Some sharp inequalities of Mizohata–Takeuchi-type

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

for all $g\in L^2(\Sigma )$ and all weights $w:{\mathbb{R}}^n\to [0,+\infty )$, 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 $R\ge 1$ and all weights $w:{\mathbb{R}}^n\to [0,+\infty )$, we have

where $T$ ranges over the family of tubes in ${\mathbb{R}}^n$ of dimensions $R^{1/2}\times \cdots \times R^{1/2}\times R$. From this we deduce the Mizohata–Takeuchi conjecture with an $R^{(n-1)/(n+1)}$-loss; i.e., that

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.