Sharp Strichartz estimate for the 1D periodic Schr\"odinger equation

TL;DR

This paper establishes a sharp logarithmic Strichartz estimate for the 1D periodic Schrödinger equation, demonstrating the precise growth rate of the bound with respect to frequency support size.

Contribution

It proves a sharp logarithmic bound for the Strichartz estimate on the 1D periodic Schrödinger equation, matching Bourgain's lower bound.

Findings

The estimate  e^{it\u2202_x^2}f _{L^6( ext{T}^2)} \, extlessa C (\log N)^{1/6} a  f _{L^2( ext{T})} for functions with Fourier support in [-N,N].

The bound (a \u220a (\log N)^{1/6}) is proven to be sharp.

The result confirms the optimality of the logarithmic growth rate in the estimate.

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}.