A sharpened $L^8_t L^4_x$-Strichartz estimate for even initial data

Abstract

It is well known that $u_0 \in L^2(\mathbb{R}) \setminus \{0\}$ is an extremizer for the $L^8_t L^4_x$-Strichartz estimate if and only if it is a Gaussian. We show that every extremizing sequence of even functions $\{ u_n \} \subset L^2(\mathbb{R}) \setminus \{0\}$ approaches a centered Gaussian in an appropriate sense. This result follows from a certain inequality for coordinate-wise even $L^2$ functions on $\mathbb{R}^4$ conjectured by Gonçalves (2019) which we prove in this paper with the optimal constant.