极大非退化Radon-like变换的局部曲率
Local curvature of maximally nondegenerate Radon-like transforms
摘要：
本文给出了一类Radon-like变换在所有维数和余维数上的完整的几何刻画，这些Radon-like变换直到端点满足二次型标度所允许的局部$L^p \rightarrow L^q$不等式的最大可能范围.必要和充分的曲率类型的标准是措辞在一个相关的牛顿图。在多项式图族的平均数的情况下，曲率条件也意味着尖锐的端点估计。证明依赖于最近开发的多线性Radon-Brascamp-Lieb测试标准和改进版本的微分不等式的多项式首次出现在工作中的Oberlin仿射曲率条件。
英文摘要：
This paper gives a complete geometric characterization in all dimensions and codimensions of those Radon-like transforms which, up to endpoints, satisfy the largest possible range of local $L^p \rightarrow L^q$ inequalities permitted by quadratic-type scaling. The necessary and sufficient curvature-type criterion is phrased in terms of an associated Newton-like diagram. In the case of averages over families of polynomial graphs, the curvature condition implies sharp endpoint estimates as well. The proof relies on the recently-developed multilinear Radon-Brascamp-Lieb testing criterion and a refined version of differential inequalities for polynomials first appearing in work on the Oberlin affine curvature condition.
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