A restricted $2$-plane transform related to Fourier Restriction for surfaces of codimension $2$

We draw a connection between the affine invariant surface measures constructed by P. Gressman and the boundedness of a certain geometric averaging operator associated to surfaces of codimension $2$ and related to the Fourier Restriction Problem for such surfaces. For a surface given by $(\xi, Q_1(\xi), Q_2(\xi))$, with $Q_1,Q_2$ quadratic forms on $\mathbb{R}^d$, the particular operator in question is the $2$-plane transform restricted to directions normal to the surface, that is $$\mathcal{T}f(x,\xi) := \iint_{|s|,|t| \leq 1} f(x - s \nabla Q_1(\xi) - t \nabla Q_2(\xi), s, t)\,ds\,dt,$$ where $x,\xi \in \mathbb{R}^d$. We show that when the surface is well-curved in the sense of Gressman (that is, the associated affine invariant surface measure does not vanish) the operator satisfies sharp $L^p \to L^q$ inequalities for $p,q$ up to the critical point. We also show that the well-curvedness assumption is necessary to obtain the full range of estimates. The proof relies on two main ingredients: a characterisation of well-curvedness in terms of properties of the polynomial $\det(s \nabla^2 Q_1 + t \nabla^2 Q_2)$, obtained with Geometric Invariant Theory techniques, and Christ's Method of Refinements. With the latter, matters are reduced to a sublevel set estimate, which is proven by a linear programming argument.