  We establish lower bounds for the operator norms of the Fourier
restriction/extension operators associated to monomial curves with affine
arclength measure. Furthermore, we prove that the set of all extremizing
sequences of such an operator is precompact modulo the operator's symmetry
group if and only if the operator norm is strictly larger than this threshold.
For the proof, we introduce a number of new ingredients, some of which may be
applicable to analogous questions on more general manifolds.
