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| Main Authors: | , |
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| Format: | Preprint |
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2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2512.08064 |
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| _version_ | 1866908699613724672 |
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| author | Cairo, Hannah Zhang, Ruixiang |
| author_facet | Cairo, Hannah Zhang, Ruixiang |
| contents | We find a family of compact $C^k$ hypersurfaces where the local Mizohata-Takeuchi Conjecture fails with a power loss of $R^α$ for any $α<\frac{n-1}{n-1+k}$. Moreover, this family is dense in the $C^k$ topology, and so the local Mizohata-Takeuchi conjecture fails for many convex hypersurfaces. In particular, the local Mizohata-Takeuchi Conjecture fails with a power loss of $R^α$ for any $α<\frac{n-1}{n+1}$ for many $C^2$ convex hypersurfaces. This power matches the best known upper bound in a paper by Tony Carbery, Marina Iliopoulou and Hong Wang up to the endpoint. For the proof, our weight is positive definite as in the first author's recent $\log(R)$-loss counterexample, and our construction is based on a projection of a higher rank lattice. As a by-product, we also construct compact convex $C^2$ hypersurfaces whose rescaling contains many lattice points in any dimension. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_08064 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Power loss for the Mizohata-Takeuchi conjecture on $C^k$ convex hypersurfaces Cairo, Hannah Zhang, Ruixiang Classical Analysis and ODEs 42B37 We find a family of compact $C^k$ hypersurfaces where the local Mizohata-Takeuchi Conjecture fails with a power loss of $R^α$ for any $α<\frac{n-1}{n-1+k}$. Moreover, this family is dense in the $C^k$ topology, and so the local Mizohata-Takeuchi conjecture fails for many convex hypersurfaces. In particular, the local Mizohata-Takeuchi Conjecture fails with a power loss of $R^α$ for any $α<\frac{n-1}{n+1}$ for many $C^2$ convex hypersurfaces. This power matches the best known upper bound in a paper by Tony Carbery, Marina Iliopoulou and Hong Wang up to the endpoint. For the proof, our weight is positive definite as in the first author's recent $\log(R)$-loss counterexample, and our construction is based on a projection of a higher rank lattice. As a by-product, we also construct compact convex $C^2$ hypersurfaces whose rescaling contains many lattice points in any dimension. |
| title | Power loss for the Mizohata-Takeuchi conjecture on $C^k$ convex hypersurfaces |
| topic | Classical Analysis and ODEs 42B37 |
| url | https://arxiv.org/abs/2512.08064 |