Discrete paraboloid ($P=\{(v,v\cdot v):v\in {\mathbb{F}}_p^{d-1}\}$)

The discrete paraboloid is the graph of the sum-of-squares quadratic form in a vector space over a finite field. Its surface measure is normalized counting measure, of total mass one. A Fourier extension operator takes a function on this graph to its character sum in the ambient finite vector space. Ambient counting measure and normalized surface measure must be distinguished when defining a restriction constant.

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 $q<4$.

For the three-dimensional discrete paraboloid over an odd prime field, the surface transform away from zero has magnitude at most $p^{-1}$, by completing squares and evaluating Quadratic Gauss sums. The adjoint-composition convolution kernel is the transform of the surface measure. Subtract its unit mass at the origin. Convolution by the remainder has ${\ell }^1\to {\ell }^{\infty }$ norm at most $p^{-1}$ and ${\ell }^2\to {\ell }^2$ norm at most $p$. The Riesz-Thorin interpolation theorem gives ${\ell }^{4/3}\to {\ell }^4$ norm at most one; adding the identity kernel gives at most two. The adjoint-composition identity and duality therefore give the displayed extension constant for normalized surface measure and ambient counting measure. This argument works for either quadratic character of minus one.