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Heat kernel estimates for Dirichlet fractional Laplacian with gradient perturbation
Peng Chen,Renming Song,Longjie Xie,Yingchao Xie 대한수학회 2019 대한수학회지 Vol.56 No.1
We give a direct proof of the sharp two-sided estimates, recently established in \cite{C-K-S-1, P-R}, 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 $\theta\in (\alpha/2, 1]$.
HEAT KERNEL ESTIMATES FOR DIRICHLET FRACTIONAL LAPLACIAN WITH GRADIENT PERTURBATION
Chen, Peng,Song, Renming,Xie, Longjie,Xie, Yingchao Korean Mathematical Society 2019 대한수학회지 Vol.56 No.1
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 ${\theta}{\in}({\alpha}/2,1]$.