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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.

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 odd $p$, write $e_p(t)=e^{2\pi it/p}$ and use the discrete paraboloidHere surface integration means $p^{-2}{\sum }_P$, whereas ambient norms below use unnormalized counting measure on ${\mathbb{F}}_p^3$. With the negative-phase transform convention,When $x_3=0$, additive-orthogonality of characters makes this one at $x=0$ and zero at all other points of that plane. When $x_3\ne 0$, completing the two squares givesThe Quadratic Gauss sum identity is $G(t{)}^2=\chi (-1)p$, where $\chi$ is the Legendre symbol. For completeness, counting the preimages of a square gives $G(t)=\chi (t)G(1)$ for nonzero $t$. Also $|G(1){|}^2={\sum }_{u,v}{e}_p(u^2-v^2)=p$, by the invertible change $(u,v)\mapsto (u-v,u+v)$ and orthogonality of characters. Since $\overline{G(1)}=G(-1)=\chi (-1)G(1)$, the asserted square identity follows. Thus for $p\equiv 3(\mathrm{mod}4)$,Using positive phase instead changes the sign inside the character, not the magnitude or any norm estimate.

Define $Eg(x)=p^{-2}{\sum }_{\xi \in P}g(\xi )e_p(x\cdot \xi )$, and define $R^{\ast }(2\to q)$ as the smallest constant forThe adjoint relative to these measures is $E^{\ast }h(\xi )={\sum }_{x}h(x)e_p(-x\cdot \xi )$. Consequently $E{E}^{\ast }h=h\ast \hat{\sigma }(-\cdot )$, where convolution is the unnormalized ambient sum. Put $K=\hat{\sigma }(-\cdot )-1_{\{0\}}$. Its supremum is at most $p^{-1}$, so convolution by $K$ has ${\ell }^1\to {\ell }^{\infty }$ norm at most $p^{-1}$.

Under the unnormalized ambient Fourier transform on a finite group, the convolution multiplier of $\hat{\sigma }(-\cdot )$ is $p^3\sigma =p{1}_P$, up to the harmless reflection convention. Thus the multiplier of $K$ is $p{1}_P-1$, with maximum absolute value at most $p$. The Plancherel theorem gives ${\ell }^2\to {\ell }^2$ norm at most $p$. The Riesz-Thorin interpolation theorem between these two bounds gives ${\ell }^{4/3}\to {\ell }^4$ norm at most one for convolution by $K$. Convolution by $1_{\{0\}}$ is the identity, whose ${\ell }^{4/3}\to {\ell }^4$ norm is at most one for counting measure. Hence $\Vert E{E}^{\ast }h{\Vert }_4\le 2\Vert h{\Vert }_{4/3}$.

By Lp duality and Hölder's inequality, $\Vert E^{\ast }h{\Vert }_{L^2(\sigma )}^2=⟨E{E}^{\ast }h,h⟩\le 2\Vert h{\Vert }_{4/3}^2$. Taking adjoints proves the finite-field paraboloid fourth-moment extension estimateIn fact this argument works for both residue classes of odd primes.

If $p\equiv 1(\mathrm{mod}4)$, choose $i\in {\mathbb{F}}_p$ with $i^2=-1$. The complete line $\ell =\{(t,it,0):t\in {\mathbb{F}}_p\}$ lies in $P$. For $g=1_{\ell }$ the normalized surface $L^2$ norm is $p^{-1/2}$, and orthogonality of characters gives $Eg(x)=p^{-1}{1}_{\{x_1+i{x}_2=0\}}$. That annihilator plane has $p^2$ points, so the isotropic-line obstruction to finite-field restriction givesThis tends to infinity along these primes for every $q<4$, as required.