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Winter 2010 Dirichlet heat kernel estimates for $\Delta^{\alpha/2}+ \Delta^{\beta/2}$
Zhen-Qing Chen, Panki Kim, Renming Song
Illinois J. Math. 54(4): 1357-1392 (Winter 2010). DOI: 10.1215/ijm/1348505533


For $d\geq1$ and $0<\beta<\alpha<2$, consider a family of pseudo differential operators $\{\Delta^{\alpha} + a^\beta \Delta^{\beta/2}$; $a \in[0, 1]\}$ on $\mathbb{R}^d$ that evolves continuously from $\Delta^{\alpha/2}$ to $ \Delta^{\alpha/2}+ \Delta^{\beta/2}$. It gives arise to a family of Lévy processes $\{X^a, a\in[0, 1]\}$ on $\mathbb{R}^d$, where each $X^a$ is the independent sum of a symmetric $\alpha$-stable process and a symmetric $\beta$-stable process with weight $a$. For any $C^{1,1}$ open set $D\subset\mathbb{R}^d$, we establish explicit sharp two-sided estimates, which are uniform in $a\in(0, 1]$, for the transition density function of the subprocess $X^{a, D}$ of $X^a$ killed upon leaving the open set~$D$. The infinitesimal generator of $X^{a, D}$ is the nonlocal operator $\Delta^{\alpha} + a^\beta\Delta^{\beta/2}$ with zero exterior condition on $D^c$. As consequences of these sharp heat kernel estimates, we obtain uniform sharp Green function estimates for $X^{a, D}$ and uniform boundary Harnack principle for $X^a$ in $D$ with explicit decay rate.


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Zhen-Qing Chen. Panki Kim. Renming Song. "Dirichlet heat kernel estimates for $\Delta^{\alpha/2}+ \Delta^{\beta/2}$." Illinois J. Math. 54 (4) 1357 - 1392, Winter 2010.


Published: Winter 2010
First available in Project Euclid: 24 September 2012

zbMATH: 1268.60101
MathSciNet: MR2981852
Digital Object Identifier: 10.1215/ijm/1348505533

Primary: 47G20, 60J35, 60J75
Secondary: 47D07

Rights: Copyright © 2010 University of Illinois at Urbana-Champaign


Vol.54 • No. 4 • Winter 2010
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