PUB - Publikationen an der Universität Bielefeld

We provide a new formulation of Harnack's inequality for nonlocal operators. In contrast to previous versions we do not assume harmonic functions to have a sign. The version of Harnack's inequality given here generalizes Harnack's classical result from 1887 to nonlocal situations. As a consequence we derive Holder regularity estimates by an extension of Moser's method. The inequality that we propose is equivalent to Harnack's original formulation but seems to be new even for the Laplace operator. (C) 2011 Academie des sciences. Published by Elsevier Masson SAS. All rights reserved.