Fourier-side Mizohata–Takeuchi-type Conjecture (partial progress toward the local form)

Establish that for every compact C^2 hypersurface Σ ⊂ R^d with surface measure σ, every f ∈ L^2(Σ,σ), and every function h: R^d → C with Fourier support contained in B_R, the inequality ||h * (f dσ)||_{L^2(R^d)}^2 ≲_ε R^ε ||f||_{L^2(Σ,σ)}^2 ||P_ν h||_{L^2(L^1(ν^⊥))}^2 holds uniformly for every unit vector ν ∈ S^{d−1}, where P_ν h(λ)(ω)=h(λν+ω) and ||P_ν h||_{L^2(L^1(ν^⊥))}^2 = ∫_R (∫_{ν^⊥} |h(λν+ω)| dω)^2 dλ.

Background

To illuminate the structure behind the counterexample and to frame potential positive results, the paper proposes a Fourier-side convolution inequality as a partial progress target toward the local Mizohata–Takeuchi conjecture.

This formulation directly involves the convolution h * (f dσ) and the L^2–L^1 mixed norm of plane restrictions P_ν h, aiming for bounds with R^ε losses that would be compatible with localization and X-ray estimates.