A Counterexample to the Mizohata-Takeuchi Conjecture

We derive a family of $L^p$ estimates of the X-Ray transform of positive measures in ${\mathbb{R}}^d$, which we use to construct a $\log R$-loss counterexample to the Mizohata-Takeuchi conjecture for every $C^2$ hypersurface in ${\mathbb{R}}^d$ that does not lie in a hyperplane. In particular, multilinear restriction estimates at the endpoint cannot be sharpened directly by the Mizohata-Takeuchi conjecture.