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Localized elliptic gradient estimate for solutions of the heat equation on \({ RCD}^*(K,N)\) metric measure spaces

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Abstract

In this paper, we will establish a localized elliptic gradient estimate for weak solutions of the heat equation on metric measure spaces with generalized Ricci curvature bounded from below. One of its main applications is a sharp gradient estimate for the logarithm of heat kernels.

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Acknowledgements

We thanks the anonymous referees for some suggestions to simplify the argument of Lemma 3.1. Jia-Cheng Huang partially supported by China Postdoctoral Science Foundation funded Project 2017M611438. Hui-Chun Zhang partially supported by NSFC 11521101 and NSFC 11571374.

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  1. School of Mathematical Sciences, Fudan University, Shanghai, China

    Jia-Cheng Huang

  2. Department of Mathematics, Sun Yat-sen University, Guangzhou, China

    Hui-Chun Zhang

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  1. Jia-Cheng Huang
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Correspondence to Hui-Chun Zhang.

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Huang, JC., Zhang, HC. Localized elliptic gradient estimate for solutions of the heat equation on \({ RCD}^*(K,N)\) metric measure spaces. manuscripta math. 161, 303–324 (2020). https://doi.org/10.1007/s00229-018-1095-z

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