# American Institute of Mathematical Sciences

2015, 22: 46-54. doi: 10.3934/era.2015.22.46

## A sharp Sobolev-Strichartz estimate for the wave equation

 1 Department of Mathematics, Graduate School of Science and Engineering, Saitama University, Saitama 338-8570, Japan 2 School of Mathematics, University of Birmingham, Edgbaston, Birmingham, B15 2TT, United Kingdom

Received  June 2014 Revised  June 2015 Published  August 2015

We calculate the the sharp constant and characterize the extremal initial data in $\dot{H}^{\frac{3}{4}} \times \dot{H}^{-\frac{1}{4}}$ for the $L^4$ Sobolev--Strichartz estimate for the wave equation in four spatial dimensions.
Citation: Neal Bez, Chris Jeavons. A sharp Sobolev-Strichartz estimate for the wave equation. Electronic Research Announcements, 2015, 22: 46-54. doi: 10.3934/era.2015.22.46
##### References:
 [1] K. Atkinson and W. Han, Spherical Harmonics and Approximations to the Unit Sphere: An Introduction, Lecture Notes in Mathematics, 2044, Springer, Heidelberg, 2012. doi: 10.1007/978-3-642-25983-8. [2] W. Beckner, Sobolev inequalities on the sphere and the Moser-Trudinger inequality, Ann. of Math. (2), 138 (1993), 213-242. doi: 10.2307/2946638. [3] N. Bez and K. M. Rogers, A sharp Strichartz estimate for the wave equation with data in the energy space, J. Eur. Math. Soc., 15 (2013), 805-823. doi: 10.4171/JEMS/377. [4] N. Bez and M. Sugimoto, Optimal constants and extremisers for some smoothing estimates, to appear in Journal d'Analyse Mathématique, arXiv:1206.5110. [5] A. Bulut, Maximizers for the Strichartz inequalities for the wave equation, Differential Integral Equations, 23 (2010), 1035-1072. [6] E. Carneiro and D. Oliveira e Silva, Some sharp restriction inequalities on the sphere, to appear in International Mathematics Research Notices, arXiv:1404.1106. doi: 10.1093/imrn/rnu194. [7] L. Fanelli, L. Vega and N. Visciglia, Existence of maximizers for Sobolev-Strichartz inequalities, Adv. Math., 229 (2012), 1912-1923. doi: 10.1016/j.aim.2011.12.012. [8] D. Foschi, Maximizers for the Strichartz inequality, J. Eur. Math. Soc., 9 (2007), 739-774. doi: 10.4171/JEMS/95. [9] D. Foschi, Global maximizers for the sphere adjoint restriction inequality, J. Funct. Anal., 268 (2015), 690-702. doi: 10.1016/j.jfa.2014.10.015. [10] R. Frank and E. H. Lieb, A new, rearrangement-free proof of the sharp Hardy-Littlewood-Sobolev inequality, in Spectral Theory, Function Spaces and Inequalities (eds. B. M. Brown, et al.), Oper. Theory Adv. Appl., 219, Birkhäuser/Springer Basel AG, Basel, 2012, 55-67. doi: 10.1007/978-3-0348-0263-5_4. [11] C. Jeavons, A sharp bilinear estimate for the Klein-Gordon equation in arbitrary space-time dimensions, Differential Integral Equations, 27 (2014), 137-156. [12] E. H. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. of Math. (2), 118 (1983), 349-374. doi: 10.2307/2007032. [13] R. Quilodrán, Nonexistence of extremals for the adjoint restriction inequality on the hyperboloid, J. Anal. Math., 125 (2015), 37-70. doi: 10.1007/s11854-015-0002-8. [14] J. Ramos, A refinement of the Strichartz inequality for the wave equation with applications, Adv. Math., 230 (2012), 649-698. doi: 10.1016/j.aim.2012.02.020. [15] S. Shao, Maximizers for the Strichartz and the Sobolev-Strichartz inequalities for the Schrödinger equation, Electron. J. Differential Equations, (2009), 13 pp. [16] E. Carneiro, A sharp inequality for the Strichartz norm, Int. Math. Res. Not., (2009), 3127-3145. doi: 10.1093/imrn/rnp045. [17] D. Hundertmark and V. Zharnitsky, On sharp Strichartz inequalities in low dimensions, Int. Math. Res. Not., (2006), Art. ID 34080, 18 pp. doi: 10.1155/IMRN/2006/34080. [18] T. Ozawa and Y. Tsutsumi, Space-time estimates for null gauge forms and nonlinear Schrödinger equations, Differential Integral Equations, 11 (1998), 201-222. [19] R. S. Strichartz, Restrictions of Fourier transforms to quadratic surfaces and decay of solutions to wave equations, Duke Math. J., 44 (1977), 705-714. doi: 10.1215/S0012-7094-77-04430-1.

show all references

##### References:
 [1] K. Atkinson and W. Han, Spherical Harmonics and Approximations to the Unit Sphere: An Introduction, Lecture Notes in Mathematics, 2044, Springer, Heidelberg, 2012. doi: 10.1007/978-3-642-25983-8. [2] W. Beckner, Sobolev inequalities on the sphere and the Moser-Trudinger inequality, Ann. of Math. (2), 138 (1993), 213-242. doi: 10.2307/2946638. [3] N. Bez and K. M. Rogers, A sharp Strichartz estimate for the wave equation with data in the energy space, J. Eur. Math. Soc., 15 (2013), 805-823. doi: 10.4171/JEMS/377. [4] N. Bez and M. Sugimoto, Optimal constants and extremisers for some smoothing estimates, to appear in Journal d'Analyse Mathématique, arXiv:1206.5110. [5] A. Bulut, Maximizers for the Strichartz inequalities for the wave equation, Differential Integral Equations, 23 (2010), 1035-1072. [6] E. Carneiro and D. Oliveira e Silva, Some sharp restriction inequalities on the sphere, to appear in International Mathematics Research Notices, arXiv:1404.1106. doi: 10.1093/imrn/rnu194. [7] L. Fanelli, L. Vega and N. Visciglia, Existence of maximizers for Sobolev-Strichartz inequalities, Adv. Math., 229 (2012), 1912-1923. doi: 10.1016/j.aim.2011.12.012. [8] D. Foschi, Maximizers for the Strichartz inequality, J. Eur. Math. Soc., 9 (2007), 739-774. doi: 10.4171/JEMS/95. [9] D. Foschi, Global maximizers for the sphere adjoint restriction inequality, J. Funct. Anal., 268 (2015), 690-702. doi: 10.1016/j.jfa.2014.10.015. [10] R. Frank and E. H. Lieb, A new, rearrangement-free proof of the sharp Hardy-Littlewood-Sobolev inequality, in Spectral Theory, Function Spaces and Inequalities (eds. B. M. Brown, et al.), Oper. Theory Adv. Appl., 219, Birkhäuser/Springer Basel AG, Basel, 2012, 55-67. doi: 10.1007/978-3-0348-0263-5_4. [11] C. Jeavons, A sharp bilinear estimate for the Klein-Gordon equation in arbitrary space-time dimensions, Differential Integral Equations, 27 (2014), 137-156. [12] E. H. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. of Math. (2), 118 (1983), 349-374. doi: 10.2307/2007032. [13] R. Quilodrán, Nonexistence of extremals for the adjoint restriction inequality on the hyperboloid, J. Anal. Math., 125 (2015), 37-70. doi: 10.1007/s11854-015-0002-8. [14] J. Ramos, A refinement of the Strichartz inequality for the wave equation with applications, Adv. Math., 230 (2012), 649-698. doi: 10.1016/j.aim.2012.02.020. [15] S. Shao, Maximizers for the Strichartz and the Sobolev-Strichartz inequalities for the Schrödinger equation, Electron. J. Differential Equations, (2009), 13 pp. [16] E. Carneiro, A sharp inequality for the Strichartz norm, Int. Math. Res. Not., (2009), 3127-3145. doi: 10.1093/imrn/rnp045. [17] D. Hundertmark and V. Zharnitsky, On sharp Strichartz inequalities in low dimensions, Int. Math. Res. Not., (2006), Art. ID 34080, 18 pp. doi: 10.1155/IMRN/2006/34080. [18] T. Ozawa and Y. Tsutsumi, Space-time estimates for null gauge forms and nonlinear Schrödinger equations, Differential Integral Equations, 11 (1998), 201-222. [19] R. S. Strichartz, Restrictions of Fourier transforms to quadratic surfaces and decay of solutions to wave equations, Duke Math. J., 44 (1977), 705-714. doi: 10.1215/S0012-7094-77-04430-1.
 [1] Robert Schippa. Sharp Strichartz estimates in spherical coordinates. Communications on Pure and Applied Analysis, 2017, 16 (6) : 2047-2051. doi: 10.3934/cpaa.2017100 [2] Seongyeon Kim, Yehyun Kwon, Ihyeok Seo. Strichartz estimates and local regularity for the elastic wave equation with singular potentials. Discrete and Continuous Dynamical Systems, 2021, 41 (4) : 1897-1911. doi: 10.3934/dcds.2020344 [3] Vladimir Georgiev, Atanas Stefanov, Mirko Tarulli. Smoothing-Strichartz estimates for the Schrodinger equation with small magnetic potential. Discrete and Continuous Dynamical Systems, 2007, 17 (4) : 771-786. doi: 10.3934/dcds.2007.17.771 [4] Younghun Hong, Changhun Yang. Uniform Strichartz estimates on the lattice. Discrete and Continuous Dynamical Systems, 2019, 39 (6) : 3239-3264. doi: 10.3934/dcds.2019134 [5] Jin-Cheng Jiang, Chengbo Wang, Xin Yu. Generalized and weighted Strichartz estimates. Communications on Pure and Applied Analysis, 2012, 11 (5) : 1723-1752. doi: 10.3934/cpaa.2012.11.1723 [6] Robert Schippa. Generalized inhomogeneous Strichartz estimates. Discrete and Continuous Dynamical Systems, 2017, 37 (6) : 3387-3410. doi: 10.3934/dcds.2017143 [7] Gong Chen. Strichartz estimates for charge transfer models. Discrete and Continuous Dynamical Systems, 2017, 37 (3) : 1201-1226. doi: 10.3934/dcds.2017050 [8] Hyeongjin Lee, Ihyeok Seo, Jihyeon Seok. Local smoothing and Strichartz estimates for the Klein-Gordon equation with the inverse-square potential. Discrete and Continuous Dynamical Systems, 2020, 40 (1) : 597-608. doi: 10.3934/dcds.2020024 [9] Chu-Hee Cho, Youngwoo Koh, Ihyeok Seo. On inhomogeneous Strichartz estimates for fractional Schrödinger equations and their applications. Discrete and Continuous Dynamical Systems, 2016, 36 (4) : 1905-1926. doi: 10.3934/dcds.2016.36.1905 [10] Youngwoo Koh, Ihyeok Seo. Strichartz estimates for Schrödinger equations in weighted $L^2$ spaces and their applications. Discrete and Continuous Dynamical Systems, 2017, 37 (9) : 4877-4906. doi: 10.3934/dcds.2017210 [11] Younghun Hong. Strichartz estimates for $N$-body Schrödinger operators with small potential interactions. Discrete and Continuous Dynamical Systems, 2017, 37 (10) : 5355-5365. doi: 10.3934/dcds.2017233 [12] Michael Goldberg. Strichartz estimates for Schrödinger operators with a non-smooth magnetic potential. Discrete and Continuous Dynamical Systems, 2011, 31 (1) : 109-118. doi: 10.3934/dcds.2011.31.109 [13] Roberta Bosi, Jean Dolbeault, Maria J. Esteban. Estimates for the optimal constants in multipolar Hardy inequalities for Schrödinger and Dirac operators. Communications on Pure and Applied Analysis, 2008, 7 (3) : 533-562. doi: 10.3934/cpaa.2008.7.533 [14] Fabrice Planchon, John G. Stalker, A. Shadi Tahvildar-Zadeh. $L^p$ Estimates for the wave equation with the inverse-square potential. Discrete and Continuous Dynamical Systems, 2003, 9 (2) : 427-442. doi: 10.3934/dcds.2003.9.427 [15] Yongqin Liu, Weike Wang. The pointwise estimates of solutions for dissipative wave equation in multi-dimensions. Discrete and Continuous Dynamical Systems, 2008, 20 (4) : 1013-1028. doi: 10.3934/dcds.2008.20.1013 [16] Mourad Bellassoued, David Dos Santos Ferreira. Stability estimates for the anisotropic wave equation from the Dirichlet-to-Neumann map. Inverse Problems and Imaging, 2011, 5 (4) : 745-773. doi: 10.3934/ipi.2011.5.745 [17] David Cruz-Uribe, SFO, José María Martell, Carlos Pérez. Sharp weighted estimates for approximating dyadic operators. Electronic Research Announcements, 2010, 17: 12-19. doi: 10.3934/era.2010.17.12 [18] Ha Tuan Dung, Nguyen Thac Dung, Jiayong Wu. Sharp gradient estimates on weighted manifolds with compact boundary. Communications on Pure and Applied Analysis, 2021, 20 (12) : 4127-4138. doi: 10.3934/cpaa.2021148 [19] Gianluca Mola. Recovering a large number of diffusion constants in a parabolic equation from energy measurements. Inverse Problems and Imaging, 2018, 12 (3) : 527-543. doi: 10.3934/ipi.2018023 [20] Haruya Mizutani. Strichartz estimates for Schrödinger equations with variable coefficients and unbounded potentials II. Superquadratic potentials. Communications on Pure and Applied Analysis, 2014, 13 (6) : 2177-2210. doi: 10.3934/cpaa.2014.13.2177

2020 Impact Factor: 0.929