Finite-field paraboloid fourth-moment extension estimate ($R^{\ast }(2\to 4)\le \sqrt{2}$)

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.