2017, 37(9): 4877-4906. doi: 10.3934/dcds.2017210

Strichartz estimates for Schrödinger equations in weighted $L^2$ spaces and their applications

1. 

Department of Mathematics Education, Kongju National University, Kongju 32588, Republic of Korea

2. 

Department of Mathematics, Sungkyunkwan University, Suwon 440-746, Republic of Korea

* Corresponding author: Ihyeok Seo

Received  May 2016 Revised  April 2017 Published  June 2017

Fund Project: Y. Koh was supported by NRF Grant 2016R1D1A1B03932049 (Republic of Korea). I. Seo was supported by the TJ Park Science Fellowship of POSCO TJ Park Foundation and by the NRF grant funded by the Korea government(MSIP) (No. 2017R1C1B5017496)

We obtain weighted $L^2$ Strichartz estimates for Schrödinger equations $i\partial_tu+(-\Delta)^{a/2}u=F(x, t)$, $u(x, 0)=f(x)$, of general orders $a>1$ with radial data $f, F$ with respect to the spatial variable $x$, whenever the weight is in a Morrey-Campanato type class. This is done by making use of a useful property of maximal functions of the weights together with frequency-localized estimates which follow from using bilinear interpolation and some estimates of Bessel functions. As consequences, we give an affirmative answer to a question posed in [1] concerning weighted homogeneous Strichartz estimates, and improve previously known Morawetz estimates. We also apply the weighted $L^2$ estimates to the well-posedness theory for the Schrödinger equations with time-dependent potentials in the class.

Citation: Youngwoo Koh, Ihyeok Seo. Strichartz estimates for Schrödinger equations in weighted $L^2$ spaces and their applications. Discrete & Continuous Dynamical Systems - A, 2017, 37 (9) : 4877-4906. doi: 10.3934/dcds.2017210
References:
[1]

J. A. BarcelóJ. M. BennettA. CarberyA. Ruiz and M. C. Vilela, Strichartz inequalities with weights in Morrey-Campanato classes, Collect. Math., 61 (2010), 49-56. doi: 10.1007/BF03191225.

[2]

J. A. BarceloJ. M. BennettA. Ruiz and M. C. Vilela, Local smoothing for Kato potentials in three dimensions, Math. Nachr., 282 (2009), 1391-1405. doi: 10.1002/mana.200610808.

[3]

J. Bergh and J. Löfström, Interpolation Spaces, An Introduction Springer-Verlag, Berlin-New York, 1976.

[4]

T. Cazenave, Semilinear Schrödinger Equations Courant Lecture Notes in Mathematics, 10. Amer. Math. Soc. , 2003.

[5]

T. Cazenave and F. B. Weissler, Rapidly decaying solutions of the nonlinear Schrödinger equation, Comm. Math. Phys., 147 (1992), 75-100. doi: 10.1007/BF02099529.

[6]

M. ChaeS. Hong and S. Lee, Mass concentration for the $L^2$-critical nonlinear Schrödinger equations of higher orders, Discrete Contin. Dyn, Syst., 29 (2011), 909-928. doi: 10.3934/dcds.2011.29.909.

[7]

S. Chanillo and E. Sawyer, Unique continuation for $Δ+v$ and the C. Fefferman-Phong class, Trans. Amer. Math. Soc., 318 (1990), 275-300. doi: 10.2307/2001239.

[8]

C.-H. ChoY. Koh and I. Seo, On inhomogeneous Strichartz estimates for fractional Schrödinger equations and their applications, Discrete Contin. Dyn. Syst., 36 (2016), 1905-1926. doi: 10.3934/dcds.2016.36.1905.

[9]

Y. Cho and S. Lee, Strichartz estimates in spherical coordinates, Indiana Univ. Math. J., 62 (2013), 991-1020. doi: 10.1512/iumj.2013.62.4970.

[10]

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

[11]

R. Coifman and R. Rochberg, Another characterization of BMO, Proc. Amer. Math. Soc., 79 (1980), 249-254. doi: 10.1090/S0002-9939-1980-0565349-8.

[12]

E. Cordero and F. Nicola, Some new Strichartz estimates for the Schrödinger equation, J. Differential equations., 245 (2008), 1945-1974. doi: 10.1016/j.jde.2008.07.009.

[13]

P. D'AnconaV. Pierfelice and N. Visciglia, Some remarks on the Schrödinger equation with a potential in $L_t^rL_x^s$, Math. Ann., 333 (2005), 271-290. doi: 10.1007/s00208-005-0672-0.

[14]

D. Fang and C. Wang, Weighted Strichartz estimates with angular regularity and their applications, Forum Math., 23 (2011), 181-205. doi: 10.1515/FORM.2011.009.

[15]

D. Foschi, Inhomogeneous Strichartz estimates, J. Hyperbolic Differ. Equ., 2 (2005), 1-24. doi: 10.1142/S0219891605000361.

[16]

J. Ginibre and G. Velo, The global Cauchy problem for the nonlinear Schrödinger equation revisited, Ann. Inst. H. Poincaré Anal. Non Linéare, 2 (1985), 309-327. doi: 10.1016/S0294-1449(16)30399-7.

[17]

L. Grafakos, Classical Fourier analysis Springer, New York, 2008.

[18]

L. Grafakos, Modern Fourier Analysis Springer, New York, 2008.

[19]

Z. Guo and Y. Wang, Improved Strichartz estimates for a class of dispersive equations in the radial case and their applications to nonlinear Schrödinger and wave equations, J. Anal. Math., 124 (2014), 1-38. doi: 10.1007/s11854-014-0025-6.

[20]

V. Karpman, Stabilization of soliton instabilities by higher-order dispersion: fourth order nonlinear Schrödinger-type equations, Phys. Lett. A, 210 (1996), 77-84. doi: 10.1016/0375-9601(95)00752-0.

[21]

V. Karpman and A. Shagalov, Stability of soliton described by nonlinear Schrödinger type equations with higher-order dispersion, Phys. D, 144 (2000), 194-210. doi: 10.1016/S0167-2789(00)00078-6.

[22]

T. Kato, An Lq, r -theory for nonlinear Schrödinger equations, in Spectral and scattering theory and applications, Adv. Stud. Pure Math. , Math. Soc. Japan, Tokyo, 23 (1994), 223–238.

[23]

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.

[24]

Y. Ke, Remark on the Strichartz estimates in the radial case, J. Math. Anal. Appl., 387 (2012), 857-861. doi: 10.1016/j.jmaa.2011.09.039.

[25]

M. Keel and T. Tao, Endpoint Strichartz estimates, Amer. J. Math., 120 (1998), 955-980. doi: 10.1353/ajm.1998.0039.

[26]

Y. Koh, Improved inhomogeneous Strichartz estimates for the Schrödinger equation, J. Math. Anal. Appl., 373 (2011), 147-160. doi: 10.1016/j.jmaa.2010.06.019.

[27]

Y. Koh and I. Seo, On weighted $L^2$ estimates for solutions of the wave equation, Proc. Amer. Math. Soc., 144 (2016), 3047-3061. doi: 10.1090/proc/12951.

[28]

Y. Koh and I. Seo, Global well-posedness for higher-order Schrödinger equations in weighted $L^2$ spaces, Comm. Partial Differential Equations, 40 (2015), 1815-1830. doi: 10.1080/03605302.2015.1048551.

[29]

Y. Koh and I. Seo, Inhomogeneous Strichartz estimates for Schrödiner's equations, J. Math. Anal. Appl., 442 (2016), 715-725. doi: 10.1016/j.jmaa.2016.04.061.

[30]

T. S. Kopaliani, Littlewood-Paley theorem on $L^{p(t)}(\mathbb{R}^n)$ spaces, (Ukrainian) Ukrain. Mat. Zh. , 60 (2008), 1709–1715; translation in Ukrainian Math. J. , 60 (2008), 2006–2014.

[31]

D. Kurtz, Littlewood-Paley and multiplier theorems on weighted $L^p$ spaces, Trans. Amer. Math. Soc., 259 (1980), 235-254. doi: 10.2307/1998156.

[32]

N. Laskin, Fractional quantum mechanics and Lévy path integrals, Phys. Lett. A, 268 (2000), 298-305. doi: 10.1016/S0375-9601(00)00201-2.

[33]

S. Lee and I. Seo, A note on unique continuation for the Schrödinger equation, J. Math. Anal. Appl., 389 (2012), 461-468. doi: 10.1016/j.jmaa.2011.11.067.

[34]

S. Lee and I. Seo, On inhomogeneous Strichartz estimates for the Schrödinger equation, Rev. Mat. Iberoam., 30 (2014), 711-726. doi: 10.4171/RMI/797.

[35]

C. S. Morawetz, Time decay for the nonlinear Klein-Gordon equation, Proc. Roy. Soc. A, 306 (1968), 291-296. doi: 10.1098/rspa.1968.0151.

[36]

V. Naibo and A. Stefanov, On some Schrödinger and wave equations with time dependent potentials, Math. Ann., 334 (2006), 325-338. doi: 10.1007/s00208-005-0720-9.

[37]

A. Ruiz and L. Vega, On local regularity of Schrödinger equations, Internat. Math. Res. Notices, (1993), 13-27. doi: 10.1155/S1073792893000029.

[38]

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.

[39]

I. Seo, Unique continuation for the Schrödinger equation with potentials in Wiener amalgam spaces, Indiana Univ. Math. J., 60 (2011), 1203-1227. doi: 10.1512/iumj.2011.60.4824.

[40]

I. Seo, Global unique continuation from a half space for the Schrödinger equation, J. Funct. Anal., 266 (2014), 85-98. doi: 10.1016/j.jfa.2013.09.025.

[41]

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.

[42]

S. Shao, Sharp linear and bilinear restriction estimate for paraboloids in the cylinderically symmetric case, Rev. Mat. Iberoam., 25 (2009), 1127-1168. doi: 10.4171/RMI/591.

[43]

E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals Princeton Mathematical Series, 1993.

[44]

R. S. Strichartz, Restriction 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.

[45]

M. Sugimoto, Global smoothing properties of generalized Schrödinger equations, J. Anal. Math., 76 (1998), 191-204. doi: 10.1007/BF02786935.

[46]

T. Tao, Nonlinear Dispersive Equations: Local and Global Analysis CBMS 106, eds: AMS, 2006.

[47]

H. Triebel, Interpolation Theory, Function Spaces, Differential Operator North-Holland, New York, 1978.

[48]

M. C. Vilela, Regularity of solutions to the free Schrödinger equation with radial initial data, Illinois J. Math., 45 (2001), 361-370.

[49]

M. C. Vilela, Inhomogeneous Strichartz estimates for the Schrödinger equation, Trans. Amer. Math. Soc., 359 (2007), 2123-2136. doi: 10.1090/S0002-9947-06-04099-2.

[50]

G. N. Watson, A Treatise on the Theory of Bessel Functions The Macmillan Company, New York, 1944.

show all references

References:
[1]

J. A. BarcelóJ. M. BennettA. CarberyA. Ruiz and M. C. Vilela, Strichartz inequalities with weights in Morrey-Campanato classes, Collect. Math., 61 (2010), 49-56. doi: 10.1007/BF03191225.

[2]

J. A. BarceloJ. M. BennettA. Ruiz and M. C. Vilela, Local smoothing for Kato potentials in three dimensions, Math. Nachr., 282 (2009), 1391-1405. doi: 10.1002/mana.200610808.

[3]

J. Bergh and J. Löfström, Interpolation Spaces, An Introduction Springer-Verlag, Berlin-New York, 1976.

[4]

T. Cazenave, Semilinear Schrödinger Equations Courant Lecture Notes in Mathematics, 10. Amer. Math. Soc. , 2003.

[5]

T. Cazenave and F. B. Weissler, Rapidly decaying solutions of the nonlinear Schrödinger equation, Comm. Math. Phys., 147 (1992), 75-100. doi: 10.1007/BF02099529.

[6]

M. ChaeS. Hong and S. Lee, Mass concentration for the $L^2$-critical nonlinear Schrödinger equations of higher orders, Discrete Contin. Dyn, Syst., 29 (2011), 909-928. doi: 10.3934/dcds.2011.29.909.

[7]

S. Chanillo and E. Sawyer, Unique continuation for $Δ+v$ and the C. Fefferman-Phong class, Trans. Amer. Math. Soc., 318 (1990), 275-300. doi: 10.2307/2001239.

[8]

C.-H. ChoY. Koh and I. Seo, On inhomogeneous Strichartz estimates for fractional Schrödinger equations and their applications, Discrete Contin. Dyn. Syst., 36 (2016), 1905-1926. doi: 10.3934/dcds.2016.36.1905.

[9]

Y. Cho and S. Lee, Strichartz estimates in spherical coordinates, Indiana Univ. Math. J., 62 (2013), 991-1020. doi: 10.1512/iumj.2013.62.4970.

[10]

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

[11]

R. Coifman and R. Rochberg, Another characterization of BMO, Proc. Amer. Math. Soc., 79 (1980), 249-254. doi: 10.1090/S0002-9939-1980-0565349-8.

[12]

E. Cordero and F. Nicola, Some new Strichartz estimates for the Schrödinger equation, J. Differential equations., 245 (2008), 1945-1974. doi: 10.1016/j.jde.2008.07.009.

[13]

P. D'AnconaV. Pierfelice and N. Visciglia, Some remarks on the Schrödinger equation with a potential in $L_t^rL_x^s$, Math. Ann., 333 (2005), 271-290. doi: 10.1007/s00208-005-0672-0.

[14]

D. Fang and C. Wang, Weighted Strichartz estimates with angular regularity and their applications, Forum Math., 23 (2011), 181-205. doi: 10.1515/FORM.2011.009.

[15]

D. Foschi, Inhomogeneous Strichartz estimates, J. Hyperbolic Differ. Equ., 2 (2005), 1-24. doi: 10.1142/S0219891605000361.

[16]

J. Ginibre and G. Velo, The global Cauchy problem for the nonlinear Schrödinger equation revisited, Ann. Inst. H. Poincaré Anal. Non Linéare, 2 (1985), 309-327. doi: 10.1016/S0294-1449(16)30399-7.

[17]

L. Grafakos, Classical Fourier analysis Springer, New York, 2008.

[18]

L. Grafakos, Modern Fourier Analysis Springer, New York, 2008.

[19]

Z. Guo and Y. Wang, Improved Strichartz estimates for a class of dispersive equations in the radial case and their applications to nonlinear Schrödinger and wave equations, J. Anal. Math., 124 (2014), 1-38. doi: 10.1007/s11854-014-0025-6.

[20]

V. Karpman, Stabilization of soliton instabilities by higher-order dispersion: fourth order nonlinear Schrödinger-type equations, Phys. Lett. A, 210 (1996), 77-84. doi: 10.1016/0375-9601(95)00752-0.

[21]

V. Karpman and A. Shagalov, Stability of soliton described by nonlinear Schrödinger type equations with higher-order dispersion, Phys. D, 144 (2000), 194-210. doi: 10.1016/S0167-2789(00)00078-6.

[22]

T. Kato, An Lq, r -theory for nonlinear Schrödinger equations, in Spectral and scattering theory and applications, Adv. Stud. Pure Math. , Math. Soc. Japan, Tokyo, 23 (1994), 223–238.

[23]

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.

[24]

Y. Ke, Remark on the Strichartz estimates in the radial case, J. Math. Anal. Appl., 387 (2012), 857-861. doi: 10.1016/j.jmaa.2011.09.039.

[25]

M. Keel and T. Tao, Endpoint Strichartz estimates, Amer. J. Math., 120 (1998), 955-980. doi: 10.1353/ajm.1998.0039.

[26]

Y. Koh, Improved inhomogeneous Strichartz estimates for the Schrödinger equation, J. Math. Anal. Appl., 373 (2011), 147-160. doi: 10.1016/j.jmaa.2010.06.019.

[27]

Y. Koh and I. Seo, On weighted $L^2$ estimates for solutions of the wave equation, Proc. Amer. Math. Soc., 144 (2016), 3047-3061. doi: 10.1090/proc/12951.

[28]

Y. Koh and I. Seo, Global well-posedness for higher-order Schrödinger equations in weighted $L^2$ spaces, Comm. Partial Differential Equations, 40 (2015), 1815-1830. doi: 10.1080/03605302.2015.1048551.

[29]

Y. Koh and I. Seo, Inhomogeneous Strichartz estimates for Schrödiner's equations, J. Math. Anal. Appl., 442 (2016), 715-725. doi: 10.1016/j.jmaa.2016.04.061.

[30]

T. S. Kopaliani, Littlewood-Paley theorem on $L^{p(t)}(\mathbb{R}^n)$ spaces, (Ukrainian) Ukrain. Mat. Zh. , 60 (2008), 1709–1715; translation in Ukrainian Math. J. , 60 (2008), 2006–2014.

[31]

D. Kurtz, Littlewood-Paley and multiplier theorems on weighted $L^p$ spaces, Trans. Amer. Math. Soc., 259 (1980), 235-254. doi: 10.2307/1998156.

[32]

N. Laskin, Fractional quantum mechanics and Lévy path integrals, Phys. Lett. A, 268 (2000), 298-305. doi: 10.1016/S0375-9601(00)00201-2.

[33]

S. Lee and I. Seo, A note on unique continuation for the Schrödinger equation, J. Math. Anal. Appl., 389 (2012), 461-468. doi: 10.1016/j.jmaa.2011.11.067.

[34]

S. Lee and I. Seo, On inhomogeneous Strichartz estimates for the Schrödinger equation, Rev. Mat. Iberoam., 30 (2014), 711-726. doi: 10.4171/RMI/797.

[35]

C. S. Morawetz, Time decay for the nonlinear Klein-Gordon equation, Proc. Roy. Soc. A, 306 (1968), 291-296. doi: 10.1098/rspa.1968.0151.

[36]

V. Naibo and A. Stefanov, On some Schrödinger and wave equations with time dependent potentials, Math. Ann., 334 (2006), 325-338. doi: 10.1007/s00208-005-0720-9.

[37]

A. Ruiz and L. Vega, On local regularity of Schrödinger equations, Internat. Math. Res. Notices, (1993), 13-27. doi: 10.1155/S1073792893000029.

[38]

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.

[39]

I. Seo, Unique continuation for the Schrödinger equation with potentials in Wiener amalgam spaces, Indiana Univ. Math. J., 60 (2011), 1203-1227. doi: 10.1512/iumj.2011.60.4824.

[40]

I. Seo, Global unique continuation from a half space for the Schrödinger equation, J. Funct. Anal., 266 (2014), 85-98. doi: 10.1016/j.jfa.2013.09.025.

[41]

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.

[42]

S. Shao, Sharp linear and bilinear restriction estimate for paraboloids in the cylinderically symmetric case, Rev. Mat. Iberoam., 25 (2009), 1127-1168. doi: 10.4171/RMI/591.

[43]

E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals Princeton Mathematical Series, 1993.

[44]

R. S. Strichartz, Restriction 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.

[45]

M. Sugimoto, Global smoothing properties of generalized Schrödinger equations, J. Anal. Math., 76 (1998), 191-204. doi: 10.1007/BF02786935.

[46]

T. Tao, Nonlinear Dispersive Equations: Local and Global Analysis CBMS 106, eds: AMS, 2006.

[47]

H. Triebel, Interpolation Theory, Function Spaces, Differential Operator North-Holland, New York, 1978.

[48]

M. C. Vilela, Regularity of solutions to the free Schrödinger equation with radial initial data, Illinois J. Math., 45 (2001), 361-370.

[49]

M. C. Vilela, Inhomogeneous Strichartz estimates for the Schrödinger equation, Trans. Amer. Math. Soc., 359 (2007), 2123-2136. doi: 10.1090/S0002-9947-06-04099-2.

[50]

G. N. Watson, A Treatise on the Theory of Bessel Functions The Macmillan Company, New York, 1944.

Figure 1.  The region of $(s,1/p)$ for (7) particularly when $a=2$
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