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

Dendrinos, Spyridon; Mustaţă, Andrei; Vitturi, Marco

doi:10.2140/apde.2025.18.475

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

Spyridon Dendrinos, Andrei Mustaţă and Marco Vitturi

Vol. 18 (2025), No. 2, 475–526

DOI: 10.2140/apde.2025.18.475

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 $ℝ^{d}$ , the particular operator in question is the $2$ -plane transform restricted to directions normal to the surface, that is,

𝒯 f (x, ξ) : = \iint_{| s |, | t | \leq 1} f (x - s \nabla Q_{1} (ξ) - t \nabla Q_{2} (ξ), s, t) d s d t,

where $x, ξ \in ℝ^{d}$ . 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.

Keywords

harmonic analysis, geometric invariant theory, restricted 2-plane transform, $k$-plane transform, Fourier restriction, Kakeya, Mizohata–Takeuchi

Mathematical Subject Classification

Primary: 44A12

Secondary: 14L24, 42B10

Milestones

Received: 16 November 2022

Revised: 26 August 2023

Accepted: 20 October 2023

Published: 5 February 2025

Authors

Spyridon Dendrinos
School of Mathematical Sciences University College Cork Cork Ireland

Andrei Mustaţă
School of Mathematical Sciences University College Cork Cork Ireland

Marco Vitturi
Department of Mathematics Munster Technological University Cork Ireland

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Vol. 18, No. 2, 2025