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 [28] and extended by Zorin-Kranich [29], and on recent results for geometric nonconcentration inequalities [11]. 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 $k<n\le 2k$.

本文考虑建立问题 ${升}^{磷}$- 改进中间维度中类氡算子的不等式（即，对于既不是曲线也不是超曲面的子流形的平均值）。由于现有方法的局限性，该机制的先前结果相对稀疏，并且往往需要环境空间的维度n和子流形的维度k之间的特殊数值关系。本文基于由 Zhang [28] 建立并由 Zorin-Kranich [29] 扩展的 Kakeya-Brascamp-Lieb 不等式的连续统版本，以及几何非浓度不等式 [11] 的最新结果，开发了一种解决此问题的新方法. 作为这种新方法的初步应用，本文建立了锐限制强类型${升}^{磷}$- 改善范围内某些模型二次子流形的不等式 $克<n\le 2克$.