Some sharp inequalities of Mizohata–Takeuchi-type

Let Σ be a strictly convex, compact patch of a C² hypersurface in Rⁿ , with non-vanishing Gaussian curvature and surface measure dσ induced by the Lebesgue measure in Rⁿ . The Mizohata–Takeuchi conjecture states that ∫ |{g dσ}|² w ≤ C \|Xw\|∞ ∫ |g|² for all g∈ L²(Σ) and all weights w Rⁿ→ [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 ε>0 , there exists a positive constant Cε , which depends only on Σ and ε , such that for all R ≥ 1 and all weights w Rⁿ→ [0,+∞) , we have ∫B_R|{g dσ}|² w ≤ Cε Rε T (∫T w(n+1)/2)2/(n+1)∫ |g|², where T ranges over the family of tubes in Rⁿ of dimensions R1/2× ⋯ × R1/2× R . From this we deduce the Mizohata–Takeuchi conjecture with an R(n-1)/(n+1) -loss; i.e., that ∫B_R|{g dσ}|² w ≤ Cε Rn-1/n+1+ ε\|Xw\|∞ ∫ |g|² for any ball BR of radius R and any ε>0 . The power (n-1)/(n+1) here cannot be replaced by anything smaller unless properties of {g dσ} 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.