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 (Duke Math. J. 168:11 (2019), 2075–2126) 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, Q₁(ξ), Q₂(ξ)), with Q₁, Q₂ quadratic forms on (Formula presented), the particular operator in question is the 2-plane transform restricted to directions normal to the surface, that is, (Formula presented), where (Formula presented). 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 → 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∇² Q₁ + t∇² Q₂), 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.