We investigate a class of sharp Fourier extension inequalities on the planar curves s=|y|p, p>1. We identify the mechanism responsible for the possible loss of compactness of nonnegative extremizing sequences, and prove that extremizers exist if 1<p<p0 for some p0>4. In particular, this resolves the dichotomy of Jiang, Pausader, and Shao concerning the existence of extremizers for the Strichartz inequality for the fourth-order Schrödinger equation in one spatial dimension. One of our tools is a geometric comparison principle for n-fold convolutions of certain singular measures in Rd, developed in the companion paper of Oliveira e Silva and Quilodrán (Math. Proc. Cambridge Philos. Soc., (2019)). We further show that any extremizer exhibits fast L2-decay in physical space, and so its Fourier transform can be extended to an entire function on the whole complex plane. Finally, we investigate the extent to which our methods apply to the case of the planar curves s=y∣∣y∣∣p−1, p>1.
- Convolution of singular measures
- Fractional schrödinger equation
- Sharp fourier restriction theory
- Strichartz inequalities
ASJC Scopus subject areas
- Numerical Analysis
- Applied Mathematics