论文标题
$ l^p $ - ra状操作员和Kakeya-Brascamp-lieb不平等的估计
$L^p$-improving estimates for Radon-like operators and the Kakeya-Brascamp-Lieb inequality
论文作者
论文摘要
本文考虑了在中间维度中为ra样操作员建立$ l^p $ - 提示不平等的问题（即，对于既不是曲线也不是横向表面的平均超级序列）。由于现有方法的限制，该制度的先前结果相对稀疏，并且倾向于在环境空间的尺寸$ n $之间与Submanifolds的尺寸$ k $之间进行特殊的数值关系。本文基于Zhang和Zorin-Kranich扩展的Kakeya-Brascamp-Lieb不平等的连续版本开发了一种新的方法，并基于Zorin-Kranich的扩展，以及最近的几何不浓度不等式的结果。作为这种新方法的初步应用，本文建立了急剧受限的强型$ l^p $ - 改进某些模型二次submanifolds的不平等现象，范围为$ k <n \ leq 2k $。
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 $k < n \leq 2k$.
