We investigate a class of sharp Fourier extension inequalities on the planar curves
,
.
We identify the mechanism responsible for the possible loss of compactness
of nonnegative extremizing sequences, and prove that extremizers exist if
for some
. In
particular, this resolves the dichotomy of Jiang, Pausader, and Shao concerning the
existence of extremizers for the Strichartz inequality for the fourth-order Schrödinger
equation in one spatial dimension. One of our tools is a geometric comparison principle for
-fold convolutions of certain
singular measures in
,
developed in the companion paper of Oliveira e Silva and Quilodrán
(Math. Proc.Cambridge Philos. Soc., (2019)). We further show that any extremizer exhibits fast
-decay
in physical space, and so its Fourier transform can be extended to an
entire function on the whole complex plane. Finally, we investigate
the extent to which our methods apply to the case of the planar curves
,
.
