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Heat Kernel Bounds on Metric Measure Spaces and Some Applications

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Abstract

Let (X,d,μ) be a R C D ∗(K,N) space with \(K\in \mathbb {R}\) and N∈[1,∞). We derive the upper and lower bounds of the heat kernel on (X,d,μ) by applying the parabolic Harnack inequality and the comparison principle, and then sharp bounds for its gradient, which are also sharp in time. For applications, we study the large time behavior of the heat kernel, the stability of solutions to the heat equation, and show the L p boundedness of (local) Riesz transforms.

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Authors and Affiliations

  1. School of Mathematical Sciences, Bei**g Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, 100875, Bei**g, China

    Ren** Jiang

  2. School of Mathematics, Sichuan University, Chengdu, 610064, China

    Huaiqian Li

  3. Department of Mathematics, Sun Yat-sen University, Guangzhou, 510275, China

    Huichun Zhang

Authors
  1. Ren** Jiang
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  2. Huaiqian Li
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  3. Huichun Zhang
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Correspondence to Ren** Jiang.

Additional information

R.J. is partially supported by NSFC (No. 11301029); H.L. is partially supported by NSFCs (No. 11401403 and No. 11371099) and the ARC grant (DP130101302); H.Z. is partially supported by NSFC (No. 11201492) and by Guangdong Natural Science Foundation (No. S2012040007550).

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Jiang, R., Li, H. & Zhang, H. Heat Kernel Bounds on Metric Measure Spaces and Some Applications. Potential Anal 44, 601–627 (2016). https://doi.org/10.1007/s11118-015-9521-2

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