Mizohata-Takeuchi estimates in the plane

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 $(t,e^{-1/t^m})$, $0 \leq t \leq c_m$, $m \in \Bbb N$.