Analyticity of extremizers to the Airy-Strichartz inequality (Q2880384)

The authors consider the Airy equation NEWLINE\[NEWLINE \partial_t u + \partial_x^3u=0,NEWLINE\]NEWLINE which, for the initial data \(u(0)=f\), admits the solution NEWLINE\[NEWLINE e^{-t\partial_x^3}f(x)\equiv (2\pi)^{-1/2}\int_{\mathbb{R}}e^{ixk+itk^3}\hat{f}(k)dk,NEWLINE\]NEWLINE where \(\hat{f}\) is the Fourier transform of \(f\).NEWLINENEWLINEThey study the following symmetrical Airy Strichartz inequality NEWLINE\[NEWLINE ||e^{-t\partial_x^3}f||_{L^8_{t,x}(\mathbb{R}\times \mathbb{R})}\leq C||f||_{L^2(\mathbb{R})}\,. \tag{AS}NEWLINE\]NEWLINENEWLINENEWLINEA function \(f\in L^2\) is called an ``extremizer'' for (AS) if it is not equal to the zero function almost everywhere and NEWLINE\[NEWLINE ||e^{-t\partial_x^3}f||_{L^8_{t,x}(\mathbb{R}\times \mathbb{R})}=\mathcal{A}||f||_{L^2(\mathbb{R})}\,, \tag{E}NEWLINE\]NEWLINE where \(\mathcal{A}\) denotes the optimal constant for (AS).NEWLINENEWLINEThe first result they prove is Theorem 1.2 which states that there exists an extremal function \(f\in L^2\) for the Airy Strichartz inequality (AS). The proof is based on the linear profile decomposition for the Airy evolution operator \(e^{-t\partial_x^3}\) acting on a bounded sequence \(f_n\in L^2\).NEWLINENEWLINEThen, the authors turn to the characterization of the extremizers to (AS) by studying the corresponding Euler-Lagrange equation NEWLINE\[NEWLINE \omega f=\int e^{t\partial_x^3}\left[|e^{-t\partial_x^3}f|^6 e^{-t\partial_x^3} f\right]dt\,, \tag{EL}NEWLINE\]NEWLINE where \(\omega\) is a Lagrange multiplier.NEWLINENEWLINEA strong regularity result for the extremizers is contained in Theorem 1.4 which states that for any extremizer to (AS), there exists \(\mu_0>0\) such that NEWLINE\[NEWLINE k \rightarrowtail e^{\mu_0|k|^3} \hat{f}(k)\in L^2\,. \tag{R}NEWLINE\]NEWLINE In particular \(f\) can be extended to an entire function on the complex plane. The expression (R) gives a decay only in Fourier space, however this decay is even much more rapid than a Gaussian decay.NEWLINENEWLINEThe proof of Theorem 1.4 is based on a bootstrap argument, which relies on a refined bilinear Strichartz inequality for the Airy operator \(e^{-t\partial_x^3}f\), and a weighted Strichartz inequality.