  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.
