A phase-space approach to weighted Fourier extension inequalities
Jonathan Bennett, Susana Gutierrez, Shohei Nakamura, Itamar Oliveira
TL;DR
This work develops a phase-space framework for weighted Fourier extension inequalities on general codimension-1 submanifolds by introducing a surface-carried Wigner transform W_S and its X-ray pullback X_S. It derives Sobolev variants of the Stein and Mizohata–Takeuchi inequalities, with curvature-robust constants governed by a curvature quotient Q(S) and, in the plane, by a refined quantity Λ(S). The approach links harmonic analysis with optics and tomographic methods, providing a robust geometric toolkit including surface-carried bilinear fractional integrals and maximal operators, along with explicit Jacobian formulas. The results illuminate how curvature interacts with phase-space concentration while remaining largely curvature-insensitive, and they yield a plane-specific improvement and a Flandrin-type result with ε-loss, illustrating the framework’s reach toward time-frequency questions and tomographic reconstructions.
Abstract
The purpose of this paper is to expose and investigate natural phase-space formulations of two longstanding problems in the restriction theory of the Fourier transform. These problems, often referred to as the Stein and Mizohata--Takeuchi conjectures, assert that Fourier extension operators associated with rather general (codimension 1) submanifolds of Euclidean space, may be effectively controlled by the classical X-ray transform via weighted $L^2$ inequalities. Our phase-space formulations, which have their origins in recent work of Dendrinos, Mustata and Vitturi, expose close connections with a conjecture of Flandrin from time-frequency analysis, and rest on the identification of an explicit ``geometric" Wigner transform associated with an arbitrary (smooth strictly convex) submanifold $S$ of $\mathbb{R}^n$. Our main results are certain natural ``Sobolev variants" of the Stein and Mizohata--Takeuchi conjectures, and involve estimating the Sobolev norms of such Wigner transforms by geometric forms of classical bilinear fractional integrals. Our broad geometric framework allows us to explore the role of the curvature of the submanifold in these problems, and in particular we obtain bounds that are independent of any lower bound on the curvature; a feature that is uncommon in the wider restriction theory of the Fourier transform. Finally, we provide a further illustration of the effectiveness of our analysis by establishing a form of Flandrin's conjecture in the plane with an $\varepsilon$-loss. While our perspective comes primarily from Euclidean harmonic analysis, the procedure used for constructing phase-space representations of extension operators is well-known in optics.