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

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.

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