Abstract
Let $\Sigma$ be a strictly convex, compact patch of a $C^2$ hypersurface in
$\mathbb{R}^n$, with non-vanishing Gaussian curvature and surface measure
$d\sigma$ induced by the Lebesgue measure in $\mathbb{R}^n$. The
Mizohata--Takeuchi conjecture states that
\begin{equation*}
\int |\widehat{gd\sigma}|^2w \leq C \|Xw\|_\infty \int |g|^2
\end{equation*}
for all $g\in L^2(\Sigma)$ and all weights $w:\mathbb{R}^n\rightarrow
[0,+\infty)$, where $X$ denotes the $X$-ray transform. As partial progress
towards the conjecture, we show, as a straightforward consequence of
recently-established decoupling inequalities, that for every $\epsilon>0$,
there exists a positive constant $C_\epsilon$, which depends only on $\Sigma$
and $\epsilon$, such that for all $R \geq 1$ and all weights
$w:\mathbb{R}^n\rightarrow [0,+\infty)$ we have \begin{equation*}
\int_{B_R} |\widehat{gd\sigma}|^2w \leq C_\epsilon R^\epsilon \sup_T
\left(\int _T w^{\frac{n+1}{2}}\right)^{\frac{2}{n+1}}\int |g|^2,
\end{equation*}
where $T$ ranges over the family of all tubes in $\mathbb{R}^n$ of dimensions
$R^{1/2} \times \dots \times R^{1/2} \times R$. From this we deduce the
Mizohata--Takeuchi conjecture with an $R^{\frac{n-1}{n+1}}$-loss; i.e., that
\begin{equation*}
\int_{B_R} |\widehat{gd\sigma}|^2w \leq C_\epsilon R^{\frac{n-1}{n+1}+
\epsilon}\|Xw\|_\infty\int |g|^2
\end{equation*}
for any ball $B_R$ of radius $R$ and any $\epsilon>0$. The power
$(n-1)/(n+1)$ here cannot be replaced by anything smaller unless properties of
$\widehat{gd\sigma}$ beyond 'decoupling axioms' are exploited. We also provide
estimates which improve this inequality under various conditions on the weight,
and discuss some new cases where the conjecture holds.