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