We prove that synthetic lower Ricci bounds for metric measure spaces — both in the sense of Bakry–Émery and in the sense of Lott–Sturm–Villani — can be characterized by various functional inequalities including local Poincaré inequalities, local logarithmic Sobolev inequalities, dimension independent Harnack inequality, and logarithmic Harnack inequality.

中文翻译：

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时变度量空间上热流的函数不等式

我们证明了度量测度空间的合成 Ricci 下界——无论是在 Bakry-Émery 的意义上还是在 Lott-Sturm-Villani 的意义上——都可以用各种函数不等式来表征，包括局部 Poincaré 不等式、局部对数 Sobolev 不等式、维度无关Harnack 不等式和对数 Harnack 不等式。