Radon-like算子的$L^p$ -改进估计和Kakeya-Brascamp-Lieb不等式
$L^p$-improving estimates for Radon-like operators and the Kakeya-Brascamp-Lieb inequality
摘要：
本文考虑了中等维Radon-like算子的$L^p$改进不等式的建立问题（即，对于既不是曲线也不是超曲面的子流形上的平均）。由于现有方法的局限性，以前的结果在这方面是比较稀疏的，往往需要特殊的数值关系之间的维度$n$的环境空间和维度$k$的子流形。本文基于由Zhang建立并由Zorin-Kranich推广的Kakeya-Brascamp-Lieb不等式的连续形式，以及几何非集中不等式的最新结果，提出了一种新的解决方法.作为这一新方法的初步应用，本文对$k < n \leq 2k$范围内的一类模型二次子流形建立了严格的强型$L^p$ -改进不等式。
英文摘要：
This paper considers the problem of establishing $L^p$-improving inequalities for Radon-like operators in intermediate dimensions (i.e., for averages overs submanifolds which are neither curves nor hypersurfaces). Due to limitations in existing approaches, previous results in this regime are comparatively sparse and tend to require special numerical relationships between the dimension $n$ of the ambient space and the dimension $k$ of the submanifolds. This paper develops a new approach to this problem based on a continuum version of the Kakeya-Brascamp-Lieb inequality, established by Zhang and extended by Zorin-Kranich, and on recent results for geometric nonconcentration inequalities. As an initial application of this new approach, this paper establishes sharp restricted strong type $L^p$-improving inequalities for certain model quadratic submanifolds in the range $k < n \leq 2k$.
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