Status of the Local Mizohata–Takeuchi Conjecture

Determine whether the Local Mizohata–Takeuchi 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 compact C^2 hypersurfaces Σ ⊂ R^d, or construct a counterexample exhibiting an R^{(n−1)/(n+1)}-loss in the spirit of Guth’s barrier argument.

Background

After disproving the global form, the authors raise the explicit question of whether the proposed local Mizohata–Takeuchi inequality should be expected to hold at the R^ε loss level.

They also point to Guth’s barrier indicating that certain decoupling axioms cannot yield better-than-R^{(n−1)/(n+1)} losses, suggesting the possibility of a refined counterexample at that scale.