A new formulation of Harnackʼs inequality for nonlocal operators
Moritz Kaßmann · Comptes Rendus Mathématique · 2011
We provide a new formulation of Harnackʼs inequality for nonlocal operators. In contrast to previous versions we do not assume harmonic functions to have a sign. The version of Harnackʼs inequality given here generalizes Harnackʼs classical result from 1887 to nonlocal situations. As a consequence we derive Hölder regularity estimates by an extension of Moserʼs method. The inequality that we propose is equivalent to Harnackʼs original formulation but seems to be new even for the Laplace operator.