Illinois Journal of Mathematics
- Illinois J. Math.
- Volume 52, Number 2 (2008), 365-388.
Weighted HLS inequalities for radial functions and Strichartz estimates for wave and Schrödinger equations
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.
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. https://projecteuclid.org/euclid.ijm/1248355340