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

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- by Zhen-Qing Chen, Panki Kim, Renming Song and Zoran Vondraček PDF
- Trans. Amer. Math. Soc.
**364**(2012), 4169-4205 Request permission

## 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.## References

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## Additional Information

**Zhen-Qing Chen**- Affiliation: Department of Mathematics, University of Washington, Seattle, Washington 98195
- MR Author ID: 242576
- ORCID: 0000-0001-7037-4030
- Email: zqchen@uw.edu
**Panki Kim**- Affiliation: Department of Mathematical Science, Seoul National University, Seoul 151-747, South Korea
- MR Author ID: 705385
- Email: pkim@snu.ac.kr
**Renming Song**- Affiliation: Department of Mathematics, University of Illinois, Urbana, Illinois 61801
- MR Author ID: 229187
- Email: rsong@math.uiuc.edu
**Zoran Vondraček**- Affiliation: Department of Mathematics, University of Zagreb, Bijenička c. 30, Zagreb, Croatia
- MR Author ID: 293132
- Email: vondra@math.hr
- 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. - © Copyright 2012
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc.
**364**(2012), 4169-4205 - MSC (2010): Primary 31B25, 60J45; Secondary 47G20, 60J75, 31B05
- DOI: https://doi.org/10.1090/S0002-9947-2012-05542-5
- MathSciNet review: 2912450