# HEAT KERNEL ESTIMATES FOR DIRICHLET FRACTIONAL LAPLACIAN WITH GRADIENT PERTURBATION

Chen, Peng;Song, Renming;Xie, Longjie;Xie, Yingchao

• Accepted : 2018.06.26
• Published : 2019.01.01
• 16 4

#### Abstract

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 ${\theta}{\in}({\alpha}/2,1]$.

#### Keywords

isotropic stable process;fractional Laplacian;Dirichlet heat kernel;Kato class;gradient estimate

#### Acknowledgement

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