Failure of weak-type endpoint restriction estimates for quadratic manifolds

A paper of Beckner, Carbery, Semmes, and Soria proved that the Fourier extension operator associated to the sphere cannot be weak-type bounded at the restriction endpoint $q = 2d/(d-1)$. We generalize their approach to prove that the extension operator associated with any n-dimensional quadratic manifold in Rᵈ cannot be weak-type bounded at $q = 2d/n$. The key step in generalizing the proof of Beckner, Carbery, Semmes, and Soria will be replacing Kakeya sets with what we will call N-Kakeya sets, where N denotes a closed subset of the Grassmannian Gr(d-n,d). We define N-Kakeya sets to be subsets of Rᵈ containing a translate of every $d-n$-plane segment in N. We will prove that if N is closed and n-dimensional, then there exists compact, measure zero N-Kakeya sets, generalizing the same result for standard Kakeya sets.