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Existence and nonexistence of subsolutions for augmented Hessian equations
Local smoothing and Strichartz estimates for the Klein-Gordon equation with the inverse-square potential
Department of Mathematics, Sungkyunkwan University, Suwon 16419, Republic of Korea |
We prove weighted $ L^2 $ estimates for the Klein-Gordon equation perturbed with singular potentials such as the inverse-square potential. We then deduce the well-posedness of the Cauchy problem for this equation with small perturbations, and go on to discuss local smoothing and Strichartz estimates which improve previously known ones.
References:
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S. Agmon and L. Hörmander,
Asymptotic properties of solutions of differential equations with simple characteristics, J. Analyse Math, 30 (1976), 1-38.
doi: 10.1007/BF02786703. |
[2] |
J. A. Barcelò, J. M. Bennett, A. Carbery, A. Ruiz and M. C. Vilela,
A note on weighted estimates for the Schrödinger operator, Rev. Mat. Complut, 21 (2008), 481-488.
doi: 10.5209/rev_rema.2008.v21.n2.16405. |
[3] |
P. Brenner,
On space-time means and everywhere defined scattering operators for nonlinear Klein-Gordon equations, Math. Z, 186 (1984), 383-391.
doi: 10.1007/BF01174891. |
[4] |
S. Chanillo and E. Sawyer,
Unique continuation for $\Delta + v$ and the C. Fefferman-Phong class, Trans. Amer. Math. Soc., 318 (1990), 275-300.
doi: 10.2307/2001239. |
[5] |
F. Chiarenza and A. Ruiz,
Uniform $L^2$-weighted Sobolev inequalities, Proc. Amer. Math. Soc., 112 (1991), 53-64.
doi: 10.2307/2048479. |
[6] |
M. Christ and A. Kiselev,
Maximal functions associated to filtrations, J. Funct. Anal., 179 (2001), 409-425.
doi: 10.1006/jfan.2000.3687. |
[7] |
P. D'Ancona,
Kato smoothing and Strichartz estimates for wave equations with magnetic potentials, Comm. Math. Phys., 335 (2015), 1-16.
doi: 10.1007/s00220-014-2169-8. |
[8] |
P. D'Ancona,
On large potential perturbations of the Schrödinger, wave and Klein-Gordon equations, Commun. Pure Appl. Anal., 19 (2020), 609-640.
doi: 10.3934/cpaa.2020029. |
[9] |
P. D'Ancona and L. Fanelli,
Strichartz and smoothing estimates for dispersive equations with magnetic potentials, Comm. Partial Differential Equations, 33 (2008), 1082-1112.
doi: 10.1080/03605300701743749. |
[10] |
T. Kato,
Wave operators and similarity for some non-selfadjoint operators, Math. Ann., 162 (1966), 258-279.
doi: 10.1007/BF01360915. |
[11] |
T. Kato and K. Yajima,
Some examples of smooth operators and the associated smoothing effect, Rev. Math. Phys., 1 (1989), 481-496.
doi: 10.1142/S0129055X89000171. |
[12] |
M. Reed and B. Simon, Methods of Modern Mathematical Physics, II: Fourier Analysis, Self-adjointness, Academic Press, New York, 1975.
![]() ![]() |
[13] |
A. Ruiz and L. Vega,
Local regularity of solutions to wave equations with time-dependent potentials, Duke Math. J., 76 (1994), 913-940.
doi: 10.1215/S0012-7094-94-07636-9. |
[14] |
I. Seo,
From resolvent estimates to unique continuation for the Schrödinger equation, Trans. Amer. Math. Soc., 368 (2016), 8755-8784.
doi: 10.1090/tran/6635. |
[15] |
C. D. Sogge, Fourier Integrals in Classical Analysis, Cambridge Tracts in Mathematics, 105. Cambridge University Press, Cambridge, 1993.
doi: 10.1017/CBO9780511530029.![]() ![]() ![]() |
[16] |
R. Strichartz,
Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations, Duke Math. J., 44 (1977), 705-714.
doi: 10.1215/S0012-7094-77-04430-1. |
[17] |
J. Zhang and J. Zheng,
Strichartz estimate and nonlinear Klein-Gordon equation on nontrapping scattering space, J. Geom. Anal., 29 (2019), 2957-2984.
doi: 10.1007/s12220-018-00100-3. |
show all references
References:
[1] |
S. Agmon and L. Hörmander,
Asymptotic properties of solutions of differential equations with simple characteristics, J. Analyse Math, 30 (1976), 1-38.
doi: 10.1007/BF02786703. |
[2] |
J. A. Barcelò, J. M. Bennett, A. Carbery, A. Ruiz and M. C. Vilela,
A note on weighted estimates for the Schrödinger operator, Rev. Mat. Complut, 21 (2008), 481-488.
doi: 10.5209/rev_rema.2008.v21.n2.16405. |
[3] |
P. Brenner,
On space-time means and everywhere defined scattering operators for nonlinear Klein-Gordon equations, Math. Z, 186 (1984), 383-391.
doi: 10.1007/BF01174891. |
[4] |
S. Chanillo and E. Sawyer,
Unique continuation for $\Delta + v$ and the C. Fefferman-Phong class, Trans. Amer. Math. Soc., 318 (1990), 275-300.
doi: 10.2307/2001239. |
[5] |
F. Chiarenza and A. Ruiz,
Uniform $L^2$-weighted Sobolev inequalities, Proc. Amer. Math. Soc., 112 (1991), 53-64.
doi: 10.2307/2048479. |
[6] |
M. Christ and A. Kiselev,
Maximal functions associated to filtrations, J. Funct. Anal., 179 (2001), 409-425.
doi: 10.1006/jfan.2000.3687. |
[7] |
P. D'Ancona,
Kato smoothing and Strichartz estimates for wave equations with magnetic potentials, Comm. Math. Phys., 335 (2015), 1-16.
doi: 10.1007/s00220-014-2169-8. |
[8] |
P. D'Ancona,
On large potential perturbations of the Schrödinger, wave and Klein-Gordon equations, Commun. Pure Appl. Anal., 19 (2020), 609-640.
doi: 10.3934/cpaa.2020029. |
[9] |
P. D'Ancona and L. Fanelli,
Strichartz and smoothing estimates for dispersive equations with magnetic potentials, Comm. Partial Differential Equations, 33 (2008), 1082-1112.
doi: 10.1080/03605300701743749. |
[10] |
T. Kato,
Wave operators and similarity for some non-selfadjoint operators, Math. Ann., 162 (1966), 258-279.
doi: 10.1007/BF01360915. |
[11] |
T. Kato and K. Yajima,
Some examples of smooth operators and the associated smoothing effect, Rev. Math. Phys., 1 (1989), 481-496.
doi: 10.1142/S0129055X89000171. |
[12] |
M. Reed and B. Simon, Methods of Modern Mathematical Physics, II: Fourier Analysis, Self-adjointness, Academic Press, New York, 1975.
![]() ![]() |
[13] |
A. Ruiz and L. Vega,
Local regularity of solutions to wave equations with time-dependent potentials, Duke Math. J., 76 (1994), 913-940.
doi: 10.1215/S0012-7094-94-07636-9. |
[14] |
I. Seo,
From resolvent estimates to unique continuation for the Schrödinger equation, Trans. Amer. Math. Soc., 368 (2016), 8755-8784.
doi: 10.1090/tran/6635. |
[15] |
C. D. Sogge, Fourier Integrals in Classical Analysis, Cambridge Tracts in Mathematics, 105. Cambridge University Press, Cambridge, 1993.
doi: 10.1017/CBO9780511530029.![]() ![]() ![]() |
[16] |
R. Strichartz,
Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations, Duke Math. J., 44 (1977), 705-714.
doi: 10.1215/S0012-7094-77-04430-1. |
[17] |
J. Zhang and J. Zheng,
Strichartz estimate and nonlinear Klein-Gordon equation on nontrapping scattering space, J. Geom. Anal., 29 (2019), 2957-2984.
doi: 10.1007/s12220-018-00100-3. |
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