Some sharp inequalities of Mizohata–Takeuchi-type

Let be a strictly convex, compact patch of a C^2 hypersurface in R^n, with non-vanishing Gaussian curvature and surface measure d induced by the Lebesgue measure in R^n. The Mizohata–Takeuchi conjecture states that |g d|^2 w C \|Xw\|_ |g|^2 for all g L^2 () and all weights w R^n [0, +), 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 >0, there exists a positive constant C_, which depends only on and, such that for all R 1 and all weights w R^n [0, +), we have ₁ₑ|g d|^2 w C_ R^ ₓ (ₓ w^ (n+1) /2) ^2/ (n+1) |g|^2, where T ranges over the family of tubes in R^n of dimensions R^1/2 R^1/2 R. From this we deduce the Mizohata–Takeuchi conjecture with an R^ (n-1) / (n+1) -loss; i. e. , that ₁ₑ|g d|^2 w C_ R^n-1{n+1+ }\|Xw\|_ |g|^2 for any ball Bₑ of radius R and any >0. The power (n-1) / (n+1) here cannot be replaced by anything smaller unless properties of g d 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.