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Transactions of the American Mathematical Society

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ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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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.
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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