Stein’s Conjecture (X-ray controlled weighted extension)

Establish that, under the Mizohata–Takeuchi hypotheses for a compact C^2 hypersurface Σ ⊂ R^d with surface measure σ, every f ∈ L^2(Σ,σ), and every nonnegative weight w: R^d → [0,∞), the inequality ∫_{R^d} |E f(x)|^2 w(x) dx ≲ ∫_Σ |f(σ)|^2 sup_{ℓ ∥ N(σ)} Xw(ℓ) dσ(σ) holds, where N(σ) is the unit normal to Σ at σ and Xw is the X-ray transform.

Background

Stein proposed controlling Bochner–Riesz multipliers via Kakeya/Nikodym maximal functions, leading to a conjectural weighted extension inequality dominated by directional X-ray transforms along normals to the surface.

The paper observes that Stein’s conjecture (in this extension-operator formulation) would imply the Mizohata–Takeuchi conjecture. Because the present work provides a counterexample to the latter, Stein’s conjecture as stated is consequently false.