Analyticity of extremisers to the Airy Strichartz inequality

Abstract

We prove that there exists an extremal function to the Airy Strichartz inequality, $e^{-t{\partial }_x^3}:L^2(\mathbb{R})\to L_{t,x}^8({\mathbb{R}}^2)$ 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.

18 pages

References in corpus (3)

• Maximizers for the Strichartz Inequalities for the Wave Equation
• Quasiextremals for a Radon-like transform
• Existence of Extremals for a Fourier Restriction Inequality

Cited by in corpus (8)

• Some sharp restriction inequalities on the sphere
• A sharp trilinear inequality related to Fourier restriction on the circle
• Extremizers for Fourier restriction on hyperboloids
• Sharp Strichartz inequalities for fractional and higher order Schrödinger equations
• Some recent progress on sharp Fourier restriction theory
• Extremals for $\alpha$-Strichartz inequalities
• Sharp Fourier extension on fractional surfaces
• On profile decomposition for Airy type equation