Local Mizohata–Takeuchi Conjecture (reformulation)

Establish that for every compact C^2 hypersurface Σ ⊂ R^d with surface measure σ, every f ∈ L^2(Σ,σ), every nonnegative weight w: R^d → [0,∞), and every radius R ≥ 1, the local inequality ∫_{B_R} |E f(x)|^2 w(x) dx ≲_ε R^ε ||f||_{L^2(Σ,σ)}^2 sup_{ℓ ⊂ R^d line} ∫_ℓ w holds for all ε>0.

Background

Motivated by the logarithmic-loss counterexample to the global Mizohata–Takeuchi inequality, the paper proposes a local reformulation with an R^ε loss. This local version aims to capture a more realistic bound compatible with localization and tube-concentration heuristics.

The authors note uncertainty about whether this local inequality should hold, highlighting possible barriers suggested by decoupling-based arguments.