Illinois Journal of Mathematics

Weighted HLS inequalities for radial functions and Strichartz estimates for wave and Schrödinger equations

Kunio Hidano and Yuki Kurokawa

Full-text: Open access


This paper is concerned with derivation of the global or local in time Strichartz estimates for radially symmetric solutions of the free wave equation from some Morawetz-type estimates via weighted Hardy–Littlewood–Sobolev (HLS) inequalities. In the same way, we also derive the weighted end-point Strichartz estimates with gain of derivatives for radially symmetric solutions of the free Schrödinger equation.

The proof of the weighted HLS inequality for radially symmetric functions involves an application of the weighted inequality due to Stein and Weiss and the Hardy–Littlewood maximal inequality in the weighted Lebesgue space due to Muckenhoupt. Under radial symmetry, we get significant gains over the usual HLS inequality and Strichartz estimate.

Article information

Illinois J. Math., Volume 52, Number 2 (2008), 365-388.

First available in Project Euclid: 23 July 2009

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35L05: Wave equation 35Q40: PDEs in connection with quantum mechanics
Secondary: 35B65: Smoothness and regularity of solutions 35Q55: NLS-like equations (nonlinear Schrödinger) [See also 37K10]


Hidano, Kunio; Kurokawa, Yuki. Weighted HLS inequalities for radial functions and Strichartz estimates for wave and Schrödinger equations. Illinois J. Math. 52 (2008), no. 2, 365--388. doi:10.1215/ijm/1248355340.

Export citation


  • R. Agemi, Blow-up of solutions to nonlinear wave equations in two space dimensions, Man. Math. 73 (1991), 153–162.
  • M. Ben-Artzi, Regularity and smoothing for some equations of evolution, Nonlinear partial differential equations and applications (H. Brezis and J. L. Lions, eds.), vol. 11, Pittman, London, 1994, pp. 1–12.
  • M. Ben-Artzi and S. Klainerman, Decay and regularity for the Schrödinger equation, J. Anal. Math. 58 (1992), 25–37.
  • M. Christ and A. Kiselev, Maximal functions associated to filtrations, J. Funct. Anal. 179 (2001), 409–425.
  • D. Fang and C. Wang, Some remarks on Strichartz estimates for homogeneous wave equation, Nonlinear Anal. 65 (2006), 697–706.
  • D. Foschi, Inhomogeneous Strichartz estimates, J. Hyperbolic Differ. Equ. 2 (2005), 1–24.
  • J. Ginibre and G. Velo, Generalized Strichartz inequalities for the wave equation, J. Funct. Anal. 133 (1995), 50–68.
  • J. Harmse, On Lebesgue space estimates for the wave equation, Indiana Univ. Math. J. 39 (1990), 229–248.
  • K. Hidano, Small data scattering and blow-up for a wave equation with a cubic convolution, Funkcial. Ekvac. 43 (2000), 559–588.
  • K. Hidano, Morawetz-Strichartz estimates for spherically symmetric solutions to wave equations and applications to semi-linear Cauchy problems, Differential Integral Equations 20 (2007), 735–754.
  • K. Hidano, Nonlinear Schrödinger equations with radially symmetric data of critical regularity, Funkcial. Ekvac. 51 (2008), 135–147.
  • K. Hidano, Small solutions to semi-linear wave equations with radial data of critical regularity, Accepted for publication in Revista Matemática Iberoamericana.
  • T. Hoshiro, On weighted $L^2$ estimates of solutions to wave equations, J. Anal. Math. 72 (1997), 127–140.
  • H. Jiao and Z. Zhou, An elementary proof of the blow-up for semilinear wave equation in high space dimensions, J. Differential Equations 189 (2003), 355–365.
  • F. John, Plane waves and spherical means: Applied to partial differential equations, Interscience Publishers, New York, 1955.
  • T. Kato, An $L^{q,r}$-theory for nonlinear Schrödinger equations, Spectral and scattering theory and applications (K. Yajima, ed.), Adv. Stud. Pure Math., vol. 23, Math. Soc. Japan, Tokyo, 1994, pp. 223–238.
  • T. Kato and K. Yajima, Some examples of smooth operators and the associated smoothing effect, Rev. Math. Phys. 1 (1989), 481–496.
  • M. Keel and T. Tao, Endpoint Strichartz estimates, Amer. J. Math. 120 (1998), 955–980.
  • S. Klainerman and M. Machedon, Space–time estimates for null forms and the local existence theorem, Comm. Pure Appl. Math. 46 (1993), 1221–1268.
  • S. Klainerman and M. Machedon, On the algebraic properties of the $H_{n/2,1/2}$ spaces, Int. Math. Res. Not. 1998 (1998), 765–774.
  • T. T. Li and Y. Zhou, A note on the life-span of classical solutions to nonlinear wave equations in four space dimensions, Indiana Univ. Math. J. 44 (1995), 1207–1248.
  • H. Lindblad and C. D. Sogge, On existence and scattering with minimal regularity for semilinear wave equations, J. Funct. Anal. 130 (1995), 357–426.
  • H. Lindblad and C. D. Sogge, Long-time existence for small amplitude semilinear wave equations, Amer. J. Math. 118 (1996), 1047–1135.
  • W. Magnus, F. Oberhettinger and R. P. Soni, Formulas and theorems for the special functions of mathematical physics, 3rd ed., Springer-Verlag, Berlin, Heidelberg, New York, 1966.
  • C. S. Morawetz, Time decay for the Klein–Gordon equation, Proc. Roy. Soc. A 306 (1968), 291–296.
  • B. Muckenhoupt, Weighted norm inequalities for the Hardy maximal function, Trans. Amer. Math. Soc. 165 (1972), 207–226.
  • D. M. Oberlin, Convolution estimates for some distributions with singularities on the light cone, Duke Math. J. 59 (1989), 747–757.
  • H. Pecher, Nonlinear small data scattering for the wave and Klein–Gordon equation, Math. Z. 185 (1984), 261–270.
  • M. A. Rammaha, Finite-time blow-up for nonlinear wave equations in high dimensions, Comm. Partial Differential Equations 12 (1987), 677–700.
  • H. Smith and C. D. Sogge, Global Strichartz estimates for nontrapping perturbations of the Laplacian, Comm. Partial Differential Equations 25 (2000), 2171–2183.
  • C. D. Sogge, Lectures on nonlinear wave equations, Int. Press, Cambridge, MA, 1995.
  • E. M. Stein, Harmonic Analysis: Real-variable Methods, Orthogonality, and Oscillatory Integrals, Princeton University Press, Princeton, NJ, 1993.
  • E. M. Stein and G. Weiss, Fractional integrals on $n$-dimensional Euclidean space, J. Math. Mech. 7 (1958), 503–514.
  • J. Sterbenz, Angular regularity and Strichartz estimates for the wave equation, With an appendix by Igor Rodnianski, Int. Math. Res. Not. 2005 (2005), 187–231.
  • 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.
  • H. Takamura, Blow-up for semi-linear wave equations with slowly decaying data in high dimensions, Differential Integral Equations 8 (1995), 647–661.
  • H. Takamura, Nonexistence of global solutions to semilinear wave equations, Thesis, Hokkaido University, 1995.
  • M. C. Vilela, Regularity of solutions to the free Schrödinger equation with radial initial data, Illinois J. Math. 45 (2001), 361–370.
  • M. C. Vilela, Inhomogeneous Strichartz estimates for the Schrödinger equation, Trans. Amer. Math. Soc. 359 (2007), 2123–2136.