Representation of harmonic functions with respect to subordinate Brownian motion

TL;DR

This paper establishes a representation formula for non-negative harmonic functions related to subordinate Brownian motion in open sets and analyzes the oscillation behavior of Poisson integral quotients.

Contribution

It introduces a new representation formula for harmonic functions with respect to subordinate Brownian motion and studies their oscillation properties in general open sets.

Findings

Derived a representation formula for harmonic functions.

Proved uniform control over oscillation of Poisson integral quotients.

Extended results to general open sets.

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.