# American Institute of Mathematical Sciences

2012, 11(5): 1723-1752. doi: 10.3934/cpaa.2012.11.1723

## Generalized and weighted Strichartz estimates

 1 Institute of Mathematics, Academic Sinica, Taipei, Taiwan 10617, Taiwan 2 Department of Mathematics, Johns Hopkins University, Baltimore, MD 21218, United States, United States

Received  January 2011 Revised  November 2011 Published  March 2012

In this paper, we explore the relations between different kinds of Strichartz estimates and give new estimates in Euclidean space $\mathbb{R}^n$. In particular, we prove the generalized and weighted Strichartz estimates for a large class of dispersive operators including the Schrödinger and wave equation. As a sample application of these new estimates, we are able to prove the Strauss conjecture with low regularity for dimension $2$ and $3$.
Citation: Jin-Cheng Jiang, Chengbo Wang, Xin Yu. Generalized and weighted Strichartz estimates. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1723-1752. doi: 10.3934/cpaa.2012.11.1723
##### References:
 [1] J. Bergh and J. Löfström, "Interpolation Spaces: An Introduction,", Springer-Velag Berlin Heidelberg, (1976). [2] T. Cazenave, "Semilinear Schrödinger Equations,", Courant Lecture notes in Mathematics, (2003). [3] M. Christ and A. Kiselev, Maximal functions associated to filtrations,, J. Funct. Anal., 179 (2001), 409. doi: 10.1006/jfan.2000.3687. [4] D. Fang and C. Wang, Some remarks on Strichartz estimates for homogeneous wave equation,, Nonlinear Anal., 65 (2006), 697. doi: 10.1016/j.na.2005.09.040. [5] D. Fang and C. Wang, Weighted Strichartz estimates with angular regularity and their applications,, Forum Math., 23 (2011), 181. doi: 10.1515/FORM.2011.009. [6] D. Fang and C. Wang, Almost global existence for some semilinear wave equations with almost critical regularity,, \arXiv{1007.0733}., (). [7] J. Ginibre and G. Velo, On the global Cauchy problem for some nonlinear Schrödinger equations,, Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire, 1 (1984), 309. [8] J. Ginibre and G. Velo, Generalized Strichartz inequalities for the wave equation,, J. Funct. Anal., 133 (1995), 50. doi: 10.1006/jfan.1995.1119. [9] V. Georgiev, H. Lindblad and C. D. Sogge, Weighted Strichartz estimates and global existence for semilinear wave equations,, Amer. J. Math., 119 (1997), 1291. doi: 10.1353/ajm.1997.0038. [10] K. Hidano, Morawetz-Strichartz estimates for spherically symmetric solutions to wave equations and applications to semi-linear Cauchy problems,, Differential Integral Equations, 20 (2007), 735. [11] K. Hidano and Y. Kurokawa, Local existence of minimal-regularity radial solutions to semi-linear wave equations,, preprint., (). [12] K. Hidano and Y. Kurokawa, Weighted HLS inequalities for radial functions and Strichartz estimates for wave and Schrödinger equations,, Illinois J. Math., 52 (2008), 365. [13] K. Hidano, J. Metcalfe, H. F. Smith, C. D. Sogge and Y. Zhou, On abstract Strichartz estimates and the Strauss conjecture for nontrapping obstacles,, Trans. Amer. Math. Soc., 362 (2010), 2789. doi: 10.1090/S0002-9947-09-05053-3. [14] K. Hidano, C. Wang and K. Yokoyama, On almost global existence and local well-posedness for some 3-D quasi-linear wave equations,, Adv. Differential Equations, 17 (2012), 267. [15] K. Hidano and K. Yokoyama, A remark on the almost global existence theorems of Keel, Smith and Sogge,, Funkcial. Ekvac., 48 (2005), 1. doi: 10.1619/fesi.48.1. [16] J. Kato and T. Ozawa, Endpoint Strichartz estimates for the Klein-Gordon equation in two space dimensions and some applications,, J. Math. Pures Appl., 95 (2011), 48. doi: 10.1016/j.matpur.2010.10.001. [17] M. Keel, H. Smith and C. D. Sogge, Almost global existence for some semilinear wave equations,, Dedicated to the memory of Thomas H. Wolff. J. Anal. Math., 87 (2002), 265. doi: 10.1007/BF02868477. [18] M. Keel and T. Tao, Endpoint Strichartz estimates,, Amer. J. Math., 120 (1998), 955. doi: 10.1353/ajm.1998.0039. [19] S. Klainerman and M. Machedon, Space-time estimates for null forms and the local existence theorem,, Comm. Pure Appl. Math., 46 (1993), 1221. doi: 10.1002/cpa.3160460902. [20] H. Lindblad, Blow up for solutions of $\square u = |u|^p$ with small initial data,, Comm. Partial Differential Equations, 15 (1990), 757. doi: 10.1080/03605309908820708. [21] H. Lindblad and C. D. Sogge, On existence and scattering with minimal regularity for semilinear wave equations,, J. Funct. Anal., 130 (1995), 357. doi: 10.1006/jfan.1995.1075. [22] H. Lindblad and C. D. Sogge, Long-time existence for small amplitude semilinear wave equations,, Amer. J. Math., 118 (1996), 1047. doi: 10.1353/ajm.1996.0042. [23] S. Machihara, M. Nakamura, K. Nakanishi and T. Ozawa, Endpoint Strichartz estimates and global solutions for the nonlinear Dirac equation,, J. Funct. Anal., 219 (2005), 1. doi: 10.1016/j.jfa.2004.07.005. [24] J. Metcalfe, Global existence for semilinear wave equations exterior to nontrapping obstacles,, Houston J. Math., 30 (2004), 259. [25] J. Metcalfe and C. D. Sogge, Long time existence of quasilinear wave equations exterior to star-shaped obstacles via energy methods,, SIAM J. Math. Anal., 38 (2006), 188. doi: 10.1137/050627149. [26] S. J. Montgomery-Smith, Time decay for the bounded mean oscillation of solutions of the Schrödinger and wave equation,, Duke Math. J., 19 (1998), 393. doi: 10.1215/S0012-7094-98-09117-7. [27] C. D. Sogge, "Fourier Integrals in Classical Analysis,", Cambridge Tracts in Mathematics, (1993). [28] C. D. Sogge, "Lectures on Nonlinear Wave Equations," Second edition,, International Press, (2008). [29] C. D. Sogge and C. Wang, Concerning the wave equation on asymptotically Euclidean manifolds,, J. Anal. Math., 112 (2010), 1. doi: 10.1007/s11854-010-0023-2. [30] E. M. Stein and G. Weiss, Fractional integrals on n-dimensional Euclidean space,, J. Math. Mech., 7 (1958), 503. [31] E. M. Stein and G. Weiss, "Introduction to Fourier Analysis on Euclidean Spaces,", Princeton Mathematical Series, (1971). [32] J. Sterbenz, Angular regularity and Strichartz estimates for the wave equation,, With an appendix by Igor Rodnianski, 4 (2005), 187. doi: 10.1155/IMRN.2005.187. [33] R. S. Strichartz, Multipliers for spherical harmonic expansions,, Trans. Amer. Math. Soc., 167 (1972), 115. doi: 10.2307/1996130. [34] R. Strichartz, Restriction of Fourier transform to quadratic surfaces and decay of solutions to the wave equation,, Duke Math. J., 44 (1977), 705. doi: 10.1215/S0012-7094-77-04430-1. [35] T. Tao, Spherically averaged endpoint Strichartz estimates for the two-dimensional Schrödinger equation,, Comm. Partial Differential Equations, 25 (2000), 1471. doi: 10.1080/03605300008821556. [36] T. Tao, "Nonlinear Dispersive Equations: Local and Global Analysis,", CBMS Regional Conference Series in Mathematics, (2006). [37] D. Tataru, Strichartz estimates in the hyperbolic space and global existence for the semilinear wave equation,, Trans. Amer. Math. Soc., 353 (2001), 795. doi: 10.1090/S0002-9947-00-02750-1. [38] H. Triebel, "Interpolation Theory, Function Spaces, Differential Operators,", North-Holland Mathematical Library, 18 (1978). [39] G. N. Watson, A treatise on the theory of Bessel functions,, Reprint of the second (1944) edition. Cambridge Mathematical Library. Cambridge University Press, (1944). [40] X. Yu, Generalized Strichartz estimates on perturbed wave equation and applications on Strauss conjecture,, Differential Integral Equations, 24 (2011), 443. [41] Y. Zhou, Blow up of classical solutions to $\square u = |u|^{1+\alpha$ in three space dimensions,, J. Partial Differential Equations, 5 (1992), 21. [42] Y. Zhou, Life span of classical solutions to $\square u=|u|^p p$ in two space dimensions,, Chinese Ann. Math. Ser. B, 14 (1993), 225.

show all references

##### References:
 [1] J. Bergh and J. Löfström, "Interpolation Spaces: An Introduction,", Springer-Velag Berlin Heidelberg, (1976). [2] T. Cazenave, "Semilinear Schrödinger Equations,", Courant Lecture notes in Mathematics, (2003). [3] M. Christ and A. Kiselev, Maximal functions associated to filtrations,, J. Funct. Anal., 179 (2001), 409. doi: 10.1006/jfan.2000.3687. [4] D. Fang and C. Wang, Some remarks on Strichartz estimates for homogeneous wave equation,, Nonlinear Anal., 65 (2006), 697. doi: 10.1016/j.na.2005.09.040. [5] D. Fang and C. Wang, Weighted Strichartz estimates with angular regularity and their applications,, Forum Math., 23 (2011), 181. doi: 10.1515/FORM.2011.009. [6] D. Fang and C. Wang, Almost global existence for some semilinear wave equations with almost critical regularity,, \arXiv{1007.0733}., (). [7] J. Ginibre and G. Velo, On the global Cauchy problem for some nonlinear Schrödinger equations,, Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire, 1 (1984), 309. [8] J. Ginibre and G. Velo, Generalized Strichartz inequalities for the wave equation,, J. Funct. Anal., 133 (1995), 50. doi: 10.1006/jfan.1995.1119. [9] V. Georgiev, H. Lindblad and C. D. Sogge, Weighted Strichartz estimates and global existence for semilinear wave equations,, Amer. J. Math., 119 (1997), 1291. doi: 10.1353/ajm.1997.0038. [10] K. Hidano, Morawetz-Strichartz estimates for spherically symmetric solutions to wave equations and applications to semi-linear Cauchy problems,, Differential Integral Equations, 20 (2007), 735. [11] K. Hidano and Y. Kurokawa, Local existence of minimal-regularity radial solutions to semi-linear wave equations,, preprint., (). [12] K. Hidano and Y. Kurokawa, Weighted HLS inequalities for radial functions and Strichartz estimates for wave and Schrödinger equations,, Illinois J. Math., 52 (2008), 365. [13] K. Hidano, J. Metcalfe, H. F. Smith, C. D. Sogge and Y. Zhou, On abstract Strichartz estimates and the Strauss conjecture for nontrapping obstacles,, Trans. Amer. Math. Soc., 362 (2010), 2789. doi: 10.1090/S0002-9947-09-05053-3. [14] K. Hidano, C. Wang and K. Yokoyama, On almost global existence and local well-posedness for some 3-D quasi-linear wave equations,, Adv. Differential Equations, 17 (2012), 267. [15] K. Hidano and K. Yokoyama, A remark on the almost global existence theorems of Keel, Smith and Sogge,, Funkcial. Ekvac., 48 (2005), 1. doi: 10.1619/fesi.48.1. [16] J. Kato and T. Ozawa, Endpoint Strichartz estimates for the Klein-Gordon equation in two space dimensions and some applications,, J. Math. Pures Appl., 95 (2011), 48. doi: 10.1016/j.matpur.2010.10.001. [17] M. Keel, H. Smith and C. D. Sogge, Almost global existence for some semilinear wave equations,, Dedicated to the memory of Thomas H. Wolff. J. Anal. Math., 87 (2002), 265. doi: 10.1007/BF02868477. [18] M. Keel and T. Tao, Endpoint Strichartz estimates,, Amer. J. Math., 120 (1998), 955. doi: 10.1353/ajm.1998.0039. [19] S. Klainerman and M. Machedon, Space-time estimates for null forms and the local existence theorem,, Comm. Pure Appl. Math., 46 (1993), 1221. doi: 10.1002/cpa.3160460902. [20] H. Lindblad, Blow up for solutions of $\square u = |u|^p$ with small initial data,, Comm. Partial Differential Equations, 15 (1990), 757. doi: 10.1080/03605309908820708. [21] H. Lindblad and C. D. Sogge, On existence and scattering with minimal regularity for semilinear wave equations,, J. Funct. Anal., 130 (1995), 357. doi: 10.1006/jfan.1995.1075. [22] H. Lindblad and C. D. Sogge, Long-time existence for small amplitude semilinear wave equations,, Amer. J. Math., 118 (1996), 1047. doi: 10.1353/ajm.1996.0042. [23] S. Machihara, M. Nakamura, K. Nakanishi and T. Ozawa, Endpoint Strichartz estimates and global solutions for the nonlinear Dirac equation,, J. Funct. Anal., 219 (2005), 1. doi: 10.1016/j.jfa.2004.07.005. [24] J. Metcalfe, Global existence for semilinear wave equations exterior to nontrapping obstacles,, Houston J. Math., 30 (2004), 259. [25] J. Metcalfe and C. D. Sogge, Long time existence of quasilinear wave equations exterior to star-shaped obstacles via energy methods,, SIAM J. Math. Anal., 38 (2006), 188. doi: 10.1137/050627149. [26] S. J. Montgomery-Smith, Time decay for the bounded mean oscillation of solutions of the Schrödinger and wave equation,, Duke Math. J., 19 (1998), 393. doi: 10.1215/S0012-7094-98-09117-7. [27] C. D. Sogge, "Fourier Integrals in Classical Analysis,", Cambridge Tracts in Mathematics, (1993). [28] C. D. Sogge, "Lectures on Nonlinear Wave Equations," Second edition,, International Press, (2008). [29] C. D. Sogge and C. Wang, Concerning the wave equation on asymptotically Euclidean manifolds,, J. Anal. Math., 112 (2010), 1. doi: 10.1007/s11854-010-0023-2. [30] E. M. Stein and G. Weiss, Fractional integrals on n-dimensional Euclidean space,, J. Math. Mech., 7 (1958), 503. [31] E. M. Stein and G. Weiss, "Introduction to Fourier Analysis on Euclidean Spaces,", Princeton Mathematical Series, (1971). [32] J. Sterbenz, Angular regularity and Strichartz estimates for the wave equation,, With an appendix by Igor Rodnianski, 4 (2005), 187. doi: 10.1155/IMRN.2005.187. [33] R. S. Strichartz, Multipliers for spherical harmonic expansions,, Trans. Amer. Math. Soc., 167 (1972), 115. doi: 10.2307/1996130. [34] R. Strichartz, Restriction of Fourier transform to quadratic surfaces and decay of solutions to the wave equation,, Duke Math. J., 44 (1977), 705. doi: 10.1215/S0012-7094-77-04430-1. [35] T. Tao, Spherically averaged endpoint Strichartz estimates for the two-dimensional Schrödinger equation,, Comm. Partial Differential Equations, 25 (2000), 1471. doi: 10.1080/03605300008821556. [36] T. Tao, "Nonlinear Dispersive Equations: Local and Global Analysis,", CBMS Regional Conference Series in Mathematics, (2006). [37] D. Tataru, Strichartz estimates in the hyperbolic space and global existence for the semilinear wave equation,, Trans. Amer. Math. Soc., 353 (2001), 795. doi: 10.1090/S0002-9947-00-02750-1. [38] H. Triebel, "Interpolation Theory, Function Spaces, Differential Operators,", North-Holland Mathematical Library, 18 (1978). [39] G. N. Watson, A treatise on the theory of Bessel functions,, Reprint of the second (1944) edition. Cambridge Mathematical Library. Cambridge University Press, (1944). [40] X. Yu, Generalized Strichartz estimates on perturbed wave equation and applications on Strauss conjecture,, Differential Integral Equations, 24 (2011), 443. [41] Y. Zhou, Blow up of classical solutions to $\square u = |u|^{1+\alpha$ in three space dimensions,, J. Partial Differential Equations, 5 (1992), 21. [42] Y. Zhou, Life span of classical solutions to $\square u=|u|^p p$ in two space dimensions,, Chinese Ann. Math. Ser. B, 14 (1993), 225.
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