A restricted 2-plane transform related to Fourier restriction for surfaces of codimension 2
Year: 2025
Type: article
Abstract: 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, Q 1 (ξ ), Q 2 (ξ )), with Q 1 , Q 2 quadratic forms on ޒ d , the particular operator in question is the 2-plane transform restricted to directions normal to the surface, that is,where x, ξ ∈ ޒ 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 → 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 Q 1 + t∇ 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.
Source: Analysis & PDE
Institutions: University College Cork, Munster Technological University
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Funder: Irish Research Council
OpenAlex ID: W4407166203