Representation of harmonic functions with respect to subordinate Brownian motion

Abstract

In this article we prove a representation formula for non-negative generalized harmonic functions with respect to a subordinate Brownian motion in a general open set $D\subset {\mathbb{R}}^d$. We also study oscillation properties of quotients of Poisson integrals and prove that oscillation can be uniformly tamed.

36 pages, minor changes in the presentation, improved results in Section 3; accepted for publication in Journal of Mathematical Analysis and Applications