A Counterexample to the Mizohata-Takeuchi Conjecture

TL;DR

This paper constructs a counterexample to the Mizohata-Takeuchi conjecture by deriving specific $L^p$ estimates for the X-Ray transform, showing limitations of existing multilinear restriction estimates at the endpoint.

Contribution

It introduces a novel family of $L^p$ estimates for the X-Ray transform and uses them to disprove a longstanding conjecture for certain hypersurfaces in $ ^d$.

Findings

Counterexample with $ ext{log } R$-loss to the Mizohata-Takeuchi conjecture

Multilinear restriction estimates at the endpoint cannot be improved by the conjecture

Derived new $L^p$ estimates for the X-Ray transform of positive measures

Abstract

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.