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Bounds on the Green function for integral operators and fractional harmonic measure with applications to boundary Harnack


Authors: Luis A. Caffarelli and Yannick Sire
Journal: Proc. Amer. Math. Soc.
MSC (2010): Primary 35B45
DOI: https://doi.org/10.1090/proc/13815
Published electronically: October 6, 2017
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Abstract: We prove a priori bounds on the Green function for general integral operators in divergence form in the spirit of Littman, Stampacchia and Weinberger's result. For general linear integral operators with bounded measurable coefficients, we introduce the so-called fractional harmonic measure and prove several estimates on it. As an application, we prove a new boundary Harnack principle for these operators. Once the bounds on the Green function are known, the proof follows the approach of Caffarelli-Fabes-Mortola-Salsa and K. Bogdan.


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Additional Information

Luis A. Caffarelli
Affiliation: Department of Mathematics, University of Texas at Austin, 2515 Speedway Stop C1200, Austin, Texas 78712
Email: caffarel@math.utexas.edu

Yannick Sire
Affiliation: Department of Mathematics, Johns Hopkins University, 3400 N. Charles Street, Baltimore, Maryland 21218
Email: sire@math.jhu.edu

DOI: https://doi.org/10.1090/proc/13815
Received by editor(s): April 26, 2017
Received by editor(s) in revised form: May 4, 2017
Published electronically: October 6, 2017
Additional Notes: The first author was supported by NSF DMS-1540162
Communicated by: Joachim Krieger
Article copyright: © Copyright 2017 American Mathematical Society