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 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.