A remark on the Strichartz inequality in one dimension

TL;DR

This paper investigates the extremal functions for the Strichartz inequality in one dimension, proving they are Gaussian and characterizing their properties using complex analysis and Fourier decay.

Contribution

It provides a new proof that extremal functions for the Strichartz inequality are Gaussian, using Fourier decay and complex analyticity.

Findings

Extremal solutions decay exponentially in Fourier space.

Extremals are characterized as Gaussian functions.

New proof of Gaussian extremals based on complex analysis.

Abstract

In this paper, we study the extremal problem for the Strichartz inequality for the Schr\"{o}dinger equation on $\mathbb{R}^2$. We show that the solutions to the associated Euler-Lagrange equation are exponentially decaying in the Fourier space and thus can be extended to be complex analytic. Consequently we provide a new proof to the characterization of the extremal functions: the only extremals are Gaussian functions, which was investigated previously by Foschi and Hundertmark-Zharnitsky.