Abstract
We prove sharp Morawetz estimates — global in time with a singular weight in the spatial variables — for the linear wave, Klein-Gordon, and Schrödinger equations, for which we can characterise the maximisers. We also prove refined inequalities with respect to the angular integrability.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
M. Ben-Artzi and S. Klainerman, Decay and regularity for the Schrödinger equation, J. Anal. Math. 58 (1992), 25–37.
J. Bennett, N. Bez, A. Carbery, and D. Hundertmark, Heat-flow monotonicity of Strichartz norms, Anal. PDE 2 (2009), 147–158.
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.
N. Bez and M. Sugimoto, Optimal constants and extremisers for some smoothing estimates, arXiv:1206.5110 [math. AP].
A. Bulut, Maximizers for the Strichartz inequalities for the wave equation, Differential Integral Equations 23 (2010), 1035–1072.
E. Carneiro, A sharp inequality for the Strichartz norm, Int. Math. Res. Not. IMRN 2009, no. 16, 3127–3145.
Y. Cho, S. Lee, and T. Ozawa, On Hartree equations with derivatives, Nonlinear Anal. 74 (2011), 2094–2108.
P. Constantin and J. C. Saut, Local smoothing properties of dispersive equations, J. Amer. Math. Soc. 1 (1988), 413–439.
M. Cowling, L. Escauriaza, C. E. Kenig, G. Ponce, and L. Vega, The Hardy uncertainty principle revisited, Indiana U. Math. J. 59 (2010), 2007–2025.
P. D’Ancona and R. Luca’, Stein-Weiss and Caffarelli-Kohn-Nirenberg inequalties with angular integrability, J. Math. Anal. Appl. 388 (2012), 1061–1079.
T. Duyckaerts, F. Merle, and S. Roudenko, Maximizers for the Strichartz norm for small solutions of mass-critical NLS, Ann. Sc. Norm. Super. Pisa Cl. Sci.(5) 10 (2011), 427–476.
L. Fanelli, L. Vega, and N. Visciglia, On the existence of maximizers for a family of restriction theorems, Bull. Lond. Math. Soc. 43 (2011), 811–817.
L. Fanelli, L. Vega, and N. Visciglia, Existence of maximizers for Sobolev-Strichartz inequalities, Adv. Math. 229 (2012), 1912–1923.
D. Foschi, Maximizers for the Strichartz inequality, J. Eur. Math. Soc. 9 (2007), 739–774.
R. L. Frank and E. H. Lieb, A new rearrangement-free proof of the sharp Hardy-Littlewood-Sobolev inequality, Spectral Theory, Function Spaces, and Inequalities, Birkhäuser/Springer Basel AG, 2012, pp. 55–67.
T. Hoshiro, On weighted L 2 estimates of solutions to wave equations, J. Anal. Math. 72 (1997), 127–140.
D. Hundertmark and V. Zharnitsky, On sharp Strichartz inequalities in low dimensions, Int.Math. Res. Not. 2006, Art. ID 34080.
T. Kato and K. Yajima, Some examples of smooth operators and the associated smoothing effect, Rev. Math. Phys. 1 (1989), 481–496.
M. Kunze, On the existence of a maximizer for the Strichartz inequality, Comm.Math. Phys. 243 (2003), 137–162.
E. H. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Annals of Math. (2) 118 (1983), 349–374.
J. E. Lin and W. A. Strauss, Decay and scattering of solutions of a nonlinear Schrödinger equation, J. Funct. Anal. 30 (1978), 245–263.
C. S. Morawetz, Time decay for the nonlinear Klein-Gordon equations, Proc. Roy. Soc. Ser. A 306 (1968), 291–296.
B. Perthame and L. Vega, Morrey-Campanato estimates for Helmholtz equations, J. Funct. Anal. 164 (1999), 340–355.
J. Ramos, A refinement of the Strichartz inequality for the wave equation with applications, Adv. Math. 230 (2012), 649–698.
S. Shao, Maximizers for the Strichartz and the Sobolev-Strichartz inequalities for the Schrödinger equation, Electron. J. Differential Equations 2009, No. 3.
B. Simon, Best constants in some operator smoothness estimates, J. Funct. Anal. 107 (1992), 66–71.
P. Sjölin, Regularity of solutions to the Schrödinger equation, DukeMath. J. 55 (1987), 699–715.
E.M. Stein and G. Weiss, Fractional integrals on n-dimensional Euclidean space, J.Math. Mech. 7 (1958), 503–514.
R. S. Strichartz, Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations, Duke Math. J. 44 (1977), 705–714.
M. Sugimoto, Global smoothing properties of generalized Schrödinger equations, J. Anal. Math. 76 (1998), 191–204.
M. Sugimoto, A smoothing property of Schrödinger equations along the sphere, J. Anal. Math. 89 (2003), 15–30.
L. Vega, Schrödinger equations: pointwise convergence to the initial data, Proc. Amer. Math. Soc. 102 (1988), 874–878.
L. Vega and N. Visciglia, On the local smoothing for the Schrödinger equation, Proc. Amer. Math. Soc. 135 (2007), 119–128.
L. Vega and N. Visciglia, On the equipartition of energy for the critical NLW, J. Funct. Anal. 255 (2008), 726–754.
M. C. Vilela, Regularity of solutions to the free Schrödinger equation with radial initial data, Illinois J. Math. 45 (2001), 361–370.
B. G. Walther, Regularity, decay, and best constants for dispersive equations, J. Funct. Anal. 189 (2002), 325–335.
K. Watanabe, Smooth perturbations of the self-adjoint operator \({\left| \Delta \right|^{\alpha /2}}\), Tokyo J.Math. 14 (1991), 239–250.
Author information
Authors and Affiliations
Corresponding author
Additional information
Supported by the JSPS grant 11133 (Japan), the ERC grant 277778 (Europe), and the MINECO grants MTM2010-16518 and SEV-2011-0087 (Spain).
Rights and permissions
About this article
Cite this article
Ozawa, T., Rogers, K.M. Sharp Morawetz estimates. JAMA 121, 163–175 (2013). https://doi.org/10.1007/s11854-013-0031-0
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11854-013-0031-0