• Chen, Peng (Department of Mathematics University of Macau) ;
  • Song, Renming (Department of Mathematics University of Illinois) ;
  • Xie, Longjie (School of Mathematics and Statistics Jiangsu Normal University) ;
  • Xie, Yingchao (School of Mathematics and Statistics Jiangsu Normal University)
  • Received : 2018.01.30
  • Accepted : 2018.06.26
  • Published : 2019.01.01


We give a direct proof of the sharp two-sided estimates, recently established in [4, 9], for the Dirichlet heat kernel of the fractional Laplacian with gradient perturbation in $C^{1,1}$ open sets by using Duhamel's formula. We also obtain a gradient estimate for the Dirichlet heat kernel. Our assumption on the open set is slightly weaker in that we only require D to be $C^{1,{\theta}}$ for some ${\theta}{\in}({\alpha}/2,1]$.


Supported by : Simons Foundation, NNSF of China, NSF of Jiangsu


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