Some sharp inequalities of Mizohata–Takeuchi-type

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 \int |\widehat{g d\sigma}|^{2} w \leq C \|Xw\|_{\infty} \int |g|^{2} for all g\in L^{2}(\Sigma) and all weights w \colon \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 \varepsilon>0 , there exists a positive constant C_{\varepsilon} , which depends only on \Sigma and \varepsilon , such that for all R \geq 1 and all weights w \colon \mathbb{R}^{n}\rightarrow [0,+\infty) , we have \int_{B_R}|\widehat{g d\sigma}|^{2} w \leq C_{\varepsilon} R^{\varepsilon} \sup_{T} \Big(\int_{T} w^{(n+1)/2}\Big)^{2/(n+1)}\int |g|^{2}, 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 \int_{B_R}|\widehat{g d\sigma}|^{2} w \leq C_{\varepsilon} R^{\frac{n-1}{n+1}+ \varepsilon}\|Xw\|_{\infty} \int |g|^{2} for any ball B_{R} of radius R and any \varepsilon>0 . The power (n-1)/(n+1) here cannot be replaced by anything smaller unless properties of \widehat{g d\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.