# Sharp heat kernel bounds and entropy in metric measure spaces

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## Abstract

We establish the sharp upper and lower bounds of Gaussian type for the heat kernel in the metric measure space satisfying the RCD(0, *N*) (equivalently, RCD* (0, *N*)) condition with *N* ϵ *ℕ* \ {1} and having the maximum volume growth, and then show its application on the large-time asymptotics of the heat kernel, sharp bounds on the (minimal) Green function, and above all, the large-time asymptotics of the Perelman entropy and the Nash entropy, where for the former the monotonicity of the Perelman entropy is proved. The results generalize the corresponding ones in the Riemannian manifolds, and some of them appear more explicit and sharper than the ones in metric measure spaces obtained recently by Jiang et al. (2016).

### Keywords

entropy heat kernel maximum volume growth Riemannian curvature-dimension condition### MSC(2010)

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## Notes

### Acknowledgements

This work was supported by National Natural Science Foundation of China (Grant No. 11401403) and the Australian Research Council (Grant No. DP130101302). The author thanks Professor Bin Qian for suggesting him study the Perelman entropy in metric measure spaces, and thanks Professors Dejun Luo and Yongsheng Song for a nice discussion when the author gave a talk on this topic in the Institute of Applied Mathematics, Chinese Academy of Sciences on November 20, 2015. This work was started when the author was a research fellow in Macquarie University from October 2014 to October 2015. The author also thanks Professor Adam Sikora for his interest. Finally, the author is grateful to the anonymous referees for their careful reading of the manuscript and asking interesting questions which are helpful for the improvement of the quality of the final paper.

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