Extremizers for Adjoint Restriction to a Pair of Reflected Paraboloids

Abstract: 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 $d \in {1,2}$.