Extremizers for Adjoint Restriction to Pairs of Translated Paraboloids

Abstract

Consider the adjoint restriction inequality associated with the hypersurface $\{ (τ, ξ) \in \mathbb{R}^{d+1} : τ= |ξ|^2 \} \cup \{(τ, ξ) \in \mathbb{R}^{d+1} : τ- τ_0 = |ξ- ξ_0|^2\}$ for any $(τ_0, ξ_0) \neq 0$. We prove that extremizers do not exist for this inequality and fully characterize extremizing sequences in terms of extremizers for the adjoint restriction inequality for the paraboloid.