J. Korean Math. Soc. 2019; 56(1): 91-111
Online first article August 6, 2018 Printed January 1, 2019
https://doi.org/10.4134/JKMS.j180065
Copyright © The Korean Mathematical Society.
Peng Chen, Renming Song, Longjie Xie, Yingchao Xie
University of Macau, University of Illinois, Jiangsu Normal University, Jiangsu Normal University
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 $\theta\in (\alpha/2, 1]$.
Keywords: isotropic stable process, fractional Laplacian, Dirichlet heat kernel, Kato class, gradient estimate
MSC numbers: Primary 60J35, 47G20, 60J75
Supported by: Research of R. Song is supported by the Simons Foundation (#429343, Renming Song). L. Xie is supported by NNSF of China (No. 11701233) and NSF of Jiangsu (No. BK20170226). Y. Xie is supported by NNSF of China (No. 11771187). The Project Funded
by the PAPD of Jiangsu Higher Education Institutions is also gratefully acknowledged.
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