Intermittent symmetry breaking and stability of the sharp Agmon--Hörmander estimate on the sphere

Abstract

We compute the optimal constant and characterise the maximisers at all spatial scales for the Agmon--Hörmander $L^2$-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.

14 pages, 2 figures. Referee suggestions incorporated. To appear in Proceedings of the American Mathematical Society