Journal of the
Korean Mathematical Society
JKMS

ISSN(Print) 0304-9914 ISSN(Online) 2234-3008

Article

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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.

Heat kernel estimates for Dirichlet fractional Laplacian with gradient perturbation

Peng Chen, Renming Song, Longjie Xie, Yingchao Xie

University of Macau, University of Illinois, Jiangsu Normal University, Jiangsu Normal University

Abstract

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.