## Abstract / Description of output

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

∫|gdσˆ|2w≤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

∫BR|gdσˆ|2w≤CϵRϵsupT(∫Twn+12)2n+1∫|g|2,

where T ranges over the family of all tubes in Rn of dimensions R1/2×⋯×R1/2×R. From this we deduce the Mizohata--Takeuchi conjecture with an Rn−1n+1-loss; i.e., that

∫BR|gdσˆ|2w≤CϵRn−1n+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 gdσˆ 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.

∫|gdσˆ|2w≤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

∫BR|gdσˆ|2w≤CϵRϵsupT(∫Twn+12)2n+1∫|g|2,

where T ranges over the family of all tubes in Rn of dimensions R1/2×⋯×R1/2×R. From this we deduce the Mizohata--Takeuchi conjecture with an Rn−1n+1-loss; i.e., that

∫BR|gdσˆ|2w≤CϵRn−1n+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 gdσˆ 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.

Original language | English |
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Journal | Revista Matemática Iberoamericana |

Publication status | Accepted/In press - 30 Nov 2023 |