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November  2017, 16(6): 2047-2051. doi: 10.3934/cpaa.2017100

Sharp Strichartz estimates in spherical coordinates

Robert Schippa ,

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

* Corresponding author

Received  November 2016 Revised  May 2017 Published  July 2017

Fund Project: Financial support by the German Science Foundation (IRTG 2235) is gratefully acknowledged.

We prove Strichartz estimates found after adding regularity in the spherical coordinates for Schrödinger-like equations. The obtained estimates are sharp up to endpoints. The proof relies on estimates involving spherical averages, which were obtained in [5]. We discuss sharpness making use of a modified Knapp-type example.

Keywords: Dispersive equations, Strichartz estimates, spherical symmetry, spherical averages, angular regularity.
Mathematics Subject Classification: Primary: 42B37; Secondary: 35Q40.
Citation: Robert Schippa. Sharp Strichartz estimates in spherical coordinates. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2047-2051. doi: 10.3934/cpaa.2017100
References:
[1]

Y. Cho, Z. Guo and S. Lee, A Sobolev estimate for the adjoint restriction operator, Math. Ann., 362 (2015), 799-815.  doi: 10.1007/s00208-014-1130-7.  Google Scholar

[2]

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.  Google Scholar

[3]

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

[4]

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.  Google Scholar

[5]

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

[6]

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.  Google Scholar

[7]

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

[8]

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

[9]

J. Sterbenz, Angular regularity and Strichartz estimates for the wave equation, Int. Math. Res. Not., (), 187-231.  doi: 10.1155/IMRN.2005.187.  Google Scholar

[10]

R. S. Strichartz, Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations, Duke Math. J., 44 (1977), 705-714.   Google Scholar

show all references

References:
[1]

Y. Cho, Z. Guo and S. Lee, A Sobolev estimate for the adjoint restriction operator, Math. Ann., 362 (2015), 799-815.  doi: 10.1007/s00208-014-1130-7.  Google Scholar

[2]

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.  Google Scholar

[3]

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

[4]

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.  Google Scholar

[5]

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

[6]

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.  Google Scholar

[7]

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

[8]

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

[9]

J. Sterbenz, Angular regularity and Strichartz estimates for the wave equation, Int. Math. Res. Not., (), 187-231.  doi: 10.1155/IMRN.2005.187.  Google Scholar

[10]

R. S. Strichartz, Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations, Duke Math. J., 44 (1977), 705-714.   Google Scholar

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