January  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). doi: 10.1007/978-3-642-25983-8. Google Scholar

[2]

W. Beckner, Sobolev inequalities on the sphere and the Moser-Trudinger inequality,, \emph{Ann. of Math. (2)}, 138 (1993), 213. doi: 10.2307/2946638. Google Scholar

[3]

N. Bez and K. M. Rogers, A sharp Strichartz estimate for the wave equation with data in the energy space,, \emph{J. Eur. Math. Soc.}, 15 (2013), 805. doi: 10.4171/JEMS/377. Google Scholar

[4]

N. Bez and M. Sugimoto, Optimal constants and extremisers for some smoothing estimates,, to appear in \emph{Journal d'Analyse Mathématique}, (). Google Scholar

[5]

A. Bulut, Maximizers for the Strichartz inequalities for the wave equation,, \emph{Differential Integral Equations}, 23 (2010), 1035. Google Scholar

[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. Google Scholar

[7]

L. Fanelli, L. Vega and N. Visciglia, Existence of maximizers for Sobolev-Strichartz inequalities,, \emph{Adv. Math.}, 229 (2012), 1912. doi: 10.1016/j.aim.2011.12.012. Google Scholar

[8]

D. Foschi, Maximizers for the Strichartz inequality,, \emph{J. Eur. Math. Soc.}, 9 (2007), 739. doi: 10.4171/JEMS/95. Google Scholar

[9]

D. Foschi, Global maximizers for the sphere adjoint restriction inequality,, \emph{J. Funct. Anal.}, 268 (2015), 690. doi: 10.1016/j.jfa.2014.10.015. Google Scholar

[10]

R. Frank and E. H. Lieb, A new, rearrangement-free proof of the sharp Hardy-Littlewood-Sobolev inequality,, in \emph{Spectral Theory, (2012), 55. doi: 10.1007/978-3-0348-0263-5_4. Google Scholar

[11]

C. Jeavons, A sharp bilinear estimate for the Klein-Gordon equation in arbitrary space-time dimensions,, \emph{Differential Integral Equations}, 27 (2014), 137. Google Scholar

[12]

E. H. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities,, \emph{Ann. of Math. (2)}, 118 (1983), 349. doi: 10.2307/2007032. Google Scholar

[13]

R. Quilodrán, Nonexistence of extremals for the adjoint restriction inequality on the hyperboloid,, \emph{J. Anal. Math.}, 125 (2015), 37. doi: 10.1007/s11854-015-0002-8. Google Scholar

[14]

J. Ramos, A refinement of the Strichartz inequality for the wave equation with applications,, \emph{Adv. Math.}, 230 (2012), 649. doi: 10.1016/j.aim.2012.02.020. Google Scholar

[15]

S. Shao, Maximizers for the Strichartz and the Sobolev-Strichartz inequalities for the Schrödinger equation,, \emph{Electron. J. Differential Equations}, (2009). Google Scholar

[16]

E. Carneiro, A sharp inequality for the Strichartz norm,, \emph{Int. Math. Res. Not.}, (2009), 3127. doi: 10.1093/imrn/rnp045. Google Scholar

[17]

D. Hundertmark and V. Zharnitsky, On sharp Strichartz inequalities in low dimensions,, \emph{Int. Math. Res. Not.}, (2006). doi: 10.1155/IMRN/2006/34080. Google Scholar

[18]

T. Ozawa and Y. Tsutsumi, Space-time estimates for null gauge forms and nonlinear Schrödinger equations,, \emph{Differential Integral Equations}, 11 (1998), 201. Google Scholar

[19]

R. S. Strichartz, Restrictions of Fourier transforms to quadratic surfaces and decay of solutions to wave equations,, \emph{Duke Math. J.}, 44 (1977), 705. doi: 10.1215/S0012-7094-77-04430-1. Google Scholar

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). doi: 10.1007/978-3-642-25983-8. Google Scholar

[2]

W. Beckner, Sobolev inequalities on the sphere and the Moser-Trudinger inequality,, \emph{Ann. of Math. (2)}, 138 (1993), 213. doi: 10.2307/2946638. Google Scholar

[3]

N. Bez and K. M. Rogers, A sharp Strichartz estimate for the wave equation with data in the energy space,, \emph{J. Eur. Math. Soc.}, 15 (2013), 805. doi: 10.4171/JEMS/377. Google Scholar

[4]

N. Bez and M. Sugimoto, Optimal constants and extremisers for some smoothing estimates,, to appear in \emph{Journal d'Analyse Mathématique}, (). Google Scholar

[5]

A. Bulut, Maximizers for the Strichartz inequalities for the wave equation,, \emph{Differential Integral Equations}, 23 (2010), 1035. Google Scholar

[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. Google Scholar

[7]

L. Fanelli, L. Vega and N. Visciglia, Existence of maximizers for Sobolev-Strichartz inequalities,, \emph{Adv. Math.}, 229 (2012), 1912. doi: 10.1016/j.aim.2011.12.012. Google Scholar

[8]

D. Foschi, Maximizers for the Strichartz inequality,, \emph{J. Eur. Math. Soc.}, 9 (2007), 739. doi: 10.4171/JEMS/95. Google Scholar

[9]

D. Foschi, Global maximizers for the sphere adjoint restriction inequality,, \emph{J. Funct. Anal.}, 268 (2015), 690. doi: 10.1016/j.jfa.2014.10.015. Google Scholar

[10]

R. Frank and E. H. Lieb, A new, rearrangement-free proof of the sharp Hardy-Littlewood-Sobolev inequality,, in \emph{Spectral Theory, (2012), 55. doi: 10.1007/978-3-0348-0263-5_4. Google Scholar

[11]

C. Jeavons, A sharp bilinear estimate for the Klein-Gordon equation in arbitrary space-time dimensions,, \emph{Differential Integral Equations}, 27 (2014), 137. Google Scholar

[12]

E. H. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities,, \emph{Ann. of Math. (2)}, 118 (1983), 349. doi: 10.2307/2007032. Google Scholar

[13]

R. Quilodrán, Nonexistence of extremals for the adjoint restriction inequality on the hyperboloid,, \emph{J. Anal. Math.}, 125 (2015), 37. doi: 10.1007/s11854-015-0002-8. Google Scholar

[14]

J. Ramos, A refinement of the Strichartz inequality for the wave equation with applications,, \emph{Adv. Math.}, 230 (2012), 649. doi: 10.1016/j.aim.2012.02.020. Google Scholar

[15]

S. Shao, Maximizers for the Strichartz and the Sobolev-Strichartz inequalities for the Schrödinger equation,, \emph{Electron. J. Differential Equations}, (2009). Google Scholar

[16]

E. Carneiro, A sharp inequality for the Strichartz norm,, \emph{Int. Math. Res. Not.}, (2009), 3127. doi: 10.1093/imrn/rnp045. Google Scholar

[17]

D. Hundertmark and V. Zharnitsky, On sharp Strichartz inequalities in low dimensions,, \emph{Int. Math. Res. Not.}, (2006). doi: 10.1155/IMRN/2006/34080. Google Scholar

[18]

T. Ozawa and Y. Tsutsumi, Space-time estimates for null gauge forms and nonlinear Schrödinger equations,, \emph{Differential Integral Equations}, 11 (1998), 201. Google Scholar

[19]

R. S. Strichartz, Restrictions of Fourier transforms to quadratic surfaces and decay of solutions to wave equations,, \emph{Duke Math. J.}, 44 (1977), 705. doi: 10.1215/S0012-7094-77-04430-1. Google Scholar

[1]

Robert Schippa. Sharp Strichartz estimates in spherical coordinates. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2047-2051. doi: 10.3934/cpaa.2017100

[2]

Vladimir Georgiev, Atanas Stefanov, Mirko Tarulli. Smoothing-Strichartz estimates for the Schrodinger equation with small magnetic potential. Discrete & Continuous Dynamical Systems - A, 2007, 17 (4) : 771-786. doi: 10.3934/dcds.2007.17.771

[3]

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

[4]

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

[5]

Younghun Hong, Changhun Yang. Uniform Strichartz estimates on the lattice. Discrete & Continuous Dynamical Systems - A, 2019, 39 (6) : 3239-3264. doi: 10.3934/dcds.2019134

[6]

Gong Chen. Strichartz estimates for charge transfer models. Discrete & Continuous Dynamical Systems - A, 2017, 37 (3) : 1201-1226. doi: 10.3934/dcds.2017050

[7]

Roberta Bosi, Jean Dolbeault, Maria J. Esteban. Estimates for the optimal constants in multipolar Hardy inequalities for Schrödinger and Dirac operators. Communications on Pure & Applied Analysis, 2008, 7 (3) : 533-562. doi: 10.3934/cpaa.2008.7.533

[8]

Chu-Hee Cho, Youngwoo Koh, Ihyeok Seo. On inhomogeneous Strichartz estimates for fractional Schrödinger equations and their applications. Discrete & Continuous Dynamical Systems - A, 2016, 36 (4) : 1905-1926. doi: 10.3934/dcds.2016.36.1905

[9]

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

[10]

Younghun Hong. Strichartz estimates for $N$-body Schrödinger operators with small potential interactions. Discrete & Continuous Dynamical Systems - A, 2017, 37 (10) : 5355-5365. doi: 10.3934/dcds.2017233

[11]

Michael Goldberg. Strichartz estimates for Schrödinger operators with a non-smooth magnetic potential. Discrete & Continuous Dynamical Systems - A, 2011, 31 (1) : 109-118. doi: 10.3934/dcds.2011.31.109

[12]

Fabrice Planchon, John G. Stalker, A. Shadi Tahvildar-Zadeh. $L^p$ Estimates for the wave equation with the inverse-square potential. Discrete & Continuous Dynamical Systems - A, 2003, 9 (2) : 427-442. doi: 10.3934/dcds.2003.9.427

[13]

Yongqin Liu, Weike Wang. The pointwise estimates of solutions for dissipative wave equation in multi-dimensions. Discrete & Continuous Dynamical Systems - A, 2008, 20 (4) : 1013-1028. doi: 10.3934/dcds.2008.20.1013

[14]

Mourad Bellassoued, David Dos Santos Ferreira. Stability estimates for the anisotropic wave equation from the Dirichlet-to-Neumann map. Inverse Problems & Imaging, 2011, 5 (4) : 745-773. doi: 10.3934/ipi.2011.5.745

[15]

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

[16]

Gianluca Mola. Recovering a large number of diffusion constants in a parabolic equation from energy measurements. Inverse Problems & Imaging, 2018, 12 (3) : 527-543. doi: 10.3934/ipi.2018023

[17]

Haruya Mizutani. Strichartz estimates for Schrödinger equations with variable coefficients and unbounded potentials II. Superquadratic potentials. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2177-2210. doi: 10.3934/cpaa.2014.13.2177

[18]

Marco Cappiello, Fabio Nicola. Sharp decay estimates and smoothness for solutions to nonlocal semilinear equations. Discrete & Continuous Dynamical Systems - A, 2016, 36 (4) : 1869-1880. doi: 10.3934/dcds.2016.36.1869

[19]

Weiwei Ao. Sharp estimates for fully bubbling solutions of $B_2$ Toda system. Discrete & Continuous Dynamical Systems - A, 2016, 36 (4) : 1759-1788. doi: 10.3934/dcds.2016.36.1759

[20]

Nikos I. Karachalios, Nikos M. Stavrakakis. Estimates on the dimension of a global attractor for a semilinear dissipative wave equation on $\mathbb R^N$. Discrete & Continuous Dynamical Systems - A, 2002, 8 (4) : 939-951. doi: 10.3934/dcds.2002.8.939

2018 Impact Factor: 0.263

Metrics

  • PDF downloads (23)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]