Mizohata–Takeuchi Conjecture (global weighted extension inequality)

Establish that for every compact C^2 hypersurface Σ ⊂ R^d with surface measure σ, every function f ∈ L^2(Σ,σ), and every nonnegative weight w: R^d → [0,∞), the inequality ∫_{R^d} |E f(x)|^2 w(x) dx ≲ ||f||_{L^2(Σ,σ)}^2 ||Xw||_{L^∞} holds, where the extension operator is E f(x) = ∫_Σ e^{-2π i ⟨ξ, x⟩} f(ξ) dσ(ξ) and Xw denotes the X-ray transform of w on lines ℓ ⊂ R^d given by Xw(ℓ) = ∫_ℓ w.

Background

The Mizohata–Takeuchi conjecture originated in the paper of well-posedness for dispersive PDE and later became a central theme in Fourier restriction theory. It proposes a sharp weighted L^2 estimate for the extension operator controlled by the X-ray transform of the weight.

This paper constructs a logarithmic-loss counterexample for every nonplanar C^2 hypersurface, showing that the global inequality as stated cannot hold in full generality. Nevertheless, the conjecture is stated explicitly in the introduction for context and for its historical significance.