2017, 37(6): 3387-3410. doi: 10.3934/dcds.2017143

Generalized inhomogeneous Strichartz estimates

Fakultät für Mathematik, Universität Bielefeld, Postfach 10 01 31, 33501 Bielefeld, Germany

Received  September 2016 Revised  December 2016 Published  February 2017

We prove new inhomogeneous generalized Strichartz estimates, which do not follow from the homogeneous generalized estimates by virtue of the Christ-Kiselev lemma. Instead, we make use of the bilinear interpolation argument worked out by Keel and Tao and refined by Foschi presented in a unified framework. Finally, we give a sample application.

Citation: Robert Schippa. Generalized inhomogeneous Strichartz estimates. Discrete & Continuous Dynamical Systems - A, 2017, 37 (6) : 3387-3410. doi: 10.3934/dcds.2017143
References:
[1]

J. Bergh and J. Löfström, Interpolation Spaces. An Introduction, Springer-Verlag, Berlin-New York, 1976, Grundlehren der Mathematischen Wissenschaften, No. 223.

[2]

J.-M. Bouclet and H. Mizutani, Uniform resolvent and Strichartz estimates for Schrödinger equations with critical singularities, preprint, arXiv: 1607.01187.

[3]

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.

[4]

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.

[5]

Y. ChoT. Ozawa and S. Xia, Remarks on some dispersive estimates, Commun. Pure Appl. Anal., 10 (2011), 1121-1128. doi: 10.3934/cpaa.2011.10.1121.

[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]

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

[8]

L. Grafakos, Classical Fourier Analysis, vol. 249 of Graduate Texts in Mathematics, 2nd edition, Springer, New York, 2008.

[9]

L. Grafakos, Modern Fourier Analysis, vol. 250 of Graduate Texts in Mathematics, 2nd edition, Springer, New York, 2009. doi: 10.1007/978-0-387-09434-2.

[10]

Z. Guo, Sharp spherically averaged Stichartz estimates for the Schrödinger equation, Nonlinearity, 29 (2016), 1668-1686. doi: 10.1088/0951-7715/29/5/1668.

[11]

Z. GuoS. LeeK. Nakanishi and C. Wang, Generalized Strichartz estimates and scattering for 3D Zakharov system, Comm. Math. Phys., 331 (2014), 239-259. doi: 10.1007/s00220-014-2006-0.

[12]

Z. GuoL. Peng and B. Wang, Decay estimates for a class of wave equations, J. Funct. Anal., 254 (2008), 1642-1660. doi: 10.1016/j.jfa.2007.12.010.

[13]

J.-C. JiangC. Wang and X. Yu, Generalized and weighted Strichartz estimates, Commun. Pure Appl. Anal., 11 (2012), 1723-1752. doi: 10.3934/cpaa.2012.11.1723.

[14]

M.Keel and T.Tao, Endpoint Strichartz estimates,Amer. J. Math., 120 (1998), 955-980, URL http://muse.jhu.edu/journals/american_journal_of_mathematics/v120/120.5keel.pdf. doi: 10.1353/ajm.1998.0039.

[15]

H. Mizutani, Remarks on endpoint Strichartz estimates for Schrödinger equations with the critical inverse-square potential, preprint, arXiv: 1607.02848.

[16]

E. Y. Ovcharov, Radial Strichartz estimates with application to the 2-D Dirac-Klein-Gordon system, Comm. Partial Differential Equations, 37 (2012), 1754-1788. doi: 10.1080/03605302.2011.632047.

[17]

I. Rodnianski and W. Schlag, Time decay for solutions of Schrödinger equations with rough and time-dependent potenetials, Invent. Math., 155 (2004), 451-513. doi: 10.1007/s00222-003-0325-4.

[18]

C. D. Sogge, Fourier Integrals in Classical Analysis vol. 105 of Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, 1993. doi: 10.1017/CBO9780511530029.

[19]

J. Sterbenz, Angular regularity and Strichartz estimates for the wave equation, Int. Math. Res. Not., (2005), 187-231, With an appendix by Igor Rodnianski. doi: 10.1155/IMRN.2005.187.

[20]

T. Tao, Spherically averaged endpoint Strichartz estimates for the two-dimensional Schrödinger equation, Comm. Partial Differential Equations, 25 (2000), 1471-1485. doi: 10.1080/03605300008821556.

show all references

References:
[1]

J. Bergh and J. Löfström, Interpolation Spaces. An Introduction, Springer-Verlag, Berlin-New York, 1976, Grundlehren der Mathematischen Wissenschaften, No. 223.

[2]

J.-M. Bouclet and H. Mizutani, Uniform resolvent and Strichartz estimates for Schrödinger equations with critical singularities, preprint, arXiv: 1607.01187.

[3]

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.

[4]

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.

[5]

Y. ChoT. Ozawa and S. Xia, Remarks on some dispersive estimates, Commun. Pure Appl. Anal., 10 (2011), 1121-1128. doi: 10.3934/cpaa.2011.10.1121.

[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]

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

[8]

L. Grafakos, Classical Fourier Analysis, vol. 249 of Graduate Texts in Mathematics, 2nd edition, Springer, New York, 2008.

[9]

L. Grafakos, Modern Fourier Analysis, vol. 250 of Graduate Texts in Mathematics, 2nd edition, Springer, New York, 2009. doi: 10.1007/978-0-387-09434-2.

[10]

Z. Guo, Sharp spherically averaged Stichartz estimates for the Schrödinger equation, Nonlinearity, 29 (2016), 1668-1686. doi: 10.1088/0951-7715/29/5/1668.

[11]

Z. GuoS. LeeK. Nakanishi and C. Wang, Generalized Strichartz estimates and scattering for 3D Zakharov system, Comm. Math. Phys., 331 (2014), 239-259. doi: 10.1007/s00220-014-2006-0.

[12]

Z. GuoL. Peng and B. Wang, Decay estimates for a class of wave equations, J. Funct. Anal., 254 (2008), 1642-1660. doi: 10.1016/j.jfa.2007.12.010.

[13]

J.-C. JiangC. Wang and X. Yu, Generalized and weighted Strichartz estimates, Commun. Pure Appl. Anal., 11 (2012), 1723-1752. doi: 10.3934/cpaa.2012.11.1723.

[14]

M.Keel and T.Tao, Endpoint Strichartz estimates,Amer. J. Math., 120 (1998), 955-980, URL http://muse.jhu.edu/journals/american_journal_of_mathematics/v120/120.5keel.pdf. doi: 10.1353/ajm.1998.0039.

[15]

H. Mizutani, Remarks on endpoint Strichartz estimates for Schrödinger equations with the critical inverse-square potential, preprint, arXiv: 1607.02848.

[16]

E. Y. Ovcharov, Radial Strichartz estimates with application to the 2-D Dirac-Klein-Gordon system, Comm. Partial Differential Equations, 37 (2012), 1754-1788. doi: 10.1080/03605302.2011.632047.

[17]

I. Rodnianski and W. Schlag, Time decay for solutions of Schrödinger equations with rough and time-dependent potenetials, Invent. Math., 155 (2004), 451-513. doi: 10.1007/s00222-003-0325-4.

[18]

C. D. Sogge, Fourier Integrals in Classical Analysis vol. 105 of Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, 1993. doi: 10.1017/CBO9780511530029.

[19]

J. Sterbenz, Angular regularity and Strichartz estimates for the wave equation, Int. Math. Res. Not., (2005), 187-231, With an appendix by Igor Rodnianski. doi: 10.1155/IMRN.2005.187.

[20]

T. Tao, Spherically averaged endpoint Strichartz estimates for the two-dimensional Schrödinger equation, Comm. Partial Differential Equations, 25 (2000), 1471-1485. doi: 10.1080/03605300008821556.

Figure 1.  This pictorial representation generalizes [7,Figure 2,p.5]. The axes refer to the spatial integrability coefficients. The rectangle $ABCD$ corresponds to estimates found from factorization and the application of the Christ-Kiselev lemma up to endpoints. The origin relates to the dispersive estimate; one finds local estimates to hold in the wedge $AOCD$ by virtue of interpolation, restrictions on global estimates cut off estimates with too large spatial integrability coefficients
Figure 2.  We give a pictorial representation similar to [3,Figure 1,p.1907]. In the setting of [3,Corollary 1.,p.1907] we find $A=(\frac{n-a}{2n},\frac{1}{2}), \, B= (\frac{n}{n+a}-\frac{n}{2(n+1)},\frac{n}{n+a}-\frac{n}{2(n+1)}), \, C=(\frac{n}{2(n+1)},\frac{n}{2(n+1)}), \, D=(\frac{1}{2},0)$, where the open line $\overline{BC}$ corresponds to the range from [3,Theorem 1.2.,p.1908] and the closed line $\overline{AD}$ corresponds to estimates found from factorization and application of the Christ-Kiselev lemma
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