On a Riemannian metric-measure space, we establish an Alexandrov-Bakelman-Pucci type measure estimate connecting Bakry-\'Emery Ricci curvature lower bound, modified Laplacian and the measure of certain special sets. We apply this estimate to prove Harnack inequalities for the modified Laplacian operator and fully non-linear operators. These inequalities seem not available in the literature; And our proof, solely based on the ABP estimate, does not involve any Sobolev inequalities nor gradient estimate. We also propose a question regarding the characterization of Ricci lower bound by the Harnack inequality.

中文翻译：

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测量估计值、Harnack 不等式和 Ricci 下界

在黎曼度量空间上，我们建立了连接 Bakry-\'Emery Ricci 曲率下界、修正的 Laplacian 和某些特殊集合的测度的 Alexandrov-Bakelman-Pucci 型测度估计。我们应用这个估计来证明修正拉普拉斯算子和完全非线性算子的哈纳克不等式。这些不等式在文献中似乎不存在；我们的证明完全基于 ABP 估计，不涉及任何 Sobolev 不等式或梯度估计。我们还提出了一个关于 Harnack 不等式表征 Ricci 下界的问题。