Sharp Strichartz estimate for the 1D periodic Schrödinger equation

Abstract

We prove the following estimate \[ \|{e^{it\partial_x^2}f}\|_{L_{(t,x)\in \mathbb{T}^2}^6}\leq C (\log N)^{1/6} \|f\|_{L^2_x(\mathbb{T})}, \] assuming $\mbox{supp} (\hat f)\subset [-N,N]$ for $N>1$. The bound $(\log N{)}^{1/6}$ is sharp in view of the lower bound by Bourgain \cite{Bourgain}.

There is a serious gap in the proof