Fourier extension for extremal quadratic submanifolds

TL;DR

This paper determines the full range of Fourier extension estimates for a specific extremal quadratic submanifold, using an inflation-based proof that involves significant overinflation to achieve sharp results.

Contribution

It establishes the complete set of $L^p$--$L^q$ Fourier extension estimates for an extremal quadratic submanifold in high-dimensional space, introducing an inflation technique with overinflation.

Findings

Full range of Fourier extension estimates established

Inflation-type argument with overinflation used successfully

Results are sharp and optimal for the model submanifold

Abstract

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.