Intermittent symmetry breaking and stability of the sharp Agmon--H\"ormander estimate on the sphere

TL;DR

This paper determines the optimal constants and maximisers for the Agmon--H"ormander Fourier restriction estimate on the sphere, revealing an intermittent symmetry-breaking behavior across scales and zeros of Bessel functions.

Contribution

It characterizes the maximisers at all scales, identifies their symmetry properties, and establishes a sharpened stability estimate with an unusual intermittent pattern.

Findings

Maximisers switch between symmetric and non-symmetric states.

Optimal constants are computed explicitly at all scales.

Stability estimates exhibit intermittent behavior linked to Bessel function zeros.

Abstract

We compute the optimal constant and characterise the maximisers at all spatial scales for the Agmon--H\"ormander $L^2$-Fourier adjoint restriction estimate on the sphere. The maximisers switch back and forth from being constants to being non-symmetric at the zeros of two Bessel functions. We also study the stability of this estimate and establish a sharpened version in the spirit of Bianchi--Egnell. The corresponding stability constant and maximisers again exhibit a curious intermittent behaviour.