Mizohata-Takeuchi estimates in the plane

Suppose $S$ is a smooth compact hypersurface in ${\mathbb{R}}^n$ and $\sigma$ is an appropriate measure on $S$. If $Ef=\widehat{fd\sigma }$ is the extension operator associated with $(S,\sigma )$, then the Mizohata-Takeuchi conjecture asserts that $\int \mathrm{\mid }Ef(x){\mathrm{\mid }}^{2}w(x)dx\le C({\sup }_{T}w(T))\mathrm{\parallel }f{\mathrm{\parallel }}_{L^2(\sigma )}^2$ for all functions $f\in L^2(\sigma )$ and weights $w:{\mathbb{R}}^n\to [0,\mathrm{\infty })$, where the $\sup$ is taken over all tubes $T$ in ${\mathbb{R}}^n$ 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 ${\mathbb{R}}^2$ if we only take the decay properties of $\widehat{\sigma }$ 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 $(t,e^{-1\mathrm{/}{t}^m})$, $0\le t\le c_m$, $m\in \mathbb{N}$.