz estimates for Schr¨ odinger operator and their applications to infinitely many particle systems.J. Math. Phys., 66(7):19, 2025. Id/No 071505

    2025

  14. [22]

    A. Hoshiya. Semiclassical limit of orthonormal Strichartz estimates on scattering manifolds.Pure Appl. Anal., 7(4):1173–1207, 2025

    2025

  15. [23]

    A. Hoshiya. Uniform resolvent and orthonormal strichartz estimates for repulsive hamiltonian.Ann. Henri Poincar´ e, 2025

    2025

  16. [24]

    C. E. Kenig, G. Ponce, and L. Vega. Oscillatory integrals and regularity of dispersive equations.Indiana Univ. Math. J., 40(1):33–69, 1991

    1991

  17. [25]

    Kishimoto

    N. Kishimoto. Remark on the periodic mass critical nonlinear Schr¨ odinger equation.Proc. Am. Math. Soc., 142(8):2649–2660, 2014

    2014

  18. [26]

    Lewin and J

    M. Lewin and J. Sabin. The Hartree equation for infinitely many particles. II: Dispersion and scattering in 2d. Anal. PDE, 7(6):1339–1363, 2014. ORTHONORMAL STRICHARTZ ESTIMATES FOR RENORMALISED DENSITIES 27

    2014

  19. [27]

    Lewin and J

    M. Lewin and J. Sabin. The Hartree equation for infinitely many particles. I: Well-posedness theory.Commun. Math. Phys., 334(1):117–170, 2015

    2015

  20. [28]

    Y. Luo, X. Yu, H. Yue, and Z. Zhao. On well-posedness results for the cubic-quintic NLS onT 3.Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods, 257:9, 2025. Id/No 113806

    2025

  21. [29]

    I. Maktouf. On the schatten exponent in orthonormal strichartz estimates for the weinstein operator.Complex Variables and Elliptic Equations, 0(0):1–25, 2026

    2026

  22. [30]

    S. S. Mondal and M. Song. Orthonormal Strichartz inequalities for the (k, a)-generalized Laguerre operator and Dunkl operator.Isr. J. Math., 269(2):697–729, 2025

    2025

  23. [31]

    Nakamura

    S. Nakamura. The orthonormal Strichartz inequality on torus.Trans. Am. Math. Soc., 373(2):1455–1476, 2020

    2020

  24. [32]

    T. T. Nguyen and C. You. Modified scattering for long-range Hartree equations of infinite rank near vacuum. SIAM J. Math. Anal., 57(6):6486–6497, 2025

    2025

  25. [33]

    T. T. Nguyen and C. You. Plasmons for the Hartree equations with Coulomb interaction.Probab. Math. Phys., 6(3):913–960, 2025

    2025

  26. [34]

    Pusateri and I

    F. Pusateri and I. M. Sigal. Long-time behaviour of time-dependent density functional theory.Arch. Ration. Mech. Anal., 241(1):447–473, 2021

    2021

  27. [35]

    P. J. K. Senapati and P. Boggarapu. Strichartz inequality for orthonormal functions associated with Dunkl Laplacian and Hermite–Schr¨ odinger operators.J. Anal., 2025

    2025

  28. [36]

    P. J. K. Senapati, P. Boggarapu, S. S. Mondal, and H. Mejjaoli. Restriction theorem for the Fourier-Dunkl transform and its applications to Strichartz inequalities.J. Geom. Anal., 34(3):36, 2024. Id/No 74

    2024

  29. [37]

    Simon.Trace ideals and their applications, volume 120 ofMath

    B. Simon.Trace ideals and their applications, volume 120 ofMath. Surv. Monogr.Providence, RI: American Mathematical Society (AMS), 2nd ed. edition, 2005

    2005

  30. [38]

    Tao.Nonlinear dispersive equations

    T. Tao.Nonlinear dispersive equations. Local and global analysis, volume 106 ofCBMS Reg. Conf. Ser. Math. Providence, RI: American Mathematical Society (AMS), 2006

    2006

  31. [39]

    X. Wang, A. Zhang, and C. Zhang. Strichartz estimates for orthonormal systems on compact manifolds. Preprint, arXiv:2503.08504 [math.AP] (2025), 2025

    2025 arXiv

  32. [40]

    X. Zhang. On the Cauchy problem of 3d energy-critical Schr¨ odinger equations with subcritical perturbations. J. Differ. Equations, 230(2):422–445, 2006

    2006

  33. [41]

    A. Zygmund. On Fourier coefficients and transforms of functions of two variables.Stud. Math., 50:189–201, 1974. Research Institute for Mathematical Sciences, Kyoto University, Kita-Shirakawa, Sakyo-ku, Kyoto, Japan 606-8502. Email address:hadama@kurims.kyoto-u.ac.jp Dipartimen...

    1974

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