A restricted 2-plane transform related to Fourier restriction for surfaces of codimension 2
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Accepted Version
Date
2025
Authors
Dendrinos, Spyridon
Mustaţă, Andrei
Vitturi, Marco
Journal Title
Journal ISSN
Volume Title
Publisher
Mathematical Sciences Publishers
Published Version
Abstract
We draw a connection between the affine invariant surface measures constructed by P. Gressman in [30] 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 Rd, the particular operator in question is the 2-plane transform restricted to directions normal to the
surface, that is
T f(x, ξ) := Z Z
|s|,|t|≤1
f(x − s∇Q1(ξ) − t∇Q2(ξ), s, t) ds dt, where x, ξ ∈ Rd. 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 Lp → Lq 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∇2Q1 + t∇2Q2), 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
Description
Keywords
Harmonic analysis , Geometric invariant theory , Restricted 2-plane transform , k-Plane transform , Fourier restriction , Kakeya , Mizohata–Takeuchi
Citation
Dendrinos, S., Mustaţă, A. and Vitturi, M. (2025) ‘A restricted 2-plane transform related to Fourier restriction for surfaces of codimension 2’, Analysis & PDE, 18(2), pp. 475–526. https://doi.org/10.2140/apde.2025.18.475