Mizohata–Takeuchi conjecture

In harmonic analysis, a branch of mathematics, the Mizohata–Takeuchi conjecture proposed a weighted ${\displaystyle L^{2}}$ inequality for the Fourier extension operator associated with a smooth hypersurface in Euclidean space. It asserted that the ${\displaystyle L^{2}}$ norm of the extension of a function ${\displaystyle f}$ from the hypersurface to ${\displaystyle \mathbb {R} ^{n}}$ could be bounded, for any nonnegative weight function, by a constant multiple of the ${\displaystyle L^{2}}$ norm of ${\displaystyle f}$, with the constant depending only on the supremum of the weight over certain tube-shaped regions.^{[a][citation needed]} The conjecture was disproven in 2025 by Hannah Cairo.^[1][2]

The conjecture^[3] 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^[4] a well-posed condition in ${\displaystyle L^{2}(\mathbb {R} ^{n})}$ that was both necessary and sufficient for the associated Cauchy problem. Sigeru Mizohata noticed^[5] that Takeuchi’s argument was not compelling and showed that Takeuchi’s condition is necessary, but whether it is also sufficient remained open.

Notes

1. Here a “tube” means a long, thin cylindrical region in ${\displaystyle \mathbb {R} ^{n}}$, typically of fixed radius and arbitrary length, as in the Kakeya problem.