Some sharp inequalities of Mizohata–Takeuchi-type

Let [Formula presented] be a strictly convex, compact patch of a [Formula presented] hypersurface in [Formula presented], with non-vanishing Gaussian curvature and surface measure dσ induced by the Lebesgue measure in [Formula presented]. The Mizohata–Takeuchi conjecture states that [Formula presented] for all [Formula presented] and all weights [Formula presented], 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 [Formula presented], there exists a positive constant [Formula presented], which depends only on [Formula presented] and ", such that for all [Formula presented] and all weights [Formula presented], we have [Formula presented]; where T ranges over the family of tubes in [Formula presented] of dimensions [Formula presented]. From this we deduce the Mizohata–Takeuchi conjecture with an [Formula presented]-loss; i.e., that [Formula presented] for any ball [Formula presented] of radius R and any [Formula presented]. The power [Formula presented] here cannot be replaced by anything smaller unless properties of [Formula presented] 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.