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Mizohata–Takeuchi conjecture

In harmonic analysis, a branch of mathematics, the Mizohata–Takeuchi conjecture proposed a weighted L2L^{2} inequality for the Fourier extension operator associated with a smooth hypersurface in Euclidean space. It asserted that the L2L^{2} norm of the extension of a function ff from the hypersurface to Rn\mathbb {R} ^{n} could be bounded, for any nonnegative weight function, by a constant multiple of the L2L^{2} norm of ff, with the constant depending only on the supremum of the weight over certain tube-shaped regions. The conjecture was disproved in 2025 by Hannah Cairo. The conjecture originally arose in the study of well-posedness for dispersive partial differential equations. In the 1970s and 1980s Jiro Takeuchi was studying the initial value problem associated with a perturbed version of the linear Schrödinger equation. He at one point claimed a well-posed condition in L2(Rn)L^{2}(\mathbb {R} ^{n}) that was both necessary and sufficient for the associated Cauchy problem. Sigeru Mizohata noticed that Takeuchi’s argument was not compelling and showed that Takeuchi’s condition is necessary, but whether it is also sufficient remained open.

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Progress summary

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Hannah Cairo’s 2025 counterexample shows that the conjectured inequality is false in general, ending the original problem.

The Mizohata–Takeuchi conjecture proposed a weighted L2L^{2} bound for Fourier extension operators on smooth hypersurfaces. It grew from Takeuchi’s work on well-posedness for perturbed Schrödinger equations and Mizohata’s subsequent criticism of the claimed sufficiency condition.

Known results

  • Takeuchi (1974, 1980) claimed a necessary-and-sufficient condition for L2(Rn)L^{2}(\mathbb{R}^{n}) well-posedness.
  • Mizohata (1985) identified an error in Takeuchi’s sufficiency argument and established necessity.
  • Later work proved only special cases and weaker estimates, including losses such as Rn−1n+1+εR^{\frac{n-1}{n+1}+\varepsilon} and Sobolev variants.

February 2025 counterexample

Hannah Mira Cairo’s preprint, posted February 10, constructs a logarithmic-loss counterexample: for every nonplanar C2C^{2} hypersurface, suitable ff and nonnegative weights violate the conjectured estimate by a factor of log⁡R\log R. A later independent account reports the result as a disproof and notes its implication that Stein’s conjecture is false as stated.

Current status (as of August 2026): The original conjecture is settled negatively by Cairo’s counterexample; related weakened, restricted, and Sobolev-type inequalities remain active questions.

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