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arXiv · 1101.0012

Analyticity of extremisers to the Airy Strichartz inequality

Dirk Hundertmark·Shuanglin Shao
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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.

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Dirk Hundertmark, Shuanglin Shao. 2010-12-29. Analyticity of extremisers to the Airy Strichartz inequality. https://doi.org/10.1112/blms%2Fbdr098

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