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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Boundary Harnack principle for $ \Delta+ \Delta^{\alpha/2}$


Authors: Zhen-Qing Chen, Panki Kim, Renming Song and Zoran Vondraček
Journal: Trans. Amer. Math. Soc. 364 (2012), 4169-4205
MSC (2010): Primary 31B25, 60J45; Secondary 47G20, 60J75, 31B05
Published electronically: March 19, 2012
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Abstract: For $ d\geq 1$ and $ \alpha \in (0, 2)$, consider the family of pseudo-differential operators $ \{\Delta + b \Delta ^{\alpha /2}; b\in [0, 1]\}$ on $ \mathbb{R}^d$ that evolves continuously from $ \Delta $ to $ \Delta + \Delta ^{\alpha /2}$. In this paper, we establish a uniform boundary Harnack principle (BHP) with explicit boundary decay rate for non-negative functions which are harmonic with respect to $ \Delta +b \Delta ^{\alpha /2}$ (or, equivalently, the sum of a Brownian motion and an independent symmetric $ \alpha $-stable process with constant multiple $ b^{1/\alpha }$) in $ C^{1, 1}$ open sets. Here a ``uniform'' BHP means that the comparing constant in the BHP is independent of $ b\in [0, 1]$. Along the way, a uniform Carleson type estimate is established for non-negative functions which are harmonic with respect to $ \Delta + b \Delta ^{\alpha /2}$ in Lipschitz open sets. Our method employs a combination of probabilistic and analytic techniques.


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Additional Information

Zhen-Qing Chen
Affiliation: Department of Mathematics, University of Washington, Seattle, Washington 98195
Email: zqchen@uw.edu

Panki Kim
Affiliation: Department of Mathematical Science, Seoul National University, Seoul 151-747, South Korea
Email: pkim@snu.ac.kr

Renming Song
Affiliation: Department of Mathematics, University of Illinois, Urbana, Illinois 61801
Email: rsong@math.uiuc.edu

Zoran Vondraček
Affiliation: Department of Mathematics, University of Zagreb, Bijenička c. 30, Zagreb, Croatia
Email: vondra@math.hr

DOI: http://dx.doi.org/10.1090/S0002-9947-2012-05542-5
PII: S 0002-9947(2012)05542-5
Keywords: Boundary Harnack principle, harmonic function, sub- and super-harmonic function, fractional Laplacian, Laplacian, symmetric $𝛼$-stable process, Brownian motion, Ito’s formula, Lévy system, martingales, exit distribution
Received by editor(s): September 23, 2010
Published electronically: March 19, 2012
Additional Notes: The first author’s research was supported by NSF Grants DMS-0600206 and DMS-0906743.
The second author’s research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) grant funded by the Korean government (MEST) (2011-00001251)
The fourth author’s research was supported by MZOS grant 037-0372790-2801 of the Republic of Croatia.
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.