  We consider the adjoint restriction inequality associated to the hypersurface
$\{(\tau, \xi) : \tau = \pm|\xi|^2, \;\xi \in \mathbb{R}^d\}$ at the
Stein-Tomas exponent. Extremizers exist in all dimensions and extremizing
sequences are precompact modulo symmetries conditional on a certain inequality,
which we verify in the case $d \in \{1,2\}$.

Classical Analysis and ODEs;Analysis of PDEs