Source
Cairo (2025), “A Counterexample to the Mizohata-Takeuchi Conjecture”, arXiv:2502.06137v2, LIT-030, read in full in NOTE-051.
What was actually shown
The conjecture. Mizohata–Takeuchi (Conj. 1.1) says ∫|Ef|²w ≲ ‖f‖²₂ · sup_ℓ ∫_ℓ w for every non-negative weight w, where E is the extension operator of a hypersurface Σ and the sup is over lines. It had been proved for radial weights and with power losses (R^{(n−1)/(n+1)+ε} in general, Carbery–Iliopoulou–Wang).
The counterexample (Thm 1.2 via Prop. 3.1). For each R ≥ 1 there are f ∈ L²(Σ) and h with supp ĥ ⊂ B_{CR} such that ‖h * f dσ‖²₂ ≳ log R · ‖f‖²₂ · ‖P_νh‖²_{L²(L¹)} for every direction ν. With w = |ĥ|² and the X-ray estimate of Theorem 2.1, this violates the conjecture by a factor of order log R. The construction takes N ≍ log R points on Σ, places R⁻¹-balls on the middle-layer subset sums Q of those points, and chooses the points so that no hyperplane meets more than 2^{d−1} of the balls (Lemma 3.5). The ratio of the two sides is then N. NOTE-051 checked this bookkeeping by hand.
Consequences. Stein’s conjecture (Conj. 1.4) implies Mizohata–Takeuchi in one line (p. 2), so it fails too (C4). Carbery, Hänninen and Valdimarsson had shown that Mizohata–Takeuchi plus endpoint multilinear Kakeya gives endpoint multilinear restriction with no R^ε loss; that route is now closed (C5).
Why curvature. It lets the projections of the chosen points lie in separated dyadic scales in every direction at once. A flat piece projects to a point in its normal direction.
What could have come out otherwise. Some hyperplane could have met many of the balls around Q, which would have cancelled the gain. Lemma 3.5 says none does.
What this does not say
- That the theorem holds for every surface not in a hyperplane. The paper states it so, but the proof (Lemma 4.4) needs a point where the second fundamental form is non-zero. A connected surface not in a hyperplane has one. A compact union of flat pieces in different hyperplanes does not, and for it the conjecture holds with constant equal to the number of pieces (NOTE-051’s argument, C6).
- That the loss is a power. This paper gives log R. Cairo and Zhang (arXiv:2512.08064, read as an abstract only) give power-loss counterexamples to the local version on a C^k-dense family of hypersurfaces. Whether the sphere and paraboloid have a power loss was not checked.
- That a single (f, w) violates the inequality by an infinite factor. The counterexample is a family indexed by R.
- That endpoint multilinear restriction is false. The paper says only that it cannot be sharpened directly through Mizohata–Takeuchi.
- That every step is checked. Lemma 4.3’s induction tracks its constants only up to ∼, and the overlap bound (3.8) is argued as a probability (NOTE-051, C3 and Method). The paper had no journal version when read.
Connections
- No other work or THEORY in the record bears on Fourier restriction.