Mizohata-Takeuchi猜想在$C^k$凸超曲面上的功率损失
Power loss for the Mizohata-Takeuchi conjecture on $C^k$ convex hypersurfaces
AI总结：
本文构造了一族$C^k$凸超曲面，证明局部Mizohata-Takeuchi猜想在特定条件下存在功率损失，并展示了其在不同维度中的应用。
摘要：
我们发现了一族紧致的$C^k$超曲面，其中局部Mizohata-Takeuchi猜想在任何$α<\ rac{n-1}{n-1+k}$的情况下都存在功率损失$R^α$。此外，这一族在$C^k$拓扑下是密集的，因此许多凸超曲面的局部Mizohata-Takeuchi猜想都失效。特别地，对于许多$C^2$凸超曲面，局部Mizohata-Takeuchi猜想在任何$α<\ rac{n-1}{n+1}$的情况下都存在功率损失$R^α$。这一功率与Tony Carbery、Marina Iliopoulou和Hong Wang在一篇论文中所知的最佳上界相符，直到端点。证明中我们的权重是正定的，如同第一作者最近的$\log(R)$-loss反例一样，我们的构造基于更高秩晶格的投影。作为副产品，我们还构造了紧致凸的$C^2$超曲面，其缩放包含任意维度中的许多晶格点。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
