In this paper we prove mean curvature comparisons and volume comparisons on a smooth metric measure space when the integral radial Bakry-Émery Ricci tensor and the potential function or its gradient are bounded. As applications, we prove diameter estimates and eigenvalue estimates on smooth metric measure spaces. These results not only give a supplement of the author’s previous results under integral Bakry-Émery Ricci tensor bounds, but also are generalizations of the Wei-Wylie’s pointwise results.

中文翻译：

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积分径向 Bakry-Émery Ricci 张量边界的比较几何

在本文中，我们证明了当积分径向 Bakry-Émery Ricci 张量和势函数或其梯度有界时，平滑度量空间上的平均曲率比较和体积比较。作为应用，我们证明了平滑度量空间上的直径估计和特征值估计。这些结果不仅是对作者在积分 Bakry-Émery Ricci 张量界下的先前结果的补充，而且是 Wei-Wylie 逐点结果的推广。