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Decay and Strichartz estimates for critical electromagnetic wave equations on conic manifolds

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Pith's one-line read This paper proves decay and Strichartz estimates for the wave equation with large, scaling-critical electromagnetic potentials on conical manifolds, closing the gap left by earlier results when the cross-section is a sphere.

desk verdict Strong result, but the central decay estimate in Section 4.3 only gives (1+|t|)^{-1/2} where (1+|t|)^{-(n-1)/2} is claimed; the Strichartz theorems rest on that missing power. read the letter →

arxiv 2506.09635 v1 pith:WEJHHC33 submitted 2025-06-11 math.AP math.DG

Qiuye Jia , Junyong Zhang This is my paper · ORCID
classification math.APmath.DG MSC 42B3735Q4035Q41
keywords waveequationconicmanifoldsscaling-criticalelectromagneticpotentialsStrichartzestimatesdecayspectralmeasurenon-focusingconditionCoulomb-type
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 establishes that the decay and Strichartz estimates for the wave equation survive when a magnetic potential with Coulomb-type decay (decaying like $1/r$) and an inverse-square electric potential are present on a conical manifold $X=C(Y)$ of dimension $n\ge 3$, provided the cone's cross-section $Y$ has a non-focusing geodesic flow up to time $\pi$. For the model case $Y=\mathbb{S}^{n-1}$, this covers potentials that are large and only scaling critical, whereas previous results required small or faster-decaying potentials. The admissible integrability pairs are exactly those of the free wave equation, with a restriction $p

What carries the argument

The engine is a localized spectral measure for $\sqrt{L_{A,a}}$. Theorem 3.1 writes the kernel as an oscillatory integral built from $\cos(s\sqrt P)$ and $\sin(\pi\sqrt P)e^{-s\sqrt P}$, where $P=P_{A,a}$ is the angular operator lifted to the cross-section; the Bessel-function expansion (3.37) supplies the leading terms. Hadamard parametrices from Lemma 2.2 and Lemma 2.3 express the wave and Poisson propagators on $Y$, and the non-focusing condition makes the sheets of the propagating Lagrangian project diffeomorphically onto $Y\times Y$, so conjugate point pairs can be separated by a partition of unity. A dyadic stationary-phase analysis of the resulting kernels yields the pointwise decay, which feeds into an abstract Keel-Tao Strichartz argument.

What would settle it

Compute, for $Y=\mathbb{S}^{n-1}$ with $0<\nu_0<(n-2)/2$, the solution from initial data $f=\psi_0(\hat x)[H_{\nu_0}\chi](r)$ with $\chi\in C_c^\infty([1,2])$ and test whether $\|e^{it\sqrt{L_{A,a}}}f\|_{L^q(\mathbb{R};L^p(X))}$ is finite at $p=p(\alpha)=n/|\alpha|$; the paper's Proposition 5.5 predicts divergence, so any finite value would disprove the necessity of $p<p(\alpha)$. Alternatively, on a flat torus cross-section containing a closed geodesic of length $\pi$, check whether the boundary cancellation between the $s=\pi$ and $s=0$ terms in Lemma 6.1 yields the claimed $(1+|t|)^{-(n-1)/2}$ decay; failure there would mean the removal of the technical NREC assumption in Section 6 is not justified.

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Extended reading notes

Core claim

The central claim, Theorem 1.2, is that if $P_{A,a}=L_{A,a}+(n-2)^2/4$ is strictly positive with lowest eigenvalue root $\nu_0$, then for every admissible pair $(q,p)\in\Lambda_{s,\alpha}(\nu_0)$ the solution of $\partial_t^2 u + L_{A,a}u=0$ satisfies $\|u\|_{L^q(\mathbb{R};L^p(X))} \le C(\|u_0\|_{\dot H^s_{A,a}(X)}+\|u_1\|_{\dot H^{s-1}_{A,a}(X)})$, with the boundary $p(\alpha)=\infty$ if $\alpha\ge 0$ and $p(\alpha)=n/|\alpha|$ if $\alpha=\nu_0-(n-2)/2<0$. The companion estimate, Theorem 1.1, gives a pointwise $(1+2^k|t|)^{-(n-1)/2}$ decay bound for the frequency-localized half-wave kernel away from the cone tip. The paper also proves the inhomogeneous Strichartz estimates and shows that the condition $p<p(\alpha)$ is necessary by exhibiting data whose $L^qL^p$ norm diverges at the boundary.

Load-bearing premise

The load-bearing premise is that the cross-section $Y$ is non-focusing up to time $\pi$: every pair of nearby points is joined by finitely many geodesics whose families project smoothly onto $Y\times Y$; if this fails, the parametrix and the resulting estimates are not justified.

Editorial extensions

If this is right

  • For $Y=\mathbb{S}^{n-1}$, the decay and Strichartz picture now covers large, scaling-critical electromagnetic potentials, extending the earlier results of [21] and [24] which required small or super-critical decay.
  • When $\alpha\ge 0$ (for instance when the electric potential is non-negative), the admissible set is the full wave-admissible set $\Lambda_s$; when $\alpha<0$ the range shrinks to $p<p(\alpha)$.
  • The inhomogeneous Strichartz estimates (1.35) hold for dual pairs in $\Lambda_{s,\alpha}(\nu_0)\times\Lambda_{1-s,\alpha}(\nu_0)$, including the endpoint case.
  • The restriction $p<p(\alpha)$ is not an artifact of the proof: the counterexample in Proposition 5.5 shows the estimate fails for $p\ge p(\alpha)$ when $0<\nu_0<(n-2)/2$.

Reading between the lines

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

  • Extension the paper leaves implicit: if the non-focusing condition is the sharp criterion, then a cone over a cross-section with a conjugate pair within time $\pi$ should fail the uniform decay estimate (1.23), which gives a concrete geometric test of the boundary.
  • The red line $p<p(\alpha)$ should affect nonlinear applications: the threshold for global well-posedness of critical nonlinear wave equations on such cones will likely need to stay below this boundary, since the linear estimate controls the Duhamel term.
  • The spectral measure identity (3.5) is independent of the Strichartz framework and could be reused to derive resolvent bounds or to treat Klein-Gordon equations with the same potentials, by replacing the half-wave phase $e^{it\lambda}$ with the appropriate dispersion relation.
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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

1 major / 6 minor

Summary. This paper studies the wave equation ∂_t^2 u + L_{A,a}u = 0 on n-dimensional product cones X=C(Y) with metric g=dr^2+r^2h, where the electromagnetic potentials have scaling-critical decay A=A(\hat x)/r and V=a(\hat x)/r^2. Under the non-focusing condition (NFC) on the geodesic flow of the cross-section Y and strict positivity of P_{A,a}=L_{A,a}+(n-2)^2/4, the authors prove frequency-localized pointwise decay estimates for the half-wave propagator (Theorem 1.1) and then use them through abstract Keel-Tao type estimates to prove global Strichartz estimates (Theorem 1.2) for the admissible set Λ_{s,α}(ν0), where α=ν0-(n-2)/2 and p<p(α). They also show by an explicit Hankel-transform counterexample that the restriction p<p(α) is necessary. The proof constructs the spectral measure of L_{A,a}, combines it with a Hadamard parametrix for cos(s√P) and Poisson-wave operators, and splits the kernel into propagating, residual, and diffractive contributions; Section 6 removes the auxiliary non-resonant endpoint condition (NREC).

Significance. If correct, this is the first higher-dimensional Strichartz estimate for wave equations with large scaling-critical electromagnetic potentials, closing the Euclidean gap left by Cuccagna-Schirmer and D'Ancona-Fanelli, and extending the authors' earlier Schrödinger results to wave equations on product cones. The paper is structured as a first-principles proof: the admissible-range restriction is derived from the bottom of the spectrum and is verified necessary by an explicit counterexample, with no fitted parameters. I checked the specific concern raised about Section 4.3: the displayed bounds (4.35)-(4.40) are indeed (1+|t|)^{-1/2} for the inner integral, but because z=r1r2/|t| and the original prefactor is (r1r2)^{-(n-2)/2}=z^{-(n-2)/2}|t|^{-(n-2)/2}, those bounds do imply the claimed (1+|t|)^{-(n-1)/2}. The stress-test concern about Section 4.3 therefore does not land. The main remaining risk is the deferred proof of the endpoint-cancellation identity in Section 6, which is load-bearing for the removal of NREC.

major comments (1)
  1. [Section 6, Lemma 6.1 and Proposition 6.1] The identity (6.6) is load-bearing for the removal of the NREC assumption, yet no proof is supplied; the text says only 'See [49, Lemma 4.4] for details of the computation.' This is not a routine integration-by-parts identity: after 2m integrations by parts, boundary terms at s=π from the first integral and at s=0 from the second must cancel through the relation between |m_{π-u}| and |n_u|, and the argument must handle derivatives of W(t,·) as well as the amplitude matching in (2.31). Since [49] is an unpublished preprint treating a different (Schrödinger) problem, please include a self-contained proof of (6.6), or a precise statement of the contour-deformation and symbol-matching argument, before the non-NREC results can be considered established.
minor comments (6)
  1. [Section 4.3, after (4.40)] To make the final exponent transparent, state explicitly that (r1r2)^{-(n-2)/2} = z^{-(n-2)/2}|t|^{-(n-2)/2}, so the displayed (1+|t|)^{-1/2} bounds for the inner integral yield (1+|t|)^{-(n-1)/2} for the full kernel.
  2. [Section 4.3, Case 2] In the dyadic estimates for d_h(\hat x,\hat y)≥C1z^{-1/2}, the text twice refers to 'the integral in (6.29)'; this should refer to the corresponding display in Section 4.3 (e.g., (4.40)), not to the later equation number.
  3. [Proposition 5.3] The quantifier 'for 2≤q<q(σ)' does not match the estimate (5.18), which is an L^p norm in r1; it should presumably read 'for 2≤p<p(σ)'.
  4. [Theorem 1.1] The phrase 'positive square roof' should read 'positive square root'.
  5. [Abstract and keywords] The word 'Strihcartz' is a typo for 'Strichartz'.
  6. [Section 5.4] In the displayed estimate for ∥P∥, the Lebesgue exponents for the time and radial variables appear to be interchanged; the expression should likely be L^q_t([0,1/4];L^p_{rn-1dr}[ϵ,1]).

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the derivation is a first-principles spectral and parametrix proof; self-citations supply independent tools rather than fitted outputs.

full rationale

The paper's central claims (Theorem 1.1 dispersive estimates and Theorem 1.2 Strichartz estimates) are derived by a genuine proof chain: the spectral measure representation (3.5) is obtained from Stone's formula and resolvent identities, the small-frequency bound (3.7) is established from the Bessel/eigenfunction expansion (3.37), and the large-frequency estimate (1.23) is attacked by a stationary-phase/parametrix argument. The admissible restriction p < p(alpha) is not an input: alpha = -(n-2)/2 + nu0 is computed from the lowest eigenvalue, and Proposition 5.5 independently tests necessity against a Hankel/Bessel counterexample. No parameter is fitted to data, and no estimate is renamed as a prediction. The reliance on the authors' previous works [49,50] for the Schroedinger propagator representation, Hadamard parametrix, and Bernstein/square-function inequalities is real but not circular: those statements are parameter-free, are stated with assumptions (NFC, strict positivity of P_{A,a}) that do not include the target Strichartz estimate, and are not derived from Theorem 1.1 or 1.2 in this paper. The manuscript even notes the new ingredients needed for wave versus the previous Schroedinger work (Remark 1.4). The contextual objection about Section 4.3 proving only (1+|t|)^{-1/2} rather than (1+|t|)^{-(n-1)/2} in (4.35)-(4.36) and (4.39)-(4.40) is a possible correctness gap in the proof of (1.23); if valid it would weaken or invalidate Theorem 1.2, but it is not an example of a conclusion being equivalent to an input by construction, and therefore does not affect the circularity score. No self-definitional, fitted-input, renamed-knowledge, or uniqueness-forbidding step appears.

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

The proof relies on the NFC hypothesis, the strict positivity of P_{A,a}, and the parametrix machinery from the authors' prior work [49,50]; no fitted constants or ad hoc invented objects appear.

assumptions (5)
  • domain assumption The cross-section (Y,h) satisfies the non-focusing condition (NFC) or the non-resonant endpoint condition (NREC) in Definition 1.1 / Definition 2.1.
    Theorem 1.1 is stated under NFC; Section 4 adds NREC as a technical condition removed in Section 6. Proposition 1.1 supplies curvature classes where NFC holds.
  • domain assumption The operator P_{A,a} = L_{A,a} + (n-2)^2/4 is strictly positive on Y, with smallest eigenvalue ν0^2 > 0.
    Theorem 1.1 and Theorem 1.2 assume strict positivity; the range of admissible p in (1.32) depends on ν0.
  • standard math The parametrix construction of the half-wave and Poisson-wave propagators from [49,50] is valid for the operator P_{A,a} on Y.
    Lemmas 2.1-2.3 in Section 2.3 rely on the parametrix results and the NFC property; these are the same authors' previously established results, quoted as [49,50].
  • standard math Bessel function estimates and spectral kernel expansions (Lemma A.1, formulas (3.37), (3.46)-(3.48)) hold with constants uniform over the relevant eigenfunction range.
    Used in Section 3.2 and Proposition 5.3 for the pointwise bounds and the p-range restriction; standard references are cited.
  • standard math The generalized Keel-Tao abstract Strichartz machinery (Proposition 5.1) applies to the microlocalized half-wave operators U_{k,j}.
    Used in Section 5.1 to convert the dispersive bound into Strichartz estimates; follows the usual bilinear argument in [52].

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Pith. "Pith review of Decay and Strichartz estimates for critical electromagnetic wave equations on conic manifolds." pith.science (2026). https://pith.science/paper/WEJHHC33

@misc{pith2026250609635,
  author       = {Pith},
  title        = {Pith review of: Decay and Strichartz estimates for critical electromagnetic wave equations on conic manifolds},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WEJHHC33}},
  note         = {Machine review of arXiv:2506.09635}
}
abstract

We establish the decay and Strichartz estimates for the wave equation with large scaling-critical electromagnetic potentials on a conical singular space $(X,g)$ with dimension $n\geq3$, where the metric $g=dr^2+r^2 h$ and $X=C(Y)=(0,\infty)\times Y$ is a product cone over the closed Riemannian manifold $(Y,h)$ with metric $h$. The decay assumption on the magnetic potentials is scaling critical and includes the decay of Coulomb type. The main technical innovation lies in proving localized pointwise estimates for the half-wave propagator by constructing a localized spectral measure, which effectively separates contributions from conjugate point pairs on $\CS$. In particular, when $Y=\mathbb{S}^{n-1}$, our results, which address the case of large critical electromagnetic potentials, extend and improve upon those in [21], which considered sufficiently decaying, and small potentials and that of [24], which considered potentials decaying faster than scaling critical ones.

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Works this paper leans on

77 extracted references · 75 canonical work pages

  1. [49]

    Jia and J

    Q. Jia and J. Zhang, Pointwise dispersive estimates for Schr¨ odinger and wave equations in a conical singular space, Preprint, arXiv:2411.16029. 2, 8, 12, 13, 15, 16, 17, 18, 19, 21, 33, 54

    arXiv

  2. [1]

    Aharonov and D

    Y. Aharonov and D. Bohm, Significance of electromagnetic potentials in the quantum theory, Phys. Rev., 115(1959), 485-491. 5

    work page 1959

  3. [2]

    Avron, I

    J. Avron, I. Herbst and B. Simon, Schr¨ odinger operators with magnetic fields, I. General interac- tions, Duke Math. J. 45 (1978), 847-883. 3

    work page 1978

  4. [3]

    Avron, I

    J. Avron, I. Herbst and B. Simon, Schr¨ odinger operators with magnetic fields, II. Separation of center of mass in homogeneous magnetic fields, Ann. Phys. 114 (1978), 431-451. 3

    work page 1978

  5. [4]

    Avron, I

    J. Avron, I. Herbst and B. Simon, Schr¨ odinger operators in magnetic fields, III. Atoms in homo- geneous magnetic field, Comm. Math. Phys. 79 (1981), 529-572. 3

    work page 1981

  6. [5]

    Baskin, J

    D. Baskin, J. L. Marzuola, The radiation field on product cones Adv. Math (408), 2022 12

    work page 2022

  7. [6]

    Baskin, J

    D. Baskin, J. Gell-Redman and J. L. Marzuola, Price’s law for the massless Dirac–Coulomb system, Pac. J. Math (335) 2, 211 - 227, 2025 12

    work page 2025

  8. [7]

    Beals, OptimalL ∞ decay for solutions to the wave equation with a potential, Comm

    M. Beals, OptimalL ∞ decay for solutions to the wave equation with a potential, Comm. Partial Differential Equations 19 (1994), 1319–1369. 11

    work page 1994

Show all 77 references
  1. [8]

    Beceanu, M

    M. Beceanu, M. Goldberg, Dispersive estimates for the Schr¨ odinger equation with scaling-critical potential, Comm. Math. Phys. 314(2012), 471–481. 10, 11

    2012

  2. [9]

    Beceanu, H.-K

    M. Beceanu, H.-K. Kwon, Decay estimates for Schr¨ odinger’s equation with magnetic potentials in three dimensions, arXiv:2411.11787. 10, 11 [10]M. D. Blair, G. A. Ford, and J. L. Marzuola,Strichartz estimates for the wave equation on flat cones, IMRN, 2012, 30 pages, doi:10.10...

    2012 arXiv

  3. [11]

    N. Burq, F. Planchon, J. Stalker, and A. S. Tahvildar-Zadeh, Strichartz estimates for the wave and Schr¨ odinger equations with the inverse-square potential, J. Funct. Anal., 203 (2003), 519-549. 10, 45

    2003

  4. [12]

    N. Burq, F. Planchon, J. G. Stalker, and A. S. Tahvildar-Zadeh, Strichartz estimates for the wave and Schr¨ odinger equations with potentials of critical decay, Indiana Univ. Math. J., 53(2004), 1665-1680. 10

    2004

  5. [13]

    Carron, Le saut en z´ ero de la fonction de d´ ecalage spectral, J

    G. Carron, Le saut en z´ ero de la fonction de d´ ecalage spectral, J. Funct. Anal., 212 (2004), 222-260. 11

    2004

  6. [14]

    Cacciafesta, P

    F. Cacciafesta, P. D’Ancona, Z. Yin and J. Zhang, Dispersive estimates for Dirac equations in Aharonov-Bohm magnetic fields: massless case, arXiv:2407.12369. 43, 68

    arXiv

  7. [15]

    Chen, The semiclassical resolvent on conic manifolds and application to Schr¨ odinger equations

    X. Chen, The semiclassical resolvent on conic manifolds and application to Schr¨ odinger equations. Comm. Math. Phy., 2(2022),757-826. 11

    2022

  8. [16]

    Cheeger, On the spectral geometry of spaces with cone-like singularities, Proceedings of the National Academy of Sciences of the United States of America, 76(1979), 2103-2106

    J. Cheeger, On the spectral geometry of spaces with cone-like singularities, Proceedings of the National Academy of Sciences of the United States of America, 76(1979), 2103-2106

    1979

  9. [17]

    Cheeger, Spectral geometry of singular Riemannian spaces, J

    J. Cheeger, Spectral geometry of singular Riemannian spaces, J. Diff. Geo. 18(1983), 575-657

    1983

  10. [18]

    Cheeger, D

    J. Cheeger, D. Gromoll, On the lower bound for the injectivity radius of 1/4-pinched Riemannian manifolds, J. Differential Geom. 15(1980) 437–442. 67 70 QIUYE JIA AND J. ZHANG

    1980

  11. [19]

    Christ, and A

    M. Christ, and A. Kiselev, Maximal functions associated to filtrations, J. Funct. Anal., 179(2001), 409-425. 51

    2001

  12. [20]

    Colton and R

    D. Colton and R. Kress, Inverse acoustic and electromagnetic scattering theory, volume 93 of Applied Mathematical Sciences. Springer-Verlag, Berlin, 1992. 24

    1992

  13. [21]

    Cuccagna, and Schirmer, On the wave equation with a magnetic potential, Comm

    S. Cuccagna, and Schirmer, On the wave equation with a magnetic potential, Comm. Pure Appl. Math., 54(2001), 135-152. 1, 2, 5, 10, 11

    2001

  14. [22]

    Cheeger, M

    J. Cheeger, M. Taylor, Diffraction of waves by Conical Singularities parts I, Comm. Pure Appl. Math., 35(1982), 275-331. 11, 13, 21

    1982

  15. [23]

    Cheeger, M

    J. Cheeger, M. Taylor, Diffraction of waves by Conical Singularities parts II, Comm. Pure Appl. Math., 35(1982), 487-529. 11, 13, 21

    1982

  16. [24]

    D’Ancona, and L

    P. D’Ancona, and L. Fanelli, Decay estimates for the wave and Dirac equations with a magnetic potential, Comm. Pure Appl. Math., 60(2007), 357-392. 1, 2, 5, 10, 11

    2007

  17. [25]

    D’Ancona, L

    P. D’Ancona, L. Fanelli, L. Vega, and N. Visciglia, Endpoint Strichartz estimates for the magnetic Schr¨ odinger equation, J. Funct. Anal., 258(2010), 3227-3240. 2, 3, 5, 10, 11

    2010

  18. [26]

    Deser and R

    S. Deser and R. Jackiw R, Classical and quantum scattering on a cone, Commun. Math. Phys. 118(1988), 495-509. 3

    1988

  19. [27]

    Ehrenberg, R.E

    W. Ehrenberg, R.E. Siday, The refractive index in electron optics and the principles of dynamics, Proceedings of the Physical Society. Section B, 62(1949), 8-21. 5

    1949

  20. [28]

    Erdogan, M

    M.B. Erdogan, M. Goldberg and W. Schlag, Strichartz and Smoothing Estimates for Schr¨ odinger Operators with Almost Critical Magnetic Potentials in Three and Higher Dimensions, Forum Math., 21(2009), 687-722. 2, 3, 5, 10, 11

    2009

  21. [29]

    Erdogan, M

    M.B. Erdogan, M. Goldberg and W. Schlag, Strichartz and smoothing estimates for Schr¨ odinger operators with large magnetic potentials inR 3, J. European Math. Soc., 10(2008), 507-531. 2, 3, 5, 10, 11

    2008

  22. [30]

    Erd´ elyi, W

    A. Erd´ elyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi. Higher transcendental functions. Vol. II. (1953). 68

    1953

  23. [31]

    Fanelli, Spherical Schr¨ odinger Hamiltonians: Spectral Analysis and Time Decay, A

    L. Fanelli, Spherical Schr¨ odinger Hamiltonians: Spectral Analysis and Time Decay, A. Michelan- geli, G. Dell’Antonio (eds.), Advances in Quantum Mechanics, Springer INdAM Series 18, DOI 10.1007/978-3-319-58904-6 8. 2

    doi

  24. [32]

    G. A. Ford, The fundamental solution and Strichartz estimates for the Schr¨ odinger equation on flat Euclidean cones, Comm. Math. Phys., 299(2010), 447-467. 11

    2010

  25. [33]

    G. A. Ford, and J. Wunsch, The diffractive wave trace on manifolds with conic singularities, Adv. in Math. 304 (2017), 1330-1385. 11

    2017

  26. [34]

    Fanelli, V

    L. Fanelli, V. Felli, M. A. Fontelos, and A. Primo, Time decay of scaling critical electromagnetic Schr¨ odinger flows, Comm. Math. Phys., 324(2013), 1033-1067. 3, 5, 10, 12

    2013

  27. [35]

    Fanelli, V

    L. Fanelli, V. Felli, M. A. Fontelos, and A. Primo, Time decay of scaling invariant electromagnetic Schr¨ odinger equations on the plane, Comm. Math. Phys., 337(2015), 1515-1533. 3, 5, 10, 12

    2015

  28. [36]

    Fanelli, and L

    L. Fanelli, and L. Vega, Magnetic virial identities, weak dispersion and Strichartz inequalities, Math. Ann. 344(2009), 249-278. 2, 3, 5, 11

    2009

  29. [37]

    Felli, A

    V. Felli, A. Ferrero, S. Terracini, Asymptotic behavior of solutions to Schr¨ odinger equations near an isolated singularity of the electromagnetic potential. J. Eur. Math. Soc. 13(2011), 119-174. 5, 8, 13

    2011

  30. [38]

    X. Gao, Z. Yin, J. Zhang, J. Zheng, Decay and Strichartz estimates in critical electromagnetic fields. Journal of Functional Analysis. 2022;282(5). 3, 5, 10, 12

    2022

  31. [39]

    Fanelli, J

    L. Fanelli, J. Zhang, J. Zheng, Dispersive estimates for 2D-wave equations with critical potentials. Advances in Mathematics. 2022;400. 3, 5, 10, 12

    2022

  32. [40]

    G ˚ arding On the asymptotic distribution of the eigenvalues and eigenfunctions of elliptic differ- ential operators

    L. G ˚ arding On the asymptotic distribution of the eigenvalues and eigenfunctions of elliptic differ- ential operators. Mathematica Scandinavica. 1953, 237-255. 28, 46

    1953

  33. [41]

    de Sousa Gerbert and R

    P. de Sousa Gerbert and R. Jackiw, Classical and quantum scattering on a spinning cone, Commun. Math. Phys. 124(1989); 229-260. 3

    1989

  34. [42]

    Guillarmou, and A

    C. Guillarmou, and A. Hassell, Uniform Sobolev estimates for non-trapping metrics, Math. Ann, (2008),341: 859–896. 27

    2008

  35. [43]

    Hassell and P

    A. Hassell and P. Lin, The Riesz transform for homogeneous Schr¨ odinger operators on metric cones. Rev. Mat.Iberoamericana 30(2014),477-522. 11, 26, 27 STRICHARTZ ESTIMATES FOR ELECTROMAGNETIC W A VE 71

    2014

  36. [44]

    G.’t Hooft, Nonperturbative 2 particle scattering amplitudes in 2+1-dimensional quantum gravity, Comm. Math. Phys. 117(1988), 685-700. 3

    1988

  37. [45]

    H¨ ormander, The analysis of linear partial differential operators III: pseudo-differential operators

    L. H¨ ormander, The analysis of linear partial differential operators III: pseudo-differential operators. Springer Berlin-Heidelberg 2009

    2009

  38. [46]

    H¨ ormander, Fourier integral operators I, Acta

    L. H¨ ormander, Fourier integral operators I, Acta. Math., 127(1971), 79-183. 17

    1971

  39. [47]

    Huang and J

    X. Huang and J. Zhang, Heat kernel estimate in a conical singular space, J. Geometry Anal., 33(2023), 284. 19

    2023

  40. [48]

    Journe, A

    J.-L. Journe, A. Soffer, and C. D. Sogge, Decay estimates for Schr¨ odinger operators, Comm. Pure Appl. Math. 44 (1991), 573-604. 11

    1991

  41. [50]

    Jia and J

    Q. Jia and J. Zhang, Strichartz estimates for Schr¨ odinger equation in high dimensional critical electromagnetic fields, Preprint, arXiv:2412.02296. 2, 5, 8, 12, 13, 16, 19, 21, 33, 54

    arXiv

  42. [51]

    Keeler and J

    B. Keeler and J. L. Marzuola, Pointwise dispersive estimates for Schr¨ odinger operators on product cones, J. Diff. Equ., 320(2022), 419-468. 11

    2022

  43. [52]

    Keel and T

    M. Keel and T. Tao, Endpoint Strichartz estimates, Amer. J. Math., 120(1998), 955-980. 8, 43, 44

    1998

  44. [53]

    Klingenberg, ¨Uber Riemannsche Mannigfaltigkeiten mit nach oben beschr¨ ankter Kr¨ ummung, Ann

    W. Klingenberg, ¨Uber Riemannsche Mannigfaltigkeiten mit nach oben beschr¨ ankter Kr¨ ummung, Ann. Mat. Pura Appl. 60 (1962) 49-59. 67

    1962

  45. [54]

    Klingenberg, T

    W. Klingenberg, T. Sakai, Injectivity radius estimate for 1/4-pinched manifolds, Archiv der Math- ematik, 34(1980) 371–376. 67

    1980

  46. [55]

    Abresch, W

    U. Abresch, W. T. Meyer, A sphere theorem with a pinching constant below 1/4, J. Differential Geom. 44(2): 214-261 (1996) 7

    1996

  47. [56]

    Berger, Sur les vari´ et´ es riemanniennes pinc´ ees juste au-dessous de 1/4, Annales de l’institut Fourier, 33(1983) 135-150

    M. Berger, Sur les vari´ et´ es riemanniennes pinc´ ees juste au-dessous de 1/4, Annales de l’institut Fourier, 33(1983) 135-150. 7

    1983

  48. [57]

    Petersen, Riemannian geometry, 2006, Springer

    P. Petersen, Riemannian geometry, 2006, Springer. 7

    2006

  49. [58]

    N. N. Lebedev. Special functions and their applications. Dover Publications Inc., New York, 1972. Revised edition, translated from the Russian and edited by Richard A. Silverman, Unabridged and corrected republication. 24

    1972

  50. [59]

    M¨ uller and A

    D. M¨ uller and A. Seeger, Regularity properties of wave propagation on conic manifolds and appli- cations to spectral multipliers, Adv. in Math. 161(2001), 41-130. 11

    2001

  51. [60]

    Reed, and B

    M. Reed, and B. Simon, Methods of modern mathematical physics. II. Fourier analysis, self- adjointness. Academic Press, New York-London, 1975. 3

    1975

  52. [61]

    Schlag, On pointwise decay of waves

    W. Schlag, On pointwise decay of waves. J. Math. Phys. 62(2021), 061509. 10

    2021

  53. [62]

    Schlag, A

    W. Schlag, A. Soffer and W. Staubach, Decay for the wave and Schr¨ odinger evolutions on manifolds with conical ends, Part I, Trans. AMS 362(2010), 19-52. 11

    2010

  54. [63]

    Schlag, A

    W. Schlag, A. Soffer and W. Staubach, Decay for the wave and Schr¨ odinger evolutions on manifolds with conical ends, Part II, Trans. AMS 362(2010), 289-318. 11

    2010

  55. [64]

    Schlag, Dispersive estimates for Schr¨ odinger operators: a survey

    W. Schlag, Dispersive estimates for Schr¨ odinger operators: a survey. Mathematical aspects of nonlinear dispersive equations, 255-285, Ann. of Math. Stud. 163, Princeton Univ. Press, Princeton, NJ, 2007. 10

    2007

  56. [65]

    C. D. Sogge, Fourier Integrals in Classical Analysis, Cambridge Tracts in Mathematics, vol. 105, Cambridge University Press, Cambridge, 1993. 26

    1993

  57. [66]

    C. D. Sogge, Hangzhou lectures on eigenfunctions of the Laplacian. Annals of Mathematics Studies,

  58. [67]

    Stein (with the assistance of M.S

    E.M. Stein (with the assistance of M.S. Timothy), Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals. 1993. 28, 29

    1993

  59. [68]

    Taira, Dispersive estimates and optimality for Schr¨ odinger equations on product cones, arXiv:2503.21527

    K. Taira, Dispersive estimates and optimality for Schr¨ odinger equations on product cones, arXiv:2503.21527. 4

    arXiv

  60. [69]

    Taylor, Partial Differential Equations, Vol II, Springer, 1996

    M. Taylor, Partial Differential Equations, Vol II, Springer, 1996. 14

    1996

  61. [70]

    G. N. Watson, A Treatise on the Theory of Bessel Functions. Second Edition, Cambridge University Press, 1944. 27, 28

    1944

  62. [71]

    Wang, Asymptotic expansion in time of the Schr¨ odinger group on conical manifolds, Ann

    X. Wang, Asymptotic expansion in time of the Schr¨ odinger group on conical manifolds, Ann. Inst. Fourier, 56(2006), 1903-1945. 11 72 QIUYE JIA AND J. ZHANG

    2006

  63. [72]

    Sitenko Yu, The Aharonov-Bohm effect and the inducing of vacuum charge by a singular magnetic string, Nuclear Physics B, 372(1992), 622-634

    A. Sitenko Yu, The Aharonov-Bohm effect and the inducing of vacuum charge by a singular magnetic string, Nuclear Physics B, 372(1992), 622-634. 3

    1992

  64. [73]

    Yang, Diffraction of the Aharonov–Bohm Hamiltonian, Annales Henri Poincar´ e, 22(2021),3619–3640

    M. Yang, Diffraction of the Aharonov–Bohm Hamiltonian, Annales Henri Poincar´ e, 22(2021),3619–3640. 5

    2021

  65. [74]

    Yang, The wave trace and resonances of the magnetic Hamiltonian with singular vector poten- tials, Comm

    M. Yang, The wave trace and resonances of the magnetic Hamiltonian with singular vector poten- tials, Comm. Math. Phys. 389(2022), 1099–1133. 5

    2022

  66. [75]

    Zhang, Resolvent and spectral measure for Schrodinger operators on flat Euclidean cones, J

    J. Zhang, Resolvent and spectral measure for Schrodinger operators on flat Euclidean cones, J. Funct. Anal. 282(2022), 109311. 11

    2022

  67. [76]

    Zhang and J

    J. Zhang and J. Zheng, Global-in-time Strichartz estimates and cubic Schr¨ odinger equation in a conical singular space, arXiv:1702.05813v3 11, 26

    arXiv

  68. [77]

    Zhang and J

    J. Zhang and J. Zheng, Strichartz estimates and wave equation in a conic singular space, Math. Ann., 376(2020),525–581. 11 Mathematical Sciences Institute, the Australian National University; Email address:Qiuye.Jia@anu.edu.au; Junyong Zhang Department of Mathematics, Beijing ...

    2020

  69. [188]

    Princeton University Press, Princeton, NJ, 2014. 28

    2014

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