Fourier extension for extremal quadratic submanifolds

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 $\gamma (x_1,\dots ,x_n):=(x_1,\dots ,x_n,(x_i{x}_j{)}_{1\le i\le j\le n})$. 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 $d\le \frac{n(n+3)}{2}$. 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.

This paper has been withdrawn by the author as this problem has been previously solved by D. M. Oberlin (Canad. Math. Bull. Vol. 48 (2), 2005 pp. 260--266) using very similar methods