Analyticity of extremizers to the Airy-Strichartz inequality

Abstract

We prove that there exists an extremal function to the Airy–Strichartz inequality ‖ e − t ∂ x 3 f ‖ L t , x 8 ( ℝ × ℝ ) ⩽ C ‖ f ‖ L 2 ( ℝ ) , using the linear profile decomposition. Furthermore we show that, if f is an extremizer, 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 extremizers is established with a bootstrap argument which relies on a refined bilinear Airy–Strichartz estimate and a weighted Strichartz inequality.

Locations

• Bulletin of the London Mathematical Society - View
• arXiv (Cornell University) - View - PDF
• DataCite API - View