Band-Limited Maximizers for a Fourier Extension Inequality on the Circle

Abstract

Among the class of functions with Fourier modes up to degree 30, constant functions are the unique real-valued maximizers for the endpoint Tomas–Stein inequality on the circle.

Keywords:

• Sharp Fourier restriction theory
• Bessel functions
• circle

Acknowledgments

We thank Emanuel Carneiro, Damiano Foschi and Felipe Gonçalves for stimulating discussions. The software Mathematica was used to perform the numerical tasks described in §3 and §4. The authors acknowledge support by the Hausdorff Center for Mathematics and the Deutsche Forschungsgemeinschaft through the Collaborative Research Center 1060.

Notes

1 We record a typo on the first formula in [Oliveira e Silva and Thiele 17, Corollary 2.7], which should read as follows:

$\left|J_0^{\pm }(z)-{\left(\frac{2}{\pi z}\right)}^{\frac{1}{2}}\hspace{0.17em}\text{cos}\hspace{0.17em}({\omega }_0)-\frac{1}{8z}{\left(\frac{2}{\pi z}\right)}^{\frac{1}{2}}\hspace{0.17em}\text{sin}\hspace{0.17em}({\omega }_0)\right|\le \frac{9}{128|z{|}^2}{\left(\frac{2}{\pi \left|z\right|}\right)}^{\frac{1}{2}}\cosh (|\mathfrak{I}(z)|){\left(\frac{\left|z\right|}{|\mathfrak{R}(z)|}\right)}^{\frac{5}{2}}.$