Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-27T09:30:35.102Z Has data issue: false hasContentIssue false
  • English
  • Français

SOME SHARP BILINEAR SPACE–TIME ESTIMATES FOR THE WAVE EQUATION

Part of: Hyperbolic equations and systems Harmonic analysis in several variables Partial differential equations

Published online by Cambridge University Press:  07 March 2016

Neal Bez ,
Chris Jeavons  and
Tohru Ozawa
Neal Bez
Affiliation:
Department of Mathematics, Graduate School of Science and Engineering, Saitama University, Saitama 338-8570, Japan email nealbez@mail.saitama-u.ac.jp
Chris Jeavons
Affiliation:
School of Mathematics, University of Birmingham, Edgbaston, Birmingham B15 2TT, U.K. email jeavonsc@maths.bham.ac.uk
Tohru Ozawa
Affiliation:
Department of Applied Physics, Waseda University, Tokyo 169-8555, Japan email txozawa@waseda.jp
Get access

Abstract

We prove a family of sharp bilinear space–time estimates for the half-wave propagator $\text{e}^{\text{i}t\sqrt{-\unicode[STIX]{x1D6E5}}}$ . As a consequence, for radially symmetric initial data, we establish sharp estimates of this kind for a range of exponents beyond the classical range.

MSC classification

Primary: 35B45: A priori estimates
Secondary: 35L05: Wave equation 42B37: Harmonic analysis and PDE
Type
Research Article
Information
Mathematika , Volume 62 , Issue 3 , 2016 , pp. 719 - 737
Copyright
Copyright © University College London 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Beals, M., Self-spreading and strength of singularities for solutions to semilinear wave equations. Ann. of Math. (2) 118 1983, 187–214.CrossRefGoogle Scholar
Beckner, W., Functionals for multilinear fractional embedding. Acta Math. Appl. Sin. Engl. Ser. 31 2015, 1–28.Google Scholar
Belazzini, J., Ghimenti, M. and Ozawa, T., Sharp lower bounds for Coulomb energy. Math. Res. Lett. (to appear); arXiv:1410.0958.Google Scholar
Bennett, J., Bez, N. and Iliopoulou, M., Flow monotonicity and Strichartz inequalities. Int. Math. Res. Not. IMRN 2015(19) 2015, 9415–9437.CrossRefGoogle Scholar
Bennett, J., Bez, N., Jeavons, C. and Pattakos, N., On sharp bilinear Strichartz estimates of Ozawa–Tsutsumi type. J. Math. Soc. Japan (to appear); arXiv:1404.2466.Google Scholar
Bez, N. and Jeavons, C., A sharp Sobolev–Strichartz estimate for the wave equation. Electron. Res. Announc. Math. Sci. 22 2015, 46–54.Google Scholar
Bez, N. and Rogers, K. M., A sharp Strichartz estimate for the wave equation with data in the energy space. J. Eur. Math. Soc. (JEMS) 15 2013, 805–823.CrossRefGoogle Scholar
Bez, N. and Sugimoto, M., Optimal forward and reverse estimates of Morawetz and Kato–Yajima type with angular smoothing index. J. Fourier Anal. Appl. 21 2015, 318–341.CrossRefGoogle Scholar
Bez, N. and Sugimoto, M., Optimal constants and extremisers for some smoothing estimates. J. Anal. Math. (to appear).Google Scholar
Bourgain, J., Estimates for cone multipliers. In Geometric Aspects of Functional Analysis (Operator Theory: Advances and Applications 77 ), Birkhäuser (Basel, 1995), 41–60.Google Scholar
Bourgain, J., Global solutions of nonlinear Schrödinger equations. Amer. Math. Soc. Colloq. Publ. 46 1999, 86–90.Google Scholar
Carneiro, E., A sharp inequality for the Strichartz norm. Int. Math. Res. Not. IMRN 16 2009, 3127–3145.Google Scholar
Cho, Y. and Ozawa, T., On radial solutions of semi-relativistic Hartree equations. Discrete Contin. Dyn. Syst. Ser. S 1 2008, 71–82.Google Scholar
Fang, D. and Wang, C., Some remarks on Strichartz estimates for homogeneous wave equation. Nonlinear Anal. 65 2006, 697–706.Google Scholar
Foschi, D., On an endpoint case of the Klainerman–Machedon estimates. Comm. Partial Differential Equations 25 2000, 1537–1547.CrossRefGoogle Scholar
Foschi, D., Maximizers for the Strichartz inequality. J. Eur. Math. Soc. (JEMS) 9 2007, 739–774.Google Scholar
Foschi, D. and Klainerman, S., Bilinear space–time estimates for homogeneous wave equations. Ann. Sci. Éc. Norm. Supér. (4) 33 2000, 211–274.CrossRefGoogle Scholar
Hidano, K., On weighted Strichartz estimates and NLS for radial data in Sobolev spaces of negative indices. RIMS Kôkyûroku Bessatsu B22 2010, 1–12.Google Scholar
Hidano, K. and Kurokawa, Y., Weighted HLS inequalities for radial functions and Strichartz estimates for wave and Schrödinger equations. Illinois J. Math. 52 2008, 365–388.CrossRefGoogle Scholar
Hundertmark, D. and Zharnitsky, V., On sharp Strichartz inequalities in low dimensions. Int. Math. Res. Not. IMRN 2006, Art. ID 34080, p. 18.CrossRefGoogle Scholar
Jeavons, C., A sharp bilinear estimate for the Klein–Gordon equation in arbitrary space–time dimensions. Differential Integral Equations 27 2014, 137–156.CrossRefGoogle Scholar
Klainerman, S. and Machedon, M., Space–time estimates for null forms and the local existence theorem. Comm. Pure Appl. Math. 46 1993, 1221–1268.CrossRefGoogle Scholar
Klainerman, S. and Machedon, M., Smoothing estimates for null forms and applications. Duke Math. J. 81 1995, 99–134.CrossRefGoogle Scholar
Klainerman, S. and Machedon, M., Remark on Strichartz–type inequalities. Int. Math. Res. Not. IMRN 5 1996, 201–220; with appendices by Jean Bourgain and Daniel Tataru.Google Scholar
Klainerman, S. and Machedon, M., Estimates for null forms and the spaces H s, 𝛿 . Int. Math. Res. Not. IMRN 17 1996, 853–865.CrossRefGoogle Scholar
Klainerman, S. and Machedon, M., On the regularity properties of a model problem related to wave maps. Duke Math. J. 87 1997, 553–589.Google Scholar
Klainerman, S. and Machedon, M., On the optimal regularity for gauge field theories. Differential Integral Equations 6 1997, 1019–1030.Google Scholar
Klainerman, S. and Selberg, S., Remark on the optimal regularity for equations of wave maps type. Comm. Partial Differential Equations 22 1997, 901–918.CrossRefGoogle Scholar
Lee, S., Bilinear restriction estimates for surfaces with curvatures of different signs. Trans. Amer. Math. Soc. 358 2006, 3511–3533.CrossRefGoogle Scholar
Lee, S., Rogers, K. and Vargas, A., Sharp null form estimates for the wave equation in ℝ3+1 . Int. Math. Res. Not. IMRN 2008, Art. ID rnn 096, p. 18.Google Scholar
Lee, S. and Vargas, A., Sharp null form estimates for the wave equation. Amer. J. Math. 130 2008, 1279–1326.Google Scholar
Lieb, E. H., Sharp constants in the Hardy–Littlewood–Sobolev and related inequalities. Ann. of Math. (2) 118 1983, 349–374.Google Scholar
Ozawa, T. and Rogers, K., Sharp Morawetz estimates. J. Anal. Math. 121 2013, 163–175.Google Scholar
Ozawa, T. and Rogers, K., A sharp bilinear estimate for the Klein–Gordon equation in ℝ1+1 . Int. Math. Res. Not. IMRN 2014 2014, 1367–1378.CrossRefGoogle Scholar
Ozawa, T. and Tsutsumi, Y., Space–time estimates for null gauge forms and nonlinear Schrödinger equations. Differential Integral Equations 11 1998, 201–222.Google Scholar
Quilodrán, R., Nonexistence of extremals for the adjoint restriction inequality on the hyperboloid. J. Anal. Math. 125 2015, 37–70.Google Scholar
Sogge, C., Lectures on Nonlinear Wave Equations, International Press (Cambridge, MA, 1995).Google Scholar
Sterbenz, J., Angular regularity and Strichartz estimates for the wave equation. Int. Math. Res. Not. IMRN 2005 2005, 187–231.CrossRefGoogle Scholar
Tao, T., Endpoint bilinear restriction theorems for the cone, and some sharp null form estimates. Math. Z. 238 2001, 215–268.Google Scholar
Wolff, T., A sharp bilinear cone restriction estimate. Ann. of Math. (2) 153 2001, 661–698.Google Scholar