Failure of weak-type endpoint restriction estimates for quadratic manifolds

TL;DR

This paper proves that the Fourier extension operator for quadratic manifolds cannot be weak-type bounded at the restriction endpoint, extending known results from paraboloids and linking to Kakeya maximal function bounds.

Contribution

It generalizes the failure of weak-type endpoint restriction estimates from paraboloids to arbitrary quadratic manifolds, introducing a novel geometric construction.

Findings

Constructed a set covering translates of all normal planes to the manifold.

Showed this set can be covered by relatively few small balls.

Established the link between this geometric property and restriction estimate failure.

Abstract

It is well-known that the Fourier extension operator for the paraboloid in $\mathbb{R}^d$ cannot be weak-type bounded at the restriction endpoint $q = 2d/(d-1)$, since such an estimate would imply bounds for the Kakeya maximal function which contradict the existence of Besicovitch sets. We generalize this approach to prove that the Fourier extension operator for an $n$-dimensional quadratic manifold $\mathcal{M}$ cannot be weak-type bounded at the restriction endpoint. The key step in this proof is constructing a set $K \subset \mathbb{R}^d$ containing a translate of every plane normal to $\mathcal{M}$ which can be covered by $\lesssim \delta^{-d}\left(\frac{\log \log (1/\delta)}{\log (1/\delta)}\right)^{n/(d-n)}$ many $\delta$-balls. Such a set rules out endpoint bounds for the associated Kakeya maximal function and hence weak-type endpoint estimates for the restriction operator.