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

TL;DR

This work introduces and analyzes a restricted $2$-plane transform $\mathcal{T}$ tied to a codimension-$2$ quadratic surface $\Sigma(Q_1,Q_2)$ and connects its $L^p\to L^q$ behavior to Fourier restriction for such surfaces. A central contribution is the algebraic characterization of well-curvedness via the determinant polynomial $\Delta(s,t)=\det(s\nabla^2 Q_1 + t\nabla^2 Q_2)$, whose root structure governs semistability under a Geometric Invariant Theory action and controls the sharpness of bounds. The authors establish sharp strong-type and restricted weak-type estimates for $\mathcal{T}$ in the well-curved case, using Christ’s Method of Refinements to reduce to sublevel-set estimates for $\det((s-s_0)\nabla^2 Q_1 + (t-t_0)\nabla^2 Q_2)$; they also prove sublevel bounds via a linear-programming argument. In the flat case, the paper provides precise obstructions and counterexamples showing failure of the endpoint range, including a complete treatment when $\det(s\nabla^2 Q_1 + t\nabla^2 Q_2)$ vanishes identically. The results extend to general well-curved codimension-2 surfaces and illuminate the deep connection between affine curvature, algebraic invariants, and Fourier-analytic restriction phenomena.

Abstract

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 $(ξ, Q_1(ξ), Q_2(ξ))$, 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 \[ \mathcal{T}f(x,ξ) := \iint_{|s|,|t| \leq 1} f(x - s \nabla Q_1(ξ) - t \nabla Q_2(ξ), s, t)\,ds\,dt, \] where $x,ξ\in \mathbb{R}^d$. 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.

Figures (3)

• Figure 1: The special form of the matrices $A,B$ in the appropriate basis.
• Figure 2: The decomposition of $A$ into rectangular blocks of dimensions as indicated. The decomposition of $B$ has the exact same shape. We remark that it might be the case that $d - k - \ell = 0$, in which case the blocks with the corresponding dimension are omitted (e.g. $A$ would contain only blocks $A_1, A_4,A_6$, which would be adjacent to each other).
• Figure 3: The shaded area corresponds to the range of boundedness afforded by Theorem \ref{['thm:flat_surfaces']}, that is, when the surface $\Sigma(Q_1,Q_2)$ is flat but $\det(s\nabla^2 Q_1 + t \nabla^2 Q_2)$ does not vanish identically. The critical lines given by \ref{['eq:mixed_necessary_conditions']} and \ref{['eq:flat_surfaces_necessary_condition_1']} are indicated: as one can see, the range of Theorem \ref{['thm:flat_surfaces']} is sharp when $2/q = 1/p$. The endpoint $(\tfrac{4}{d+4}, \tfrac{2}{d+4})$ for the well-curved case is also indicated, and one can see that for these surfaces all $L^p \to L^q$ estimates for $(1/p,1/q)$ close to this endpoint are false.

Theorems & Definitions (44)

• Remark 1
• Remark 2
• Remark 3
• Definition 1
• Theorem 1: well-curved surfaces
• Theorem \ref{main_theorem}${}^\prime$: General well-curved surfaces
• Corollary 1: mixed-norm range
• Theorem 2
• Example 1
• Theorem 3: flat surfaces