In this work, we study the extremal functions of the log-Sobolev functional on compact metric measure spaces satisfying the \(\textrm{RCD}^*(K,N)\) condition for K in \(\mathbb {R}\) and N in \((2,\infty )\). We show the existence, regularity and positivity of non-negative extremal functions. Based on these results, we prove a Li-Yau type estimate for the logarithmic transform of any non-negative extremal functions of the log-Sobolev functional. As applications, we show a Harnack type inequality as well as lower and upper bounds for the non-negative extremal functions.

中文翻译：

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$$\textrm{RCD}^*(K,N)$$ 空间上的 Log-Sobolev 泛函极值和 Li-Yau 估计

在这项工作中，我们研究了紧度量测度空间上的 log-Sobolev 函数的极值函数，满足 \( \mathbb {R}\中K的 \(\textrm{RCD}^*(K,N)\)条件)和N在\((2,\infty )\)中。我们证明了非负极值函数的存在性、正则性和正性。基于这些结果，我们证明了 log-Sobolev 泛函的任何非负极值函数的对数变换的 Li-Yau 型估计。作为应用，我们展示了 Harnack 型不等式以及非负极值函数的下界和上限。