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

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.