= Finite-field paraboloid fourth-moment extension estimate
{title2=$R^*(2\to4)\leq\sqrt2$}
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.
Back to article page
