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