Mizohata-Takeuchi estimates in the plane

TL;DR

This paper explores the Mizohata-Takeuchi conjecture in the plane, focusing on decay properties of the Fourier transform of measures on convex curves, and provides new estimates for certain flat convex curves.

Contribution

It advances understanding of the Mizohata-Takeuchi conjecture in two dimensions by deriving new estimates based on decay properties of the Fourier transform for convex curves.

Findings

New estimates for convex curves including exponentially flat ones

Progress in the Mizohata-Takeuchi conjecture in the plane using decay properties

Application to a class of convex curves with specific Fourier decay

Abstract

Suppose $S$ is a smooth compact hypersurface in $\Bbb R^n$ and $\sigma$ is an appropriate measure on $S$. If $Ef= \hat{fd\sigma}$ is the extension operator associated with $(S,\sigma)$, then the Mizohata-Takeuchi conjecture asserts that $\int |Ef(x)|^2 w(x) dx \leq C (\sup_T w(T)) \| f \|_{L^2(\sigma)}^2$ for all functions $f \in L^2(\sigma)$ and weights $w : \Bbb R^n \to [0,\infty)$, where the $\sup$ is taken over all tubes $T$ in $\Bbb R^n$ of cross-section 1, and $w(T)= \int_T w(x) dx$. This paper investigates how far we can go in proving the Mizohata-Takeuchi conjecture in $\Bbb R^2$ if we only take the decay properties of $\hat{\sigma}$ into consideration. As a consequence of our results, we obtain new estimates for a class of convex curves that include exponentially flat ones such as $(t,e^{-1/t^m})$, $0 \leq t \leq c_m$, $m \in \Bbb N$.