Analyticity of extremisers to the Airy Strichartz inequality

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