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January  2020, 40(1): 597-608. doi: 10.3934/dcds.2020024

Local smoothing and Strichartz estimates for the Klein-Gordon equation with the inverse-square potential

Hyeongjin Lee , Ihyeok Seo , and  Jihyeon Seok

Department of Mathematics, Sungkyunkwan University, Suwon 16419, Republic of Korea

* Corresponding author: Ihyeok Seo

Received  April 2019 Published  October 2019

Fund Project: This research was supported by NRF-2019R1F1A1061316.

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.

Keywords: Smoothing estimates, Strichartz estimates, Klein-Gordon equation.
Mathematics Subject Classification: Primary: 35B45; Secondary: 35Q40.
Citation: Hyeongjin Lee, Ihyeok Seo, Jihyeon Seok. Local smoothing and Strichartz estimates for the Klein-Gordon equation with the inverse-square potential. Discrete & Continuous Dynamical Systems, 2020, 40 (1) : 597-608. doi: 10.3934/dcds.2020024
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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F. Chiarenza and A. Ruiz, Uniform $L^2$-weighted Sobolev inequalities, Proc. Amer. Math. Soc., 112 (1991), 53-64.  doi: 10.2307/2048479.  Google Scholar

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M. Christ and A. Kiselev, Maximal functions associated to filtrations, J. Funct. Anal., 179 (2001), 409-425.  doi: 10.1006/jfan.2000.3687.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

T. Kato, Wave operators and similarity for some non-selfadjoint operators, Math. Ann., 162 (1966), 258-279.  doi: 10.1007/BF01360915.  Google Scholar

[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.  Google Scholar

[12] M. Reed and B. Simon, Methods of Modern Mathematical Physics, II: Fourier Analysis, Self-adjointness, Academic Press, New York, 1975.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[15] C. D. Sogge, Fourier Integrals in Classical Analysis, Cambridge Tracts in Mathematics, 105. Cambridge University Press, Cambridge, 1993.  doi: 10.1017/CBO9780511530029.  Google Scholar
[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

F. Chiarenza and A. Ruiz, Uniform $L^2$-weighted Sobolev inequalities, Proc. Amer. Math. Soc., 112 (1991), 53-64.  doi: 10.2307/2048479.  Google Scholar

[6]

M. Christ and A. Kiselev, Maximal functions associated to filtrations, J. Funct. Anal., 179 (2001), 409-425.  doi: 10.1006/jfan.2000.3687.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

T. Kato, Wave operators and similarity for some non-selfadjoint operators, Math. Ann., 162 (1966), 258-279.  doi: 10.1007/BF01360915.  Google Scholar

[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.  Google Scholar

[12] M. Reed and B. Simon, Methods of Modern Mathematical Physics, II: Fourier Analysis, Self-adjointness, Academic Press, New York, 1975.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[15] C. D. Sogge, Fourier Integrals in Classical Analysis, Cambridge Tracts in Mathematics, 105. Cambridge University Press, Cambridge, 1993.  doi: 10.1017/CBO9780511530029.  Google Scholar
[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.  Google Scholar

[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.  Google Scholar

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