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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 $\theta\in (\alpha/2, 1]$.
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참고문헌 (13)
참고문헌 신청
K. Bogdan / 2007 / Estimates of heat kernel of fractional Laplacian perturbed by gradient operators / Comm. Math. Phys 271(1):179~198
Z. -Q. Chen / 2017 / Heat kernels for non-symmetric diffusion operators with jumps / J. Differential Equations 263(10):6576~6634
Z.-Q. Chen / 2010 / Heat kernel estimates for the Dirichlet fractional Laplacian / J. Eur. Math. Soc. (JEMS) 12(5):1307~1329
Z.-Q. Chen / 2012 / Dirichlet heat kernel estimates for fractional Laplacian with gradient perturba-tion / Ann. Probab 40(6):2483~2538
Z.-Q. Chen / 2015 / Stability of Dirichlet heat kernel estimates for non-local operators under Feynman-Kac perturbation / Trans. Amer. Math. Soc 367(7):5237~5270
Z. -Q. Chen / 2017 / Heat kernels for non-symmetric diffusion operators with jumps / J. Differential Equations 263(10):6576~6634
Z.-Q. Chen / 2010 / Heat kernel estimates for the Dirichlet fractional Laplacian / J. Eur. Math. Soc. (JEMS) 12(5):1307~1329
Z.-Q. Chen / 2012 / Dirichlet heat kernel estimates for fractional Laplacian with gradient perturba-tion / Ann. Probab 40(6):2483~2538
Z.-Q. Chen / 2015 / Stability of Dirichlet heat kernel estimates for non-local operators under Feynman-Kac perturbation / Trans. Amer. Math. Soc 367(7):5237~5270
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