Analyticity of extremisers to the Airy Strichartz inequality

TL;DR

This paper proves the existence and analyticity of extremisers for the Airy Strichartz inequality, showing they decay rapidly in Fourier space and can be extended as entire functions.

Contribution

It establishes the existence of extremal functions for the Airy Strichartz inequality and proves their rapid Fourier decay and analyticity, using profile decomposition and refined estimates.

Findings

Existence of extremisers for the Airy Strichartz inequality.

Extremisers decay rapidly in Fourier space.

Extremisers can be extended as entire functions.

Abstract

We prove that there exists an extremal function to the Airy Strichartz inequality, $e^{-t\partial_x^3}: L^2(\mathbb{R})\to L^8_{t,x}(\mathbb{R}^2)$ by using the linear profile decomposition. Furthermore we show that, if $f$ is an extremiser, then $f$ is extremely fast decaying in Fourier space and so $f$ can be extended to be an entire function on the whole complex domain. The rapid decay of the Fourier transform of extremisers is established with a bootstrap argument which relies on a refined bilinear Airy Strichartz estimate and a weighted Strichartz inequality.