  This note establishes the full range of $L^p$--$L^q$ Fourier extension
estimates for the model $n$-dimensional quadratic submanifold in ${\mathbb
R}^{n(n+3)/2}$ parametrized by $\gamma(x_1,\ldots,x_n) := (x_1,\ldots,x_n, (x_i
x_j)_{1 \leq i \leq j \leq n})$. This class of submanifolds is extremal in the
sense that an $n$-dimensional quadratic submanifold of ${\mathbb R}^d$ can only
satisfy nontrivial Fourier extension inequalities when $d \leq
\frac{n(n+3)}{2}$. The proof is via an inflation-type argument, with the
unexpected twist that a significant amount of "overinflation" is necessary but
in no way limits the sharpness of the argument.
