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]

Notes

1. Here a “tube” means a long, thin cylindrical region in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle \mathbb R^n} , typically of fixed radius and arbitrary length, as in the Kakeya problem.