Isotropic-line obstruction to finite-field restriction ($R^{\ast }(2\to q)\ge p^{2/q-1/2}$)

If $i^2=-1$ in the prime field, the line $\ell =\{(t,it,0):t\in {\mathbb{F}}_p\}$ lies in the three-dimensional discrete paraboloid. Its indicator has normalized surface $L^2$ norm $p^{-1/2}$. Its extension is $p^{-1}$ on the two-dimensional annihilator plane and zero elsewhere. Thus its ambient ${\ell }^q$ norm is $p^{2/q-1}$, proving the displayed lower bound, which diverges for $q<4$.